Children's syllogistic reasoning

Most theories of the development of deductive ability propose that children acquire formal rules of inference. An alternative theory assumes that reasoning consists of constructing a mental model of the situation described in the premises, scanning the model for an informative conclusion, and then searching for alternative models that refute this conclusion. Hence, performance should reflect two principal factors: the difficulty of constructing a model, which depends on the “figure” of the premises, and the number of models that have to be evaluated to respond correctly. In Experiment 1, two groups of children (9- to 10- and 11- to 12-year-olds) drew conclusions from 20 pairs of syllogistic premises. The results confirmed that children are affected both by figure and by number of models. Experiment 2 corroborated these findings for all 64 possible forms of syllogistic premises. The development of reasoning ability may therefore depend on the acquisition, not of formal rules of logic, but of procedures for manipulating models.     

Strategies in Syllogistic Reasoning

This paper is about syllogistic reasoning, i.e., reasoning from such pairs of premises as, All the chefs are musicians; some of the musicians are painters. We present a computer model that implements the latest account of syllogisms, which is based on the theory of mental models. We also report four experiments that were designed to test this account. Experiments 1 and 2 examined the strategies revealed by the participants' use of paper and pencil as aids to reasoning. Experiment 3 used a new technique to externalize thinking. The participants had to refute, if possible, putative conclusions by constructing external models that were examples of the premises but counterexamples of the conclusions. Experiment 4 used the same techniques to examine the participants' strategies as they drew their own conclusions from syllogistic premises. The results of the experiments showed that individuals not trained in logic can construct counterexamples, that they use similar operations to those implemented in the computer model, but that they rely on a much greater variety of interpretations of premises and of search strategies than the computer model does. We re-evaluates current theories of syllogistic reasoning in the light of these results.     

Are conjunctive inferences easier than disjunctive inferences? A comparison of rules and models

We report four experiments investigating conjunctive inferences (from a conjunction and two conditional premises) and disjunctive inferences (from a disjunction and the same two conditionals). The mental model theory predicts that the conjunctive inferences, which require one model, should be easier than the disjunctive inferences, which require multiple models. Formal rule theories predict either the opposite result or no difference between the inferences. The experiments showed that the inferences were equally easy when the participants evaluated given conclusions, but that the conjunctive inferences were easier than the disjunctive inferences (1) when the participants drew their own conclusions, (2) when the conjunction and disjunction came last in the premises, (3) in the time the participants spent reading the premises and in responding to given conclusions, and (4) in their ratings of the difficulty of the inferences. The results support the model theory and demonstrate the importance of reasoners' inferential strategies.     

Illusions in modal reasoning

According to the mental model theory, models represent what is true, but not what is false. One unexpected consequence is that certain inferences should have compelling, but invalid, conclusions. Three experiments corroborated the occurrence of such illusions in reasoning about possibilities. When problems had the heading "Only one of the premises is true," the participants considered the truth of each premise in turn, but neglected the fact that when one premise is true, the others are false. When two-premise problems had the heading "One of the premises is true and one is false," the participants still neglected the falsity of one of the premises. As predicted, however, the illusions were reduced when reasoners were told to check their conclusions against the constraint that only one of the premises was true. We discuss alternative explanations for illusory inferences and their implications for current theories of reasoning.     

Transitive and pseudo-transitive inferences

Given that A is longer than B, and that B is longer than C, even 5-year-old children can infer that A is longer than C. Theories of reasoning based on formal rules of inference invoke simple axioms ("meaning postulates") to capture such transitive inferences. An alternative theory proposes instead that reasoners construct mental models of the situation described by the premises in order to draw such inferences. An unexpected consequence of the model theory is that if adult reasoners construct simple models of typical situations, then they should infer transitive relations where, in certain cases, none exists. We report four studies corroborating the occurrence of these "pseudo-transitive" fallacies. Experiment 1 established that individuals' diagrams of certain non-transitive relations yield transitive conclusions. Experiment 2 showed that these premises also give rise to fallacious transitive inferences. Experiment 3 established that when the context suggested alternatives to the simple models, the participants made fewer errors. Experiment 4 showed that tense is an important aspect of meaning which affects whether individuals draw transitive conclusions. We discuss the implications of these results for various theories of reasoning.     

The source of belief bias effects in syllogistic reasoning.

In studies of the belief bias effect in syllogistic reasoning, an interaction between logical validity and the believability of the conclusion has been found; in essence, logic has a larger effect on unbelievable than on believable conclusions. Two main explanations have been proposed for this finding. The selective scrutiny account claims that people focus on the conclusion and only engage in logical processing if this is found to be unbelievable; while the misinterpreted necessity account claims that subjects misunderstand what is meant by logical necessity and respond on the basis of believability when indeterminate syllogisms are presented. Experiments 1 and 2 compared the predictions of these two theories by examining whether the interaction would disappear if only determinate syllogisms were used. It did, thus providing strong support for the misinterpreted necessity explanation. However, the results are also consistent with a version of the mental models theory, and so Experiment 3 was carried out to compare these two explanations. The mental models theory received strong support, as it did also in the follow-up Experiments 4 and 5. It is concluded that people try to construct a mental model of the premises but, if there is a believable conclusion consistent with the first model they produce, then they fail to construct alternative models.     

A Model Theory of Modal Reasoning

This paper presents a new theory of modal reasoning, i.e. reasoning about what may or may not be the case, and what must or must not be the case. It postulates that individuals construct models of the premises in which they make explicit only what is true. A conclusion is possible if it holds in at least one model, whereas it is necessary if it holds in all the models. The theory makes three predictions, which are corroborated experimentally. First, conclusions correspond to the true, but not the false, components of possibilities. Second, there is a key interaction: it is easier to infer that a situation is possible as opposed to impossible, whereas it is easier to infer that a situation is not necessary as opposed to necessary. Third, individuals make systematic errors of omission and of commission. We contrast the theory with theories based on formal rules.     

Figural effects in a syllogistic evaluation paradigm: an inspection-time analysis

Robust biases have been found in syllogistic reasoning that relate to the figure of premises and to the directionality of terms in given conclusions. Mental models theorists (e.g., Johnson-Laird & Byrne, 1991) have explained figural bias by assuming that reasoners can more readily form integrated models of premises when their middle terms are contiguous than when they are not. Biases associated with the direction of conclusion terms have been interpreted as reflecting a natural mode of reading off conclusions from models according to a "first-in, first-out principle." We report an experiment investigating the impact of systematic figural and conclusion-direction manipulations on the processing effort directed at syllogistic components, as indexed through a novel inspection-time method. The study yielded reliable support for mental-models predictions concerning the nature and locus of figural and directionality effects in syllogistic reasoning. We argue that other accounts of syllogistic reasoning seem less able to accommodate the full breadth of inspection-time findings observed.     

Language processing,activation and reasoning: A reply to Espino,Santamaría,and García-Madruga (2000)

Espino, Santamaría, and García-Madruga (2000) report three results on the time taken to respond to a probe word occurring as end term in the premises of a syllogistic argument. They argue that these results can only be predicted by the theory of mental models. It is argued that two of these results, on differential reaction times to end-terms occurring in different premises and in different figures, are consistent with Chater and Oaksford's (1999) probability heuristics model (PHM). It is argued that the third finding, on different reaction times between figures, does not address the issue of processing difficulty where PHM predicts no differences between figures. It is concluded that Espino et al.'s results do not discriminate between theories of syllogistic reasoning as effectively as they propose.     

Generating alternatives: A key component in human reasoning?

According to mental models theory, a key aspect of deductive reasoning is the production of alternative models that can falsify provisional conclusions. In the present paper, the possibility is investigated that there are individual differences in the ability to produce alternative models. The results indicate that some people do not proceed beyond the first model when they reason with syllogisms but that others do. Furthermore, the ability to generate alternatives can be independently measured by asking participants to generate different representations of pairs of premises. These findings support the predictions of mental models theory and also indicate the potential importance of alternatives generation as a measure of individual differences in processing style.     

Unity of perception

Perceptual scientists have recently enjoyed success in constructing mathematical theories for specific perceptual capacities, capacities such as stereovision, auditory localization, and color perception. Analysis of these theories suggests that they all share a common mathematical structure. If this is true, the elucidation of this structure, the study of its properties, the derivation of its consequences, and the empirical testing of its predictions are promising directions for perceptual research. We consider a candidate for the common structure, a candidate called an "observer". Observers, in essence, perform inferences; each observer has a characteristic class of perceptual premises, a characteristic class of perceptual conclusions, and its own functional relationship between these premises and conclusions. If observers indeed capture the structure common to perceptual capacities, then each capacity, regardless of its modality or manner of instantiation, can be described as some observer. In this paper we develop the definition of an observer. We first consider two examples of perceptual capacities: the measurement of visual motion, and the perception of depth from visual motion. In each case, we review a formal theory of the capacity and abstract its structural essence. From this essence we construct the definition of observer. We then exercise the definition in discussions of transduction, perceptual illusions, perceptual uncertainty, regularization theory, the cognitive penetrability of perception, and the theory neutrality of observation.     

小学儿童一维空间方位传递性推理能力的发展

研究了小学儿童一维空间方位传递性推理能力的发展水平及认知策略 ,同时 ,对心理模型理论进行了检验。被试为城市中等小学 7岁、9岁、11岁儿童各 2 4名 ,男女各半。 4种实验任务分别为三前提单模型、三前提双模型、四前提单模型和四前提双模型。采用个别实验 ,儿童在前提呈现的情况下进行推理。主要研究结果 :(1)从小学 7岁到 11岁 ,儿童的一维空间方位传递性推理能力明显提高 ,7岁儿童初步形成了一维空间方位推理能力 ,9岁和 11岁基本具有了这种能力 ;(2 )随着年龄增长 ,使用模型建构策略解决问题的儿童人次越来越多 ,绝大部分 11岁儿童都能使用这一策略进行推理。但即使儿童使用了模型建构策略 ,他们的推理成绩也没有反映出模型数量所造成的任务难度差异 ,即不符合心理模型理论关于模型数量的主要预期。     

Falsifying mental models: Testing the predictions of theories of syllogistic reasoning

Four experiments are reported that tested the claim, drawn from mental models theory, that reasoners attempt to construct alternative representations of problems that might falsify preliminary conclusions they have drawn. In Experiment 1, participants were asked to indicate which alternative conclusion(s) they had considered in a syllogistic reasoning task. In Experiments 2-4, participants were asked to draw diagrams consistent with the premises, on the assumption that these diagrams would provide insights into the mental representation being used. In none of the experiments was there any evidence that people constructed more models for multiple-model than for single-model syllogisms, nor was there any correlation between number of models constructed and overall accuracy. The results are interpreted as showing that falsification of the kind proposed by mental models theory may not routinely occur in reasoning.     

The Role of Negation in Conditional Inference

This paper addresses the issue of how negative components affect people's ability to draw conditional inferences. The study was motivated by an attempt to resolve a difficulty for the mental models theory of Johnson-Laird and Byrne, whose account of matching bias in the selection task is apparently inconsistent with Johnson-Laird's explanation of the double negation effects in conditional inference reported by Evans, Clibbens, and Rood (1995). Two experiments are reported, which investigate frequencies of conditional inferences with task presentation similar to that of the selection task in two respects: the presence of a picture of four cards and the use of implicit negations in the premises. The latter variable was shown to be critical and demonstrated a new phenomenon: Conditional inferences of all kinds are substantially suppressed when based on implicitly negative premises. This phenomenon was shown to operate independently of and in addition to the double negation effect. A third experiment showed that the implicit negation effect could be extended to the paradigm in which people are asked to produce their own conclusions. It is argued that these two effects can be explained within either the mental models theory or the inference rule theory, of propositional reasoning, but that each will require some revision in order to offer a convincing account.     

Naïve deontics: a theory of meaning, representation, and reasoning

Deontic assertions concern what one ought to do, may do, and ought not to do. This paper proposes a theory of their meanings and of how these meanings are represented in mental models. The meanings of deontic assertions refer to sets of permissible and impermissible states. An experiment corroborated the ability of individuals to list these states. The most salient were those corresponding to the mental models of the assertions. When individuals reason, they rely on mental models, which do not make all states explicit. The theory predicts the most frequent conclusions drawn from deontic premises. It also predicts the occurrence of illusory inferences from assertions of permission, i.e., inferences that seem highly plausible but that are in fact invalid. Assertions of prohibitions, according to the theory, should reduce the illusions. Further experiments corroborated these predictions.     

A relevance theory of induction

A framework theory, organized around the principle of relevance, is proposed for category-based reasoning. According to the relevance principle, people assume that premises are informative with respect to conclusions. This idea leads to the prediction that people will use causal scenarios and property reinforcement strategies in inductive reasoning. These predictions are contrasted with both existing models and normative logic. Judgments of argument strength were gathered in three different countries, and the results showed the importance of both causal scenarios and property reinforcement in categorybased inferences. The relation between the relevance framework and existing models of category-based inductive reasoning is discussed in the light of these findings.     

Premise Misinterpretation and Syllogistic Reasoning

A study by Ceraso and Provitera (1971) found that elaboration of the premises used in syllogistic reasoning led to substantially improved performance. This finding is of considerable importance because of the implications it has for mental logic and mental models theories of reasoning. Three experiments are reported, which replicated and extended the original findings. It was found that elaboration led to a significant improvement in performance, but that this was confined to multiple model syllogisms, where the elaboration has the effect of reducing the number of models involved. A fourth experiment indicated that elaboration can vary within the same syllogism depending on the direction of the conclusion drawn. These findings are best explained under the assumption that reasoners build mental models when solving problems and that elaboration can reduce the number of possible models.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Illusions in quantified reasoning: How to make the impossible seem possible, and vice versa

The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally represent only what is true. The theory has an unexpected consequence. It predicts the existence of illusions in inferences. Certain inferences should have compelling but erroneous conclusions. Two experiments corroborated the occurrence of such illusions in inferences about what is possible from disjunctions of quantified assertions, such as, "at least some of the plastic beads are not red." Experiment 1 showed that participants erroneously inferred that impossible situations were possible, and that possible situations were impossible, but that they performed well with control problems based on the same premises. Experiment 2 corroborated these findings in inferences from assertions based on dyadic relations, such as, "all the boys played with the girls."     

Complexity of categorical syllogisms: An integration of two metrics

The complexity of categorical syllogisms was assessed using the relational complexity metric, which is based on the number of entities that are related in a single cognitive representation. This was compared with number of mental models in an experiment in which adult participants solved all 64 syllogisms. Both metrics accounted for similarly large proportions of the variance, showing that complexity depends on the number of categories that are related in a representation of the combined premises, whether represented in multiple mental models, or by a single model. This obviates the difficulty with mental models theory due to equivocal evidence for construction of more than one mental model. The “no valid conclusion” response was used for complex syllogisms that had valid conclusions. The results are interpreted as showing that the relational complexity metric can be applied to syllogistic reasoning, and can be integrated with mental models theory, which together account for a wide range of cognitive performances.