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.     

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.     

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.     

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.     

Strategies in sentential reasoning

Four experiments examined the strategies that individuals develop in sentential reasoning. They led to the discovery of five different strategies. According to the theory proposed in the paper, each of the strategies depends on component tactics, which all normal adults possess, and which are based on mental models. Reasoners vary their use of tactics in ways that have no deterministic account. This variation leads different individuals to assemble different strategies, which include the construction of incremental diagrams corresponding to mental models, and the pursuit of the consequences of a single model step by step. Moreover, the difficulty of a problem (i.e., the number of mental models required by the premises) predisposes reasoners towards certain strategies. Likewise, the sentential connectives in the premises also bias reasoners towards certain strategies, e.g., conditional premises tend to elicit reasoning step by step whereas disjunctive premises tend to elicit incremental diagrams.     

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。     

Disjunctive illusory inferences and how to eliminate them

The mental model theory of reasoning postulates that individuals construct mental models of the possibilities in which the premises of an inference hold and that these models represent what is true but not what is false. An unexpected consequence of this assumption is that certain premises should yield systematically invalid inferences. This prediction is unique among current theories of reasoning, because no alternative theory, whether based on formal rules of inference or on probabilistic considerations, predicts these illusory inferences. We report three studies of novel illusory inferences that depend on embedded disjunctions—for example, premises of this sort: A or else (B or else C). The theory distinguishes between those embedded disjunctions that should yield illusions and those that should not. In Experiment 1, we corroborated this distinction. In Experiment 2, we extended the illusory inferences to a more stringently controlled set of problems. In Experiment 3, we established a novel method for reducing illusions by calling for participants to make auxiliary inferences.     

Reconnecting interpretation to reasoning through individual differences

Computational theories of mind assume that participants interpret information and then reason from those interpretations. Research on interpretation in deductive reasoning has claimed to show that subjects' interpretation of single syllogistic premises in an “immediate inference” task is radically different from their interpretation of pairs of the same premises in syllogistic reasoning tasks (Newstead, 1989, 1995; Roberts, Newstead, & Griggs, 2001). Narrow appeal to particular Gricean implicatures in this work fails to bridge the gap. Grice's theory taken as a broad framework for credulous discourse processing in which participants construct speakers' “intended models” of discourses can reconcile these results, purchasing continuity of interpretation through variety of logical treatments. We present exploratory experimental data on immediate inference and subsequent syllogistic reasoning. Systematic patterns of interpretation driven by two factors (whether the subject's model of the discourse is credulous, and their degree of reliance on information packaging) are shown to transcend particular quantifier inferences and to drive systematic differences in subjects' subsequent syllogistic reasoning. We conclude that most participants do not understand deductive tasks as experimenters intend, and just as there is no single logical model of reasoning, so there is no reason to expect a single “fundamental human reasoning mechanism”.     

Mental model construction in linear reasoning: Evidence for the construction of initial annotated models

According to the mental model theory, reasoners build an initial model representing the information given in the premises. In the context of relational reasoning, the question arises as to which kind of representation is used to cope with indeterminate or multimodel problems. The present article presents an array of possible answers arising from the initial construction of complete explicit models, partial explicit models, partial implicit models, a single "isomeric" model, or a single annotated model. Predictions generated from these views are tested in two experiments that vary the problem structure and the number of models consistent with the premises. Analyses of the premise processing times, answering times and accuracy show that the annotated model yields the best fit of the data. Implications of these findings for the mental model theory as developed for relational reasoning are discussed.     

Reasoning with conditionals

This paper reviews the psychological investigation of reasoning with conditionals, putting an emphasis on recent work. In the first part, a few methodological remarks are presented. In the second part, the main theories of deductive reasoning (mental rules, mental models, and the probabilistic approach) are considered in turn; their content is summarised and the semantics they assume for if and the way they explain formal conditional reasoning are discussed, in particular in the light of experimental work on the probability of conditionals. The last part presents the recent shift of interest towards the study of conditional reasoning in context, that is, with large knowledge bases and uncertain premises.     

Reasoning from Suppositions

Two experiments investigated inferences based on suppositions. In Experiment 1, the subjects decided whether suppositions about individuals' veracity were consistent with their assertions—for example, whether the supposition “Ann is telling the truth and Beth is telling a lie”, is consistent with the premises: “Ann asserts: I am telling the truth and Beth is telling the truth. Beth asserts: Ann is telling the truth”. It showed that these inferences are more difficult than ones based on factual premises: “Ann asserts: I live in Dublin and Beth lives in Dublin”. There was no difference between problems about truthtellers and liars, who always told the truth or always lied, and normals, who sometimes told the truth and sometimes lied. In Experiment 2, the subjects made inferences about factual matters set in three contexts: a truth-inducing context in which friends confided their personality characteristics, a lie-inducing context in which business rivals advertised their products, and a neutral context in which computers printed their program characteristics. Given the supposition that the individuals were lying, it was more difficult to make inferences in a truth-inducing context than in the other two contexts. We discuss the implications of our results for everyday reasoning from suppositions, and for theories of reasoning based on models or inference rules.     

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.     

Mental models and deduction

According to the mental-model theory of deductive reasoning, reasoners use the meanings of assertions together with general knowledge to construct mental models of the possibilities compatible with the premises. Each model represents what is true in a possibility. A conclusion is held to be valid if it holds in all the models of the premises. Recent evidence described here shows that the fewer models an inference calls for, the easier the inference is. Errors arise because reasoners fail to consider all possible models, and because models do not normally represent what is false, even though reasoners can construct counterexamples to refute invalid conclusions.     

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.     

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.     

Supposition and representation in human reasoning

We report the results of three experiments designed to assess the role of suppositions in human reasoning. Theories of reasoning based on formal rules propose that the ability to make suppositions is central to deductive reasoning. Our first experiment compared two types of problem that could be solved by a suppositional strategy. Our results showed no difference in difficulty between problems requiring affirmative or negative suppositions and very low logical solution rates throughout. Further analysis of the error data showed a pattern of responses, which suggested that participants reason from a superficial representation of the premises in these arguments and this drives their choice of conclusion. Our second experiment employed a different set of suppositional problems but with extremely similar proofs in terms of the rules applied and number of inferential steps required. As predicted by our interpretation of reasoning strategies employed in Experiment 1, logical performance was very much higher on these problems. Our third experiment showed that problems that could be solved by constructing an initial representation of the premises were easier than problems in which this representation was not sufficient. This effect was independent of the suppositional structure of the problems. We discuss the implications of this research for theories of reasoning based on mental models and inference rules.     

Logical reasoning,world knowledge,and mental imagery: Interconnections in cognitive processes

To better understand the role of problem content in verbal reasoning, the effect of two aspects of problem representation on conditional reasoning was examined. Specifically, this study focused on the effect of availability of knowledge schemata and mental imagery on recognition of indeterminacy. Four groups of 20 adults solved syllogisms that varied in imagery value and in tendency to access knowledge schemata (assessed by ratings of the relatedness of antecedent and consequent clauses of premises). When problems both were high in imagery value and had related clauses, performance was significantly better on indeterminate syllogisms. Access to schemata may permit elaborative processing and the generation of counterexamples to invalid inferences; imagery may support representation of problems and generation of elaborative information in memory.     

HOW DIAGRAMS CAN IMPROVE REASONING

Abstract— We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions, for example:
Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows?
We confirmed that subjects find it difficult to deduce a valid conclusion, such as
Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia.
In a preliminary study, the formal of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance. In the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning.     

In the beginning was the doing: the premises of the practical syllogism

If practical reasoning deserves its name, its form must be different from that of ordinary (theoretical) reasoning. A few have thought that the conclusion of practical reasoning is an action, rather than a mental state. I argue here that if the conclusion is an action, then so too is one of the premises. You might reason your way from doing one thing to doing another: from browsing journal abstracts to reading a particular journal article. I motivate this by sympathetically re-examining Hume's claim that a conclusion about what ought to be done follows only from an argument one of whose premises is likewise about what ought to be done.     

Illusory inferences: a novel class of erroneous deductions.

The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally make explicit only what is true. The theory has an unexpected consequence: it predicts the occurrence of inferences that are compelling but invalid. They should arise from reasoners failing to take into account what is false. Three experiments corroborated the systematic occurrence of these illusory inferences, and eliminated a number of alternative explanations for them. Their results illuminate the controversy among various current theories of reasoning.