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.     

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.     

Naive probability: a mental model theory of extensional reasoning

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which it occurs. The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic 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.     

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.     

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.     

Reasoning and dyslexia: A spatial strategy may impede reasoning with visually rich information

Bacon, Handley and Newstead (2003, 2004) , have presented evidence for individual differences in reasoning strategies, with most people seeming to represent and manipulate problem information using either a verbal or a spatial strategy. There is also evidence that individuals with dyslexia are inclined to conceptualise information in a visuo‐spatial, rather than a verbal, way (e.g. von Károlyi et al., 2003 ). If so, we might expect a higher proportion of individuals with dyslexia to be spatial reasoners, compared with individuals who do not have dyslexia. The study reported here directly compared strategies reported by these two groups of participants on a syllogistic reasoning task. Moreover, problem content was manipulated so that reasoning across concrete and abstract materials could be compared. The findings suggest that whilst most individuals without dyslexia use a verbal strategy, reasoners with dyslexia do tend to adopt a spatial approach, though their performance is impaired with visually concrete materials. However, when reasoning with more abstract content, they perform comparably with non‐dyslexic controls. The paper discusses these results in the light of recent research which has suggested that visual images may impede reasoning, and considers how individuals with dyslexia may differ from other reasoners.     

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.     

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.     

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.     

On the generation and evaluation of inferences from single premises

A theory of how individuals construct mental models to draw inferences from single premises was tested in three experiments. Experiment 1 confirmed a counterintuitive prediction that it is easier to generate inferences between conditionals and disjunctions than it is to evaluate them. Experiment 2 replicated this finding, but an advantage found in the first experiment for conditional-to-disjunction over disjunction-to-conditional inferences was removed with different sentence contents. Experiment 3 showed that disjunction-to-conditional inferences were facilitated when premises expressed familiar indicative relations, whereas conditional-to-disjunction inferences were facilitated when premises expressed causal relations. The results indicate that small changes in task format can have large effects on the strategies that people use to represent and reason about different sentential connectives. We discuss the potential for theories other than mental models to account for these results. We argue that, despite the important role played by single-premise inferences in paraphrasing logical forms during inference, mental logic theories cannot account for the results reported here.     

The Effect of an Irrelevant Premise on Temporal and Spatial Reasoning

The mental model theory of reasoning postulates that individuals reason by constructing models of the situation described by premises. The more models reasoners have to build, the harder a problem will be. Byrne and Johnson-Laird (1989) confirmed this prediction with spatial problems and Schaeken et al. (1996 a, b) with temporal ones. There is, however, a problem with these studies. All the multiple-model problems and some of the one-model problems contained an irrelevant premise. As Rips (1994) argues, it is possible that an irrelevant premise would complicate the search for a derivation. The present paper reports an experiment which explicitly manipulated the presence of such an irrelevant premise in both one-model and multiple-model problems. The results corroborate the predictions of the mental model theory.     

Suppression of valid inferences and knowledge structures: the curious effect of producing alternative antecedents on reasoning with causal conditionals

These studies looked at the difficulty that reasoners have in accepting conditional ("If P then Q") major premises that are not necessarily true empirically, as a basis for deductive reasoning. Preliminary results have shown that when reasoners are asked to produce possible alternate antecedents to the major premise ("If A then Q"), they paradoxically tend to deny the modus ponens (MP) inference ("If P is true, then Q is true"). Three studies further explored these results. The first study gave university students paper-and-pencil tests in which instructions to "suppose that the major premise is true" was followed by a request to determine the next number in a sequence, to retrieve information unrelated to the premises, or to retrieve a possible case of "If A then Q." Relative to a control group, reasoners asked to produce an alternative antecedent showed a significant tendency to deny the MP inference, whereas no such tendency was observed for the two other tasks used. A second study compared performance on a condition in which reasoners were asked to produce an alternative antecedent with that when they were given an explicit alternative. Premises used in this study were such that the latter alternative antecedent was also spontaneously produced by over 70% of reasoners. Results showed that the tendency to refuse the MP premise could not be accounted for by the specific nature of the alternative produced. A third study found that the tendency to refuse the MP inference after producing an alternative antecedent was affected by the number of "disabling conditions" (i.e., conditions that allow "P to be true" and "Q to be false") available for the major premise. These results are interpreted as being consistent with a model that supposes that logical reasoning requires selective inhibition of real-world knowledge.     

Temporal and spatial relations in sentential reasoning

The mental model theory postulates that the meanings of assertions, and knowledge about their context can modulate the logical meaning of sentential connectives, such as "if" and "or". One known effect of modulation is to block the representation of possibilities to which a proposition refers. But, modulation should also add relational information, such as temporal order, to models of possibilities. Three experiments tested this prediction. Experiment 1 showed that individuals spontaneously matched the tense of their conclusions (in Portuguese) to embody implied, but unexpressed, temporal relations in conditional premises. Experiment 2 demonstrated the same phenomenon in inferences from disjunctions. Experiment 3 showed that the number of such implicit relations in inferences from conditionals affects both accuracy and the speed of reasoning. These results support the modulation hypothesis.     

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.     

Syllogistic reasoning time: Disconfirmation disconfirmed

University of Saskatchewan, Saskatoon, Saskatchewan, Canada Models of deductive reasoning typically assume that reasoners dedicate more logical analysis to unbelievable conclusions than to believable ones (e.g., Evans, Newstead, Allen, & Pollard, 1994; Newstead, Pollard, Evans, & Allen, 1992). When the conclusion is believable, reasoners are assumed to accept it without much further thought, but when it is unbelievable, they are assumed to analyze the conclusion, presumably in an attempt to disconfirm it. This disconfirmation hypothesis leads to two predictions, which were tested in the present experiment: Reasoners should take longer to reason about problems leading to unbelievable conclusions, and reasoners should consider more models or representations of premise information for unbelievable conclusions than for believable ones. Neither prediction was supported by our data. Indeed, we observed that reasoners took significantly longer to reason about believable conclusions than about unbelievable ones and generated the same number of representations regardless of the believability of the premises. We propose a model, based on a modified version of verbal reasoning theory (Polk & Newell, 1995), that does not depend on the disconfirmation assumption.     

Individual differences in strategies for syllogistic reasoning

Current theories of reasoning such as mental models or mental logic assume a universal cognitive mechanism that underlies human reasoning performance. However, there is evidence that this is not the case, for example, the work of Ford (1995), who found that some people adopted predominantly spatial and some verbal strategies in a syllogistic reasoning task. Using written and think-aloud protocols, the present study confirmed the existence of these individual differences. However, in sharp contrast to Ford, the present study found few differences in reasoning performance between the two groups, in terms of accuracy or type of conclusion generated. Hence, reasoners present an outward appearance of ubiquity, despite underlying differences in reasoning processes. These findings have implications for theoretical accounts of reasoning, and for attempts to model reasoning data. Any comprehensive account needs to account for strategic differences and how these may develop in logically untrained individuals.     

Recognition of proofs in conditional reasoning

Relatively little is known about those who consistently produce the valid response to Modus Tollens (MT) problems. In two studies, people who responded correctly to MT problems indicated how “convinced” they were by proofs of conditional reasoning conclusions. The first experiment showed that MT competent reasoners found accurate proofs of MT reasoning more convincing than similar “proofs” of invalid reasoning. Similarly, there was a tendency for MT competent reasoners to find an initial counterfactual supposition more convincing than did people who were less competent in MT. The second experiment showed that when individuals produced the correct MT response, and found correct MT proofs to be more convincing than “bogus” proofs, they were also less likely to find the conclusions to Denying the Antecedent, or Affirming the Consequent problems valid, compared to individuals who could not discriminate between valid and bogus MT proofs. These findings are discussed in terms of both their implications for the mental logic and mental models positions, and individual differences in System 1 and System 2 reasoning.     

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.