On the conflict between logic and belief in syllogistic reasoning

Three experiments are reported that investigate the weighting attached to logic and belief in syllogistic reasoning. Substantial belief biases were observed despite controls for possible conversions of the premises. Equally substantial effects of logic were observed despite controls for two possible response biases. A consistent interaction between belief and logic was also recorded; belief bias was more marked on invalid than on valid syllogisms. In all experiments, verbal protocols were recorded and analyzed. These protocols are interpreted in some cases as providing rationalizations for prejudiced decisions and, in other cases, as reflecting a genuine process of premise to conclusion reasoning. In the latter cases, belief bias was minimal but still present. Similarly, even subjects who focus primarily on the conclusion are influenced to an extent by the logic. Thus a conflict between logic and belief is observed throughout, but at several levels of extent.     

When logic and belief collide: Individual differences in reasoning times support a selective processing model

When the validity of a deductive conclusion conflicts with its believability people often respond in a belief-biased manner. This study used response times to test the selective processing model, which views belief-bias effects as arising from the interplay between superficial heuristic processes and more rigorous analytic processes. Participants were split into three response groups according to their propensity to endorse logically normative conclusions. The low-logic, high belief-bias group demonstrated rapid responding, consistent with heuristic processing. The medium-logic, moderate belief-bias group showed slower responding, consistent with enhanced analytic processing, albeit selectively biased by conclusion believability. The high-logic, low belief-bias group's relatively unbiased responses came at the cost of increased processing times, especially with invalid-believable conclusions. These findings support selective processing claims that distinct heuristic and analytic processing systems underpin reasoning, and indicate that certain individuals differentially engage one system more than the other. A minor amendment is proposed to the current selective processing model to capture the full range of observed effects.     

Necessity, possibility and belief: A study of syllogistic reasoning

The present study extended the investigation of the belief bias effect in syllogistic reasoning in two ways: (1) The effect was studied under instructions to decide whether conclusions were possible, as well as necessary, given the premises; and (2) the effect was studied for types of syllogism where people rarely endorse the conclusions as well as those (valid and fallacious) where endorsements are common. Three experiments are reported, which show first that there is a marked tendency to reject unbelievable conclusions relative to abstract or neutral controls on all kinds of syllogism and under both types of instruction. There was also significant evidence of positive belief bias (increased acceptance of believable conclusions) and of interactions between belief bias effects and logical form. The results are discussed with particular respect to accounts of belief bias offered by theorists in the mental-model tradition.     

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.     

Logic and belief across the lifespan: the rise and fall of belief inhibition during syllogistic reasoning

Popular reasoning theories postulate that the ability to inhibit inappropriate beliefs lies at the heart of the human reasoning engine. Given that people's inhibitory capacities are known to rise and fall across the lifespan, we predicted that people's deductive reasoning performance would show similar curvilinear age trends. A group of children (12‐year‐olds), young adults (20‐year‐olds), and older adults (65+‐year‐olds) were presented with a classic syllogistic reasoning task and a decision‐making questionnaire. Results indicated that on syllogisms where beliefs and logic conflicted, reasoning performance showed the expected curvilinear age trend: Reasoning performance initially increased from childhood to early adulthood but declined again in later life. On syllogisms where beliefs and logic were consistent and sound reasoning did not require belief inhibition, however, age did not affect performance. Furthermore, across the lifespan we observed that the better people were at resisting intuitive temptations in the decision‐making task, the less they were biased by their beliefs on the conflict syllogisms. As with the effect of age, one's ability to override intuitions in the decision‐making task did not mediate reasoning performance on the no‐conflict syllogisms. Results lend credence to the postulated central role of inhibitory processing in those situations where beliefs and logic conflict.     

Working memory, metacognitive uncertainty, and belief bias in syllogistic reasoning

Studies of syllogistic reasoning have shown that the size of the belief bias effect varies with manipulations of logical validity and problem form. This paper presents a mental models-based account, which explains these findings in terms of variations in the working-memory demands of different problem types. We propose that belief bias may reflect the use of a heuristic that is applied when a threshold of uncertainty in one's processing-attributable to working-memory overload-is exceeded during reasoning. Three experiments are reported, which tested predictions deriving from this account. In Experiment 1, conclusions of neutral believability were presented for evaluation, and a predicted dissociation was observed in confidence ratings for responses to valid and invalid arguments, with participants being more confident in the former. In Experiment 2, an attempt to manipulate working-memory loads indirectly by varying syllogistic figure failed to produce predicted effects upon the size of the belief bias effect. It is argued that the employment of a conclusion evaluation methodology minimized the effect of the figural manipulation in this experiment. In Experiment 3, participants' articulatory and spatial recall capacities were calibrated as a direct test of working-memory involvement in belief bias. Predicted differences in the pattern of belief bias observed between highand lowspatial recall groups supported the view that limited working memory plays a key role in belief bias.     

Early and late processes in syllogistic reasoning: evidence from eye-movements

An eye-movement monitoring experiment was carried out to examine the effects of the difficulty of the problem (simple versus complex problems) and the type of figure (figure 1 or figure 4) on the time course of processing categorical syllogisms. The results showed that the course of influence for these two factors is different. We found early processing effects for the figure but not for the difficulty of the syllogism and later processing effects for both the figure and the difficulty. These results lend support to the Model Theory (Johnson-Laird, P. N., Byrne, R. M. J. (1991). Deduction. Hillsdale, New Jersey: LEA.) as opposed to other theories of reasoning (Chater, N., Oaksford, M. (1999). The probability heuristics model of syllogistic reasoning. Cognitive Psychology, 38, 191-258; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press).     

Characterizing belief bias in syllogistic reasoning: A hierarchical Bayesian meta-analysis of ROC data

The belief-bias effect is one of the most-studied biases in reasoning. A recent study of the phenomenon using the signal detection theory (SDT) model called into question all theoretical accounts of belief bias by demonstrating that belief-based differences in the ability to discriminate between valid and invalid syllogisms may be an artifact stemming from the use of inappropriate linear measurement models such as analysis of variance (Dube et al., Psychological Review, 117(3), 831–863, 2010). The discrepancy between Dube et al.’s, Psychological Review, 117(3), 831–863 (2010) results and the previous three decades of work, together with former’s methodological criticisms suggests the need to revisit earlier results, this time collecting confidence-rating responses. Using a hierarchical Bayesian meta-analysis, we reanalyzed a corpus of 22 confidence-rating studies (N =?993). The results indicated that extensive replications using confidence-rating data are unnecessary as the observed receiver operating characteristic functions are not systematically asymmetric. These results were subsequently corroborated by a novel experimental design based on SDT’s generalized area theorem. Although the meta-analysis confirms that believability does not influence discriminability unconditionally, it also confirmed previous results that factors such as individual differences mediate the effect. The main point is that data from previous and future studies can be safely analyzed using appropriate hierarchical methods that do not require confidence ratings. More generally, our results set a new standard for analyzing data and evaluating theories in reasoning. Important methodological and theoretical considerations for future work on belief bias and related domains are discussed.     

Phonological and visual distinctiveness effects in syllogistic reasoning: Implications for mental models theory

Two experiments are reported in which the representational distinctiveness of terms within categorical syllogisms was manipulated in order to examine the assumption of mental models theory that abstract, spatially based representations underpin deduction. In Experiment 1, participants evaluated conclusion validity for syllogisms containing either phonologically distinctive terms (e.g., harks, paps, and fids) or phonologically nondistinctive terms (e.g., fuds, fods, and feds). Logical performance was enhanced with the distinctive contents, suggesting that the phonological properties of syllogism terms can play an important role in deduction. In Experiment 2, participants received either the phonological materials from Experiment 1 or syllogisms involving distinctive or nondistinctive visual contents. Logical inference was again enhanced for the distinctive contents, whether phonological or visual in nature. Our findings suggest a broad involvement of multimodal information in syllogistic reasoning and question the assumed primacy of abstract, spatially organized representations in deduction, as is claimed by mental models theorists.     

Individual differences in working memory capacity and resistance to belief bias in syllogistic reasoning

In two experiments, we investigated the possibility that individual differences in working memory capacity (WMC) would provide resistance to belief bias in syllogistic reasoning. In Experiment 1 (N?=?157), participants showed a belief bias effect in that they had longer response times and decreased accuracy on syllogisms with conflict between the validity and believability of the conclusion than on syllogisms with no such conflict. However, this effect did not differ as a function of individual differences in WMC. Experiment 2 (N?=?122) replicated this effect with the addition of decontextualized (i.e., nonsense) syllogisms as a different means of measuring the magnitude of the belief bias effect. Although individual differences in WMC and fluid intelligence were related to better reasoning overall, the magnitude of the belief bias effect was not smaller for participants with greater WMC. The present study offers two novel findings: (a) The belief bias effect is independent of individual differences in WMC and fluid intelligence, and (b) resolving conflict in verbal reasoning is not a type of conflict resolution that correlates with individual differences in WMC, further establishing boundary conditions for the role of WMC in human cognitive processes.     

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 effect of figure on syllogistic reasoning

Several investigators of reasoning with abstract categorical syllogisms have noted that the four figures of the classical syllogism, which vary in the order in which terms occur in the major and minor premises, resemble the four three-stage mediation paradigms in paired associate learning. On the basis of the analogy, figure differences have been predicted and some support has been obtained for these predictions. The present paper proposes an alternative information processing explanation for figure effects based upon contradictions between the correct conclusions that follow from forward and backward processing of the premises. This explanation, in contrast to the associative explanation, successfully predicts which premise combinations will show figure effects as well as the nature of the specific errors which will occur.     

The effects of externalization on syllogistic reasoning

An experiment is reported that investigated the effects of externalization of mental models in syllogistic reasoning. Although there was no evidence that the requirement to “externalize” mental models of syllogisms improved reasoning, an unexpected recognition test demonstrated that subjects' memory for the meaning of the premises was improved by externalization. In particular, where the correct conclusion had been deduced using the externalization procedure, responses in the recognition test reflected an appreciation of the relations between the end terms of the premises.     

Effects of perspective and belief on analytic reasoning in a scientific reasoning task

The purpose of these studies was to test the hypothesis that changing perspectives from one's own to another's promotes the engagement of analytic processing and, in turn, reduces the impact of beliefs. In two experiments participants evaluated research vignettes containing belief-consistent and belief-inconsistent conclusions, and indicated whether the data supported a correlation between two variables. Consistent with our hypothesis, the tendency to endorse correlations consistent with prior belief was reduced when participants evaluated the data from the researcher's perspective relative to their own. We also administered the Actively Open Minded Thinking (AOT) scale (Stanovich & West, 2007, 2008), which did not predict belief effects on our task. We did however observe that the AOT was reliably associated with different response strategies: high AOT scorers were more inclined to choose ambiguous response options, such as “no conclusion is warranted”, whereas low scorers evinced a preference for more determinate options (e.g., there is no relationship between the two variables). We interpret our findings in the context of dual process theories of reasoning and from a Bayesian perspective.     

The effects of belief on the spontaneous production of syllogistic conclusions

Two experiments examined the effects of subjects' beliefs on syllogistic inference. The first experiment showed that beliefs biased the spontaneous conclusions that subjects drew for themselves. These effects were more marked for indeterminate premises (which yield no non-trivial valid conclusions) than for determinate premises (which yield valid conclusions). There was also an effect of the nature of the beliefs: conclusions that were false by definition had a bigger effect on deductions than those that were false as a matter of fact. The second experiment replicated the finding for determinate syllogisms, using problems in moods in which the status of the valid conclusion could not be altered by conversion of the premises. Beliefs accordingly appear to affect the process of reasoning rather than the interpretation of premises.     

Conditional reasoning by mental models: chronometric and developmental evidence

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.     

Confidence judgments in syllogistic reasoning: the role of consistency and response cardinality

In two experiments, we examined the resolution of confidence judgments in syllogistic reasoning and their heuristic bases. Based on the assumptions of Koriat's Self-Consistency Model of confidence, we expected the confidence judgments to be related to conclusion consensuality, reflecting the role of consistency as a heuristic cue to confidence. In Experiment 1, the participants evaluated 24 syllogisms with conclusions that varied with respect to validity and consensuality. In Experiment 2, the participants produced conclusions to 64 pairs of premises. The correlation between confidence and reasoning accuracy was low. In both experiments confidence was related to the consensuality of the responses. For consensually correct items, correlation between confidence and accuracy was positive; however, for consensually incorrect items it was negative. In Experiment 2, confidence was lower for syllogisms with higher response cardinality, or syllogisms that elicited a greater variety of conclusions.     

A three-factor model of syllogistic reasoning: The study of isolable stages

Computer models of the syllogistic reasoning process are constructed. The models are used to determine the influence of three factors—the misinterpretation of the premises, the limited capacity of working memory, and the operation of the deductive strategy—on subjects’ behavior. Evidence from Experiments 1, 2, and 3 suggests that all three factors play important roles in the production of errors when “possibly true” and “necessarily false” are the two response categories. This conclusion does not agree with earlier analyses that had singled out one particular factor as crucial. Evidence from Experiment 4 suggests that the influence of the first two factors remains strong when “necessarily true” is used as an additional response category. However, the third factor appears to interact with task demands. Some concluding analyses suggest that the models offer alternative explanations for certain well established results.