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.     

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.     

Believability and syllogistic reasoning

In this paper we investigate the locus of believability effects in syllogistic reasoning. We identify three points in the reasoning process at which such effects could occur: the initial interpretation of premises, the examination of alternative representations of them (in all of which any valid conclusion must be true), and the "filtering" of putative conclusions. The effect of beliefs at the first of these loci is well established. In this paper we report three experiments that examine whether beliefs have an effect at the other two loci. In experiments 1 and 2 subjects drew their own conclusions from syllogisms that suggested believable or unbelievable ones. In the third experiment they evaluated conclusions that were presented to them. The data show that beliefs both affect the examination of alternative models and act as a filter on putative conclusions. We conclude by showing how some types of problem and some problem contents make the existence of alternative models more obvious than others.     

Working memory and conditional reasoning

Little is known about the role of working memory in conditional reasoning. This paper reports three experiments that examine the contributions of the visuo-spatial scratch pad (VSSP), the articulatory loop, and the central executive components of Baddeley and Hitch's (1974) model of working memory to conditional reasoning. The first experiment employs a spatial memory task that is presented concurrently with two putative spatial interference tasks (tapping and tracking), articulatory suppression, and a verbal memory load. Only the tracking and memory load impaired performance, suggesting that these tap the VSSP and central executive, respectively. Having established the potency of these interference tasks two further experiments examined the effects of tapping and tracking (Experiment 2) and articulation and memory load (Experiment 3) on a conditional reasoning task. Neither tracking nor tapping affected the number of inferences accepted or response latency. Articulation also failed to affect conditional reasoning but memory load selectively reduced acceptance of modus tollens inferences. These results are discussed in terms of both rule-based and mental models theories of reasoning. While these data cannot discriminate between the two perspectives they provide support for one of the central assumptions in each: that some errors in reasoning are attributable directly to working memory demands. Taken together these experiments suggest that conditional reasoning requires an abstract working memory medium for representation; it does not require either the VSSP or the articulatory loop. It is concluded that the central executive provides the necessary substrate.     

Source credibility and syllogistic reasoning

Two experiments examined whether a source credibility effect would be observed for a syllogistic reasoning task. In the experiments, people were given two statements, presented as the results from a survey, followed by a conclusion that was supposedly made by one of two sources. In Experiment 1, one of the sources was described as honest and the other as dishonest, and in Experiment 2, one of the sources was described as an expert and the other as a non-expert. Because a pilot experiment showed that credibility can be overridden by people’s experience with a source, all conclusions in Experiments 1 and 2 were ones that were likely to be accepted (i.e., necessary and possible strong conclusions). Both experiments showed a clear source credibility effect, particularly for the invalid conclusions. These results, along with the belief bias effect and previous research with conditional reasoning, suggest that people can be influenced by extraneous context, such as the honesty or expertise of a source, in a syllogistic reasoning task.     

Components of syllogistic reasoning

The present research sought to understand the components of syllogistic reasoning used in a syllogistic evaluation task. In this task, subjects must indicate whether a conclusion such as “Some Yale professors are humbugs” is definitely true, or never true of a set of premises such as “Some humbugs study syllogistic reasoning; some Yale professors study syllogistic reasoning”. A modified form of componential analysis (Sternberg 1977, 1978) was used to decompose the syllogistic evaluation task with abstract content into encoding and encoding plus combination subtasks. The response-choice data from these subtasks were used to provide (a) direct tests of a proposed theory of syllogistic reasoning, and in particular, of its assumptions about sources of error in syllogistic reasoning; and (b) direct inferences regarding the representation of relations between the subject and predicate of the premises as encoded and combined. The results supported a proposed transitive-chain model of syllogistic reasoning.     

Error processes in syllogistic reasoning

In recent years, a number of proposals have been advanced to account for the errors that subjects make in deductive inferences from invalid syllogisms. Principles such as erroneous conversion of premises, probabilistic inference, feature selection, and various other interpretation and combination processes have been suggested. The present paper focuses on the 32 invalid categorical syllogisms for which conversion of premises does not provide an explanation of subject error. An explanation is presented in terms of three error processes: the erroneous conversion of conclusions resulting from backward processing, the erroneous integration of information from the two premises, and the failure to consider hypothetical possibilities. Empirical predictions regarding the differential difficulty of the various premise combinations as well as the pattern of correlations between premise combinations are derived from this formulation, and data are presented that support these predictions.     

A transitive-chain theory of syllogistic reasoning

A new theory of syllogistic reasoning, called the transitive-chain theory, is presented. The transitive-chain theory proposes that information about set relations is represented in memory by pairs of informational components. The theory further proposes that information about set relations is integrated by applying a small set of rules to transitive chains of these set relations. The rules that are applied to these chains are specified in detail. The theory is cast in terms of information-processing models for variants of the syllogistic reasoning task, and then mathematical models that quantify each of these information-processing models are presented. In a series of five experiments, the transitive-chain theory provides a good account of the response-choice data for syllogisms with various types of content, quantifiers, and logical relations (categorical and conditional). The results of these experiments offer tentative answers to four issues in the theory of syllogistic reasoning: (a) representation of premise information; (b) combination of premise information; (c) sources of difficulty in syllogistic reasoning; and (d) generality of the processes used in syllogistic reasoning.     

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.     

The development of linear syllogistic reasoning

Between 24 and 26 children in each of grades 3, 5, 7, 9, and 11 (with mean ages of 8.5, 10.2, 13.0, 15.0, and 16.6 years, respectively) were tested in their ability to solve linear syllogisms, problems such as “John is taller than Mary. Mary is taller than Pete. Who is tallest? John, Mary, Pete.” Response latencies and error rates decreased across grade levels and sessions. Component latencies also generally decreased with increasing age. Four alternative information-processing models were fit to the group latency data at each grade level. These models were (a) a spatial model, according to which people represent the terms of a linear syllogism in the form of a linear spatial array; (b) a linguistic model, according to which linear syllogisms are solved via inferences on functional relations represented by linguistic deep structures; (c) an algorithmic model, according to which people solve linear syllogisms by applying a series of simple, essentially mechanical steps, and (d) a mixed model, which combines selected features of the spatial and linguistic models, and adds new features of its own. The latency data supported the mixed model at each grade level, although in grade 9 the model was not the preferred one until the second session of testing.     

Content and strategy in syllogistic reasoning.

Syllogistic reasoning has been investigated as a general deductive process (Johnson-Laird & Byrne, 1991; Revlis, 1975; Rips, 1994). However, several studies have demonstrated the role of cognitive strategies in this type of reasoning. These strategies focus on the method used by the participants (Ford, 1995; Gilhooly, Logie, Wetherick, & Wynn, 1993) and strategies related to different interpretations of the quantified premises (Roberts, Newstead, & Griggs, 2001). In this paper, we propose that content (as well as individual cognitive differences) is an important factor in inducing a certain strategy or method for syllogistic resolution. Specifically, we suggest that syllogisms with a causal conditional premise that can be extended by an agency premise induce the use of a conditional method. To demonstrate this, we carried out two experiments. Experiment 1 provided evidence that this type of syllogism leads participants to draw the predicted conditional conclusions, in contrast with control content syllogisms. In Experiment 2, we demonstrated that the drawing of conditional conclusions is based on a causal conditional to an agent representation of the syllogism premises. These results support the role of content as inducing a particular strategy for syllogistic resolution. The implications of these results are discussed.     

Activation of end-terms in syllogistic reasoning

We report five experiments showing that the activation of the end-terms of a syllogism is determined by their position in the composite model of the premises. We show that it is not determined by the position of the terms in the rule being applied (Ford, 1994), by the syntactic role of the terms in the premises (Polk & Newell, 1995; Wetherick & Gilhooly, 1990), by the type of conclusion (Chater & Oaksford, 1999), or by the terms from the source premise (Stenning & Yule, 1997). In our first experiment we found that after reading a categorical premise, the most active term is the last term in the premise. In Experiments 2, 3, and 4 we demonstrated that this pattern of activity is due to the position of the concepts in the model of the premises, regardless of the delay after reading the premises (150 or 2000 msec) or the quantity of the quantifiers (universal or existential). The fifth experiment showed that the pattern switches around after participants evaluate a conclusion. We propose that the last element in the model maintains a higher level of activity during the comprehension process because it is generally used to attach the incoming information. After this process, the first term becomes more active because it is the concept to which the whole representation is referred. These results are predicted by the mental model theory (Johnson-Laird & Byrne, 1991), but not by the verbal reasoning theory (Polk & Newell, 1995), the graphical methods theory (Yule & Stenning, 1992), the attachment-heuristic theory (Chater & Oaksford, 1999), or the mental rules theory (Ford, 1994).     

On belief bias in syllogistic reasoning

A multinomial model is used to disentangle the respective contributions of reasoning processes and response bias in conclusion-acceptance data that exhibit belief bias. A model-based meta-analysis of 22 studies reveals that such data are structurally too sparse to allow discrimination of different accounts of belief bias. Four experiments are conducted to obtain richer data, allowing deeper tests through the use of the multinomial model. None of the current accounts of belief bias is consistent with the complex pattern of results. A new theory of belief bias is proposed that assumes that most reasoners construct only one mental model representing the premises as well as the conclusion or, in the case of an unbelievable conclusion, its logical negation. New predictions derived from the theory are confirmed in 4 additional studies.     

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.     

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.     

Working memory and reasoning: An individual differences perspective

This article reports three experiments that investigated the relationship between working memory capacity and syllogistic and five-term series spatial inference. A series of complex and simple verbal and spatial working memory measures were employed. Correlational analyses showed that verbal and spatial working memory span tasks consistently predicted syllogistic and spatial reasoning performance. A confirmatory factor analysis showed that three factors best accounted for the data—a verbal, a spatial, and a general factor. Syllogistic reasoning performance loaded all three factors, whilst spatial reasoning loaded only the general factor. The implications of these findings are discussed in the context of reasoning theories and contemporary accounts of the structure of working memory.     

A successive-conditionalization approach to disjunctive and syllogistic reasoning

With p and q each standing for a familiar event, a disjunctive statement, “either p or q”, seems quite different from its material conditional, “if not p then q”. The notions of sufficiency and necessity seem specific to conditional statements. It is surprising, however, to find that perceived sufficiency and necessity affect disjunctive reasoning in the way they affect conditional reasoning. With B and C each standing for a category name, a universal statement, “all B are C”, seems stronger than its logically equivalent conditional statement, “if B then C”. However, the effects of perceived sufficiency or necessity were found to be as pronounced in conditional reasoning as in syllogistic reasoning. Furthermore, two experiments also showed that (a) MP (modus ponens)-comparable disjunctive reasoning was as difficult as MT (modus tollens)-comparable disjunctive reasoning, and that (b) MT-comparable syllogisms were easier to solve than MT problems in conditional reasoning.     

Belief bias and figural bias in syllogistic reasoning

Belief bias is the tendency to be influenced by the believability of the conclusion when attempting to solve a syllogistic reasoning problem. Figural bias is the tendency to be influenced by the order in which the information is presented in the premises when attempting to solve a syllogistic reasoning problem. When studied simultaneously they enable an investigation of whether participants' reasoning on the syllogistic reasoning task is guided by the conclusion (backward reasoning) or the premises (forward reasoning). Experiments 1 and 2 found evidence of belief bias but not figural bias on the syllogistic evaluation task paradigm. Experiments 3 and 4 found evidence of figural bias but not belief bias on the syllogistic production task paradigm. The findings highlight that different task characteristics influence performance dependent upon the nature of task presentation. These findings are discussed in the context of current theories of belief bias in syllogistic reasoning.