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.     

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.     

On the role of imagery in linear syllogistic reasoning

Summary The functional role of imagery in linear syllogistic reasoning has not yet been convincingly established. Shortcomings of previous theories and research are discussed. A conceptual analysis of the imagery position leads to the differentiation of modality-specific, analogical-representational, and process perspectives. To obtain evidence relevant to the first and third of these perspectives, a Stroop-type interference paradigm was used in combination with self-paced successive presentation of task components (premises and question of linear syllogisms). Effects of interference information and task components upon response time were consistent with the process reconceptualization of the imagery position.     

Complexly fractionated syllogistic quantifiers

Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantifiers, and then include further quantifiers which are modifications of such fractions (such as “almost 1/2 the S are P” and “Much more than 1/2 the S are P”). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes &; Peterson (forthcoming) for proofs of the unsoundness and incompleteness of Thompson's rules. Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., “almost 1/2 the S are P” is represented as “?(3(SP)?SP)”.     

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.     

Tree proofs for syllogistic

This paper presents a tree method for testing the validity of inferences, including syllogisms, in a simple term logic. The method is given in the form of an algorithm and is shown to be sound and complete with respect to the obvious denotational semantics. The primitive logical constants of the system, which is indebted to the logical works of Jevons, Brentano and Lewis Carroll, are term negation, polyadic term conjunction, and functors affirming and denying existence, and use is also made of a metalinguistic concept of formal synonymy. It is indicated briefly how the method may be extended to other systems.To the memory of Jerzy Supecki     

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.     

Working memory and syllogistic reasoning

The purpose of this study was to examine the relation between working memory span and syllogistic reasoning performance. In addition, performance for the reasoning task was compared to predictions made by mental model theory and the probability heuristics model. According to mental model theory, syllogisms that require the use of more mental models are more difficult. According to the probability heuristics model difficulty is related to the number of probabilistic heuristics that must be applied, or (for invalid syllogisms) inconsistencies between the derived and correct conclusion. The predictions of these theories were examined across two experiments. In general, people with larger working memory capacities reasoned better. Also, the responses made by people with larger capacities were more likely to correspond to the predictions made by both mental model theory and the probability heuristics model. Relations between working memory span and performance were also consistent with both theories.     

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.     

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.     

Three-membered domains for aristotle's syllogistic

The paper shows that for any invalid polysyllogism there is a procedure for constructing a model with a domain with exactly three members and an interpretation that assigns non-empty, non-universal subsets of the domain to terms such that the model invalidates the polysyllogism.I am grateful to T. J. Smiley and an anonymous referee for comments that improved this paper.     

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.     

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.     

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.