Conditional reasoning following contradictory evidence: A developmental analysis

Recent evidence suggests that young adults do not correctly understand the logical relationship of the conditional (if p the q) as it applies to hypothesis testing, and most training procedures have not been productive. However, the introduction of contradictory evidence following faulty inferences has led to accurate inferences with conditional statements. Third and seventh grade and college studients (9, 13, and 21 years of age, respectively) were tested to assess developmental differences in improvement following contradiction training and to test whether improved performance transfers to other conditional reasoning tasks. Significant improvement in conditional reasoning was found for the young adult group following the introduction of contradictory evidence, and the positive effect of the treatment transferred across tasks. The third grade students showed no effects from the introduction of the contradiction, but the seventh graders were often confused by the introduction of the contradiction. Seventh grade and college student performances were generally worse than that of the third graders for the positive instances (p · q), but while the contradiction training improved college studients' performance it did not affect the seventh graders. The results are discussed in terms of changes in cognitive structures.     

The mental representation of causal conditional reasoning: mental models or causal models

In this paper, two experiments are reported investigating the nature of the cognitive representations underlying causal conditional reasoning performance. The predictions of causal and logical interpretations of the conditional diverge sharply when inferences involving pairs of conditionals—such as if P₁then Q and if P₂then Q—are considered. From a causal perspective, the causal direction of these conditionals is critical: are the P_icauses of Q; or symptoms caused byQ. The rich variety of inference patterns can naturally be modelled by Bayesian networks. A pair of causal conditionals where Q is an effect corresponds to a “collider” structure where the two causes (P_i) converge on a common effect. In contrast, a pair of causal conditionals where Q is a cause corresponds to a network where two effects (P_i) diverge from a common cause. Very different predictions are made by fully explicit or initial mental models interpretations. These predictions were tested in two experiments, each of which yielded data most consistent with causal model theory, rather than with mental models.     

Updating mental models in predictive reasoning

Abstract

A mental model account for predictive judgement is proposed. According to this view, solving a predictive task involves the foundation ad updating of a mental he work as the relevant data are provided. The final model state determines the subject's predictive outcome. Two experiments examine the temporal course of reasoning in novices (Experiment 1) and experts (Experiment 2). Each task provided in succession a quantitative source (e.g. “Last year, of the vehidts that stopped at the cafe, 80% were cars and 20% were trucks”), a diagnostic source (e.g. “This vehide is mksy”) and a categorical choice (e.g. “What kind of vehicle was it?”). Two factors were manipulated: The order of sources aimed at analysing the contextual dependence of the updating processes, and the between-sources congruence (congruent or incongruent) in order to explore the integrative processes in model building. The pattern of the sources reading times, the choice times and the categorical choices suggested that: (1) the processing of a source depends on the previous one, (2) subjects try to integrste both sources into a single mental framework, and (3) experts and novices have a similar performance, although experts give more importance to the quantitative source in the diagnostic-quantitative order. These results are better accounted for by the mental model framework than both the heuristic and the formal view of prediction.     

Conditional reasoning and the competence-performance issue: A developmental analysis of a training task

Empirical investigations of conditional reasoning have generally found that both children and young adults perform poorly on tasks that require the selection or evaluation of those propositions that test the truth status of conditional statements (if p then q). Earlier work (D. O'Brien & W. F. Overton, Journal of Experimental Child Psychology 1980, 30, 44–60) suggested that poor performances with these tasks by young adults show improvement following the introduction of evidence that contradicts earlier faulty inferences, and this improvement generalizes to other conditional reasoning tasks. The effects of the contradiction training were not found with younger subjects. The present research is an extension of the contradiction training paradigm. Ten-, fourteen-, and eighteen-year-olds were tested to assess developmental differences in improvement with an evaluation and a conditional syllogism task. Significant improvement in performance was found for the twelfth grade students following the contradiction training, and this generalized across tasks. This effect was not found for the two younger groups. The usual poor performance of the oldest group is considered to be a false negative assessment of their conditional reasoning competency. Further, it is suggested that several correct performances of younger reasoners are false positive assessments of their conditional reasoning competency.     

Conditional reasoning and causation

An experiment was conducted to investigate the relative contributions of syntactic form and content to conditional reasoning. The content domain chosen was that of causation. Conditional statements that described causal relationships (if mean value of cause, then mean value of effect) were embedded in simple arguments whose entailments are governed by the rules of truth-functional logic (i.e., modus ponens, modus tollens, denying the antecedent, and affirming the consequent). The causal statements differed in terms of the number of alternative causes and disabling conditions that characterized the causal relationship. (A disabling condition is an event that prevents an effect from occurring even though a relevant cause is present). Subjects were required to judge whether or not each argument's conclusion could be accepted. Judgements were found to vary systematically with the number of alternative causes and disabling conditions. Conclusions of arguments based on conditionals with few alternative causes or disabling conditions were found to be more acceptable than conclusions based on those with many.     

Conditional reasoning and conditionalization

In solving conditional reasoning problems, reasoners are assumed to compute the probability of the conclusion, conditionalizing first on the categorical premise, giving the knowledge-based component, and conditionalizing then on the conditional-statement premise, from which the assumption-based component is derived. Because reasoners find it difficult to compute the second-step conditionalization except when the conditional-statement premise is found to be related to the result of the first-step conditionalization as for modus ponens or, possibly, for modus tollens, the knowledge-based component generally dominates reasoning performance. After representing all the possible cases in which conditional-argument forms may appear, this approach was found to be consistent with the results from the 3 experiments reported in this study, whereas 2 alternative hypotheses account for only some of the results.     

Conditional reasoning, representation, and empirical evidence on a concrete task

This study examined interactions between empirical data, internal representations, and reasoning performance on a conditional reasoning task using a concrete apparatus. Subjects were asked an initial series of questions in order to determine the pattern of inferences they made after simple exposure to the apparatus. They were subsequently shown two different experimental manipulations designed to provide data about the internal structure of the apparatus without giving information about specific inferences. Some subjects did change their reasoning in response to the new data, although most remained stable throughout the experiment. These results are consistent with the idea that reasoning may require generation of an internal representation of a problem space. It was also concluded that the relation between reasoning and empirical evidence cannot be understood without supposing that evidence is often interpreted by subjects according to their reasoning patterns.     

Sample diversity and premise typicality in inductive reasoning: evidence for developmental change

Evaluating whether a limited sample of evidence provides a good basis for induction is a critical cognitive task. We hypothesized that whereas adults evaluate the inductive strength of samples containing multiple pieces of evidence by attending to the relations among the exemplars (e.g., sample diversity), six-year-olds would attend to the degree to which each individual exemplar in a sample independently appears informative (e.g., premise typicality). To test these hypotheses, participants were asked to select between diverse and non-diverse samples to help them learn about basic-level animal categories. Across various between-subject conditions (N=133), we varied the typicality present in the diverse and non-diverse samples. We found that adults reliably selected to examine diverse over non-diverse samples, regardless of exemplar typicality, six-year-olds preferred to examine samples containing typical exemplars, regardless of sample diversity, and nine-year-olds were somewhat in the midst of this developmental transition.     

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.     

Children's solution strategies and mental rotation: evidence for a developmental shift

We examined children's ability to anticipate the appearance of transformed multicomponents using visual stimuli with variations of a mental rotation task. We hypothesized that (a) performance would depend on the presence, location, and spatial relationship of specific stimulus features, and that (b) younger children would use a single component to make their predictions, whereas older children would be more likely to use multiple components. In Study 1, 40 first-, and fifth-grade subjects were presented with 32 rotation problems consisting of varied stimulus characteristics; subjects selected the correct option from a field where foils were indicative of the strategy used. Analyses of variance and log linear analyses revealed the hypothesized stimulus effect differentially associated with the children's ages. In Study 2, sixty 5-, 7-, and 9-year olds confronted 64 mental rotation problems with multicomponent stimuli and a construction task. Results and implications confirmed those of Study 1.     

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.     

Simon-type effects: chronometric evidence for keypress schemata in typewriting

In 4 experiments, chronometric evidence for keypress schemata in typing was sought by presenting stimuli to be typed in positions that were displaced from a central fixation point. Reaction times were shorter when stimulus positions corresponded to keyboard locations of the letters to be typed, suggesting that position was an important part of the internal representation of the response. Experiment 1 presented single letters left and right of fixation. Experiment 2 presented single letters above and below fixation. Experiment 3 presented words left and right of fixation and found evidence of parallel activation of keypress schemata. Experiment 4 found no effect of the eccentricity of the keyboard locations and responding fingers, suggesting that response-location codes are categorical, not metric. The results are consistent with D. E. Rumelhart and D. A. Norman's (1982) theory of typewriting.     

Effects of belief and logic on syllogistic reasoning: Eye-movement evidence for selective processing models

Studies of syllogistic reasoning have demonstrated a nonlogical tendency for people to endorse more believable conclusions than unbelievable ones. This belief bias effect is more dominant on invalid syllogisms than valid ones, giving rise to a logic by belief interaction. We report an experiment in which participants' eye movements were recorded in order to provide insights into the nature and time course of the reasoning processes associated with manipulations of conclusion validity and believability. Our main dependent measure was people's inspection times for syllogistic premises, and we tested predictions deriving from three contemporary mental-models accounts of the logic by belief interaction. Results supported recent "selective processing" theories of belief bias (e.g., Evans, 2000; Klauer, Musch, & Naumer, 2000), which assume that the believability of a conclusion biases model construction processes, rather than biasing the search for falsifying models (e.g., Oakhill & Johnson-Laird, 1985) or a response stage of reasoning arising from subjective uncertainty (e.g., Quayle & Ball, 2000). We conclude by suggesting that the eye-movement analyses in reasoning research may provide a useful adjunct to other process-tracing techniques such as verbal protocol analysis.     

Heuristic and analytic processes in mental models for conditionals: An integrative developmental theory

This article presents a developmental dual-process theory of the understanding of conditionals that integrates Evans’ heuristic–analytic theory within the revised mental model theory of conditional proposed by Barrouillet, Gauffroy, and Lecas (2008). According to this theory, the interpretation of a conditional sentence is driven by unconscious and implicit heuristic processes that provide individuals with an initial representation that captures its meaning by representing the cases that make it true. This initial model can be enriched with additional models (a process named fleshing out within the mental model theory) through the intervention of conscious and demanding analytic processes. Being optional, these processes construct representations of cases that are only compatible with the conditional, leaving its truth-value indeterminate when they occur. Because heuristic processes are relatively immune to developmental changes, while analytic processes strongly develop with age, the initial model remains stable through development whereas the number of additional models that can be constructed increases steadily. Thus, the dual-process mental model theory predicts in which cases conditionals will be deemed true, indeterminate, or false and how these cases evolve with age. These predictions were verified in children, adolescents and adults who were asked to evaluate the truth value and the probability of several types of conditionals. The results reveal a variety of developmental trajectories in the way different conditionals are interpreted, which can all be accounted for by our revised mental model theory.