Simon J. Handley

Conditionals and conditional probability.

The authors report 3 experiments in which participants were invited to judge the probability of statements of the form if p then q given frequency information about the cases pq, p not q, not pq, and not p not q (where not = not). Three hypotheses were compared: (a) that people equate the probability with that of the material conditional, 1 - P(p not q); (b) that people assign the conditional probability, P(q/p); and (c) that people assign the conjunctive probability P(pq). The experimental evidence allowed rejection of the 1st hypothesis but provided some support for the 2nd and 3rd hypotheses. Individual difference analyses showed that half of the participants used conditional probability and that most of the remaining participants used conjunctive probability as the basis of their judgments.

The probability of causal conditionals.

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.

Frequency versus probability formats in statistical word problems

Three experiments examined peoples ability to incorporate base rate information when judging posterior probabilities. Specifically, we tested the (Cosmides, L., & Tooby, J. (1996). Are humans good intuitive statisticians after all? Rethinking some conclusions from the literature on judgement under uncertainty. Cognition, 58, 1-73) conclusion that peoples reasoning appears to follow Bayesian principles when they are presented with information in a frequency format, but not when information is presented as one case probabilities. First, we found that frequency formats were not generally associated with better performance than probability formats unless they were presented in a manner which facilitated construction of a set inclusion mental model. Second, we demonstrated that the use of frequency information may promote biases in the weighting of information. When participants are asked to express their judgements in frequency rather than probability format, they were more likely to produce the base rate as their answer, ignoring diagnostic evidence.

: Working memory, inhibitory control and the development of children's reasoning

The ability to reason independently from ones own goals or beliefs has long been recognised as a key characteristic of the development of formal operational thought. In this article we present the results of a study that examined the correlates of this ability in a group of 10-year-old children (N = 61). Participants were presented with conditional and relational reasoning items, where the content was manipulated such that the conclusion to the arguments were either congruent, neutral, or incongruent with beliefs, and either logically valid or logically invalid. Participants also received a measure of working memory capacity (the counting span task) and a measure of inhibitory control (the stop signal task). Indices of belief bias and logical reasoning on belief-based problems were predicted independently by both measures. In contrast logical reasoning on belief neutral problems was predicted by working memory alone. The findings suggest that executive functions play a key role in the development of childrens ability to decontextualise their thinking.

Individual differences in deductive reasoning

Three studies are reported, which examined individual differences in deductive reasoning as a function of intellectual ability and thinking style. Intellectual ability was a good predictor of logical performance on syllogisms, especially where there was a conflict between logic and believability. However, in the first two experiments there was no link between ability and performance on indicative selection tasks, in sharp contrast to previous research. This correlation did, however, return in the final study. Our data are consistent with the claim that the correlation with logical accuracy on abstract selection tasks is found primarily with participants of relatively high ability. At lower levels, pragmatically cued responses are given but those of slightly higher ability divorce the rule from the scenario and respond consistently (though incorrectly) across problems. Self-report questionnaires were generally poor predictors of performance, but a measure of the ability to generate alternative representations proved an excellent predictor. These results are consistent with a mental models approach to reasoning and also have implications for the debate about human rationality.

Reasoning about necessity and possibility : a test of the mental model theory of deduction

This article examined syllogistic reasoning that differs from previous research in 2 significant ways: (a) Participants were asked to decide whether conclusions were possible as well as necessary, and (b) every possible combination of syllogistic premises and conclusions was presented for evaluation with both single-premise (Experiment 1) and double-premise (Experiment 2) problems. Participants more frequently endorsed conclusions as possible than as necessary, and differences in response to the 2 forms of instruction conformed to several predictions derived from the mental model theory of deduction. Findings of Experiments 2 and 3 showed that some fallacies are consistently endorsed and others consistently resisted when people are asked to judge whether conclusions that are only possible follow necessarily. This finding was accounted for by the computational implementation of the model theory: Fallacies are made when the first mental model of the premises considered supports the conclusion presented. Traditional applications of logic concern the validity of arguments, that is, proving that some conclusion is necessary given some premises. However, in everyday reasoning it may be just as important to decide whether some proposition is possible in light of the given information. Inferences of possibility occur whenever rules and regulations constrain a persons behavior rather than determine what it must be. For example, students choosing a degree program within a modular course structure will have many degrees of freedom but will have to respect constraints owing to timetabling restrictions, availability of teaching staff, prerequisite and corequisite relations between modules, and so on. Their decision making here involves inferring what is possible and then deciding between the possibilities identified. Most of the rules by which people live their lives in society, including criminal law, operate in a similar way. It is rarely determined by such rules that people must follow a particular course of action, but they are frequently constrained to act within a set of legal possibilities. The psychological study of deductive reasoning has been steadily increasing in recent years, and there are now many hundreds of experimental studies reported in the literature (see Evans, Newstead, & Byrne, 1993, for a review). In view of the above remarks, however, there is a curious limitation

Thinking about conditionals: A study of individual differences

Recent studies have shown the existence of two qualitatively distinct groups of people based on how they judge the probability of a conditional statement. The present study was designed to test whether these differences are rooted in distinctive means of processing conditional statements and whether they are linked to differences in general intelligence. In the study, each of 120 participants completed three separate cognitive tasks involving the processing of abstract conditional statements—the probability-of-conditionals task, the conditional truth table task, and the conditional inference task—in addition to completing a test of general intelligence (AH4). The results showed a number of predicted effects: People responding with conditional (rather than conjunctive) probabilities on the first task were higher in cognitive ability, showed reasoning patterns more consistent with a suppositional treatment of the conditional, and showed a strongly “defective” truth table pattern. The results include several novel findings and post challenges to contemporary psychological theories of conditionals.

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.

Conditional reasoning and the Tower of Hanoi: The role of spatial and verbal working memory

This paper reports a study that investigated the relationships between verbal and spatial measures of working memory capacity and performance on the Tower of Hanoi and a conditional reasoning task. Four working memory measures were included, a simple word span, a simple spatial span, a complex verbal span and a complex spatial span. The participants were 70 undergraduate students. Solution time on the Tower of Hanoi task correlated highly with both measures of spatial memory capacity, but not with the verbal working memory spans. In contrast, logical performance on the conditional reasoning task correlated highly with the complex verbal working memory span, but not with the spatial spans. A confirmatory factor analysis of the data elicited a best-fit model comprising a correlated spatial and verbal factor. The results are interpreted in the context of recent work that suggests a dissociation between spatial and verbal resources at the level of the central executive.

The influence of cognitive ability and instructional set on causal conditional inference.

We report a large study in which participants are invited to draw inferences from causal conditional sentences with varying degrees of believability. General intelligence was measured, and participants were split into groups of high and low ability. Under strict deductive-reasoning instructions, it was observed that higher ability participants were significantly less influenced by prior belief than were those of lower ability. This effect disappeared, however, when pragmatic reasoning instructions were employed in a separate group. These findings are in accord with dual-process theories of reasoning. We also took detailed measures of beliefs in the conditional sentences used for the reasoning tasks. Statistical modelling showed that it is not belief in the conditional statement per se that is the causal factor, but rather correlates of it. Two different models of belief-based reasoning were found to fit the data according to the kind of instructions and the type of inference under consideration.