David E. Over

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.

Frequency illusions and other fallacies

Cosmides and Tooby (1996) increased performance using a frequency rather than probability frame on a problem known to elicit base-rate neglect. Analogously, Gigerenzer (1994) claimed that the conjunction fallacy disappears when formulated in terms of frequency rather than the more usual single-event probability. These authors conclude that a module or algorithm of mind exists that is able to compute with frequencies but not probabilities. The studies reported here found that base-rate neglect could also be reduced using a clearly stated single-event probability frame and by using a diagram that clarified the critical nested-set relations of the problem; that the frequency advantage could be eliminated in the conjunction fallacy by separating the critical statements so that their nested relation was opaque; and that the large effect of frequency framing on the two problems studied is not stable. Facilitation via frequency is a result of clarifying the probabilistic interpretation of the problem and inducing a representation in terms of instances, a form that makes the nested-set relations amongst the problem components transparent.

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.

Deduction from Uncertain Premises

We investigate how the perceived uncertainty of a conditional affects a persons choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne (1989). These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results.

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.

Reasoning, decision making and rationality

It is argued that reasoning in the real world supports decision making and is aimed at the achievement of goals. A distinction is developed between two notions of rationality: rationality which is reasoning in such a way as to achieve ones goals--within cognitive constraints--and rationality which is reasoning by a process of logic. This dichotomy is related to the philosophical distinction between practical and theoretical reasoning. It is argued that logicality (rationality) does not provide a good basis for rationality and some psychological research on deductive reasoning is re-examined in this light. First, we review belief bias effects in syllogistic reasoning, and argue that the phenomena do not support the interpretations of irrationality that are often placed upon them. Second, we review and discuss recent studies of deontic reasoning in the Wason selection task, which demonstrate the decision making, and rational nature of reasoning in realistic contexts. The final section of the paper examines contemporary decision theory and shows how it fails, in comparable manner to formal logic, to provide an adequate model for assessing the rationality of human reasoning and decision making.

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.

New paradigm psychology of reasoning

Oaksford and Chater (O&C) have written a book of great interest, which has advanced the new paradigm in the psychology of reasoning. As their title implies, they are Bayesians and see an essential connection between reasoning and probability judgement. They attack the claim that rationality is about following the rules of logic, and argue for Bayesian probability theory as the correct normative and computation theory of human reasoning. Logic is correctly used in pure mathematics, but ordinary human beings cannot escape uncertainty. Why would people simply assume that they win a fortune if they buy a lottery ticket? By making this assumption, and then buying a ticket, they can validly infer that they will win the fortune. But that inference is useless for a rational belief and decision making about the lottery and buying a ticket. From this perspective, O&C propose new theories of conditional reasoning, Wason’s selection task (both indicative and deontic), and syllogistic inference. O&C’s book has already been extensively reviewed in Behavioral and Brain Sciences, and they have extended andmodified their proposals in replies to the commentaries (Oaksford & Chater, 2009a, 2009b). I will not repeat here reviewing points that have been covered adequately in those commentaries. But I believe that fundamental aspects of O&C’s approach have not yet had sufficient attention. These are the relation between Bayesian rationality and logic, and the analysis of conditional reasoning in termsof the centralBayesian concept of conditional probability. The bestway to introduce these topics, and explain O&C’s contribution to the new paradigm, is by describing the old ‘‘logicist’’ paradigm, as O&C call it. I will argue that their use of ‘‘logicist’’ is misleading and prefer ‘‘binary paradigm’’ for the old one. The best example of a book in the binary paradigm is Johnson-Laird and Byrne (1991). In their mental model theory, every proposition is only THINKING & REASONING, 2009, 15 (4), 431–438

The probability of conditionals: The psychological evidence

The two main psychological theories of the ordinary conditional were designed to account for inferences made from assumptions, but few premises in everyday life can be simply assumed true. Useful premises usually have a probability that is less than certainty. But what is the probability of the ordinary conditional and how is it determined? We argue that people use a two stage Ramsey test that we specify to make probability judgements about indicative conditionals in natural language, and we describe experiments that support this conclusion. Our account can explain why most people give the conditional probability as the probability of the conditional, but also why some give the conjunctive probability. We discuss how our psychological work is related to the analysis of ordinary indicative conditionals in philosophical logic.

Betting on conditionals

A study is reported testing two hypotheses about a close parallel relation between indicative conditionals, if A then B, and conditional bets, I bet you that if A then B. The first is that both the indicative conditional and the conditional bet are related to the conditional probability, P(B|A). The second is that de Finettis three-valued truth table has psychological reality for both types of conditional—true, false, or void for indicative conditionals and win, lose, or void for conditional bets. The participants were presented with an array of chips in two different colours and two different shapes, and an indicative conditional or a conditional bet about a random chip. They had to make judgements in two conditions: either about the chances of making the indicative conditional true or false or about the chances of winning or losing the conditional bet. The observed distributions of responses in the two conditions were generally related to the conditional probability, supporting the first hypothesis. In addition, a majority of participants in further conditions chose the third option, “void”, when the antecedent of the conditional was false, supporting the second hypothesis.