The conjunction fallacy: a misunderstanding about conjunction?

It is easy to construct pairs of sentences X, Y that lead many people to ascribe higher probability to the conjunction X-and-Y than to the conjuncts X, Y. Whether an error is thereby committed depends on reasoners’ interpretation of the expressions “probability” and “and.” We report two experiments designed to clarify the normative status of typical responses to conjunction problems.     

The significance of unusual acts in sexual homicide

In addition to being an unusual form of crime, sexual homicide (SH) sometimes includes unusual crime scene behaviours, such as carving on the victim, evisceration (i.e., removal of internal organs), skinning the victim, cannibalism and vampirism. The current study investigates these unusual crime scene behaviours to better understand their meaning as well as to explore whether such behaviours are associated with a specific crime-commission process. Using a sample of 762 SH cases, the study uses a combination of chi-square and stepwise forward logistic regression analyses to identify the differences between cases with and without unusual acts. Findings reveal that SH cases with unusual acts present a specific crime-commission process. This crime-commission process is characterised by a greater level of sadism—as demonstrated by the presence of mutilation and foreign object insertion—as well as a greater level of organisation from the selection of a contact location where the risk of being seen or heard was low. These findings provide a better understanding of these behaviours and could help investigators facing such cases.     

Instrumental and expressive violence in Belgian homicide perpetrators

The present study aimed to examine offender types in a sample of Belgian single‐perpetrator/single‐victim homicide cases. First, it was investigated if the distinction between instrumental and expressive aggression could be documented in crime scene and offender background characteristics. Second, the instrumental and expressive themes from the first analysis were examined in relation to the motives the perpetrators themselves provided for their offence. A sample of 97 solved homicide cases was analysed using the non‐metric multidimensional scaling procedure Proxscal. The results revealed that 62% of the homicide crime scenes and 67% of the offender backgrounds could be classified as either expressive or instrumental. The self‐reported motives did not correspond with the themes according to the Proxscal analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Source reliability and the conjunction fallacy

Information generally comes from less than fully reliable sources. Rationality, it seems, requires that one take source reliability into account when reasoning on the basis of such information. Recently, Bovens and Hartmann (2003) proposed an account of the conjunction fallacy based on this idea. They show that, when statements in conjunction fallacy scenarios are perceived as coming from such sources, probability theory prescribes that the "fallacy" be committed in certain situations. Here, the empirical validity of their model was assessed. The model predicts that statements added to standard conjunction problems will change the incidence of the fallacy. It also predicts that statements from reliable sources should yield an increase in fallacy rates (relative to unreliable sources). Neither the former (Experiment 1) nor the latter prediction (Experiment 3) was confirmed, although Experiment 2 showed that people can derive source reliability estimates from the likelihood of statements in a manner consistent with the tested model. In line with the experimental results, model fits and sensitivity analyses also provided very little evidence in favor of the model. This suggests that Bovens and Hartmann's present model fails to explain fully people's judgements in standard conjunction fallacy tasks.     

Serial homicide: an investigation of behavioural consistency

To examine if serial homicide offenders are consistent across their crimes, a model was developed empirically that could be used to distinguish between crimes that were instrumental and those that were expressive. The first known three offences in each series of 69 US serial homicides committed by 23 offenders, were examined and the instrumental and expressive themes determined. Three models were then explored that test for consistency across these themes. The most liberal model was found to classify all of the offences effectively and to reveal complete consistency across the three crimes for all offenders. The implications of these results for offender profiling and further study of serial homicide are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

How probability theory explains the conjunction fallacy

The conjunction fallacy occurs when people judge a conjunctive statement B‐and‐A to be more probable than a constituent B, in contrast to the law of probability that P(B ∧ A) cannot exceed P(B) or P(A). Researchers see this fallacy as demonstrating that people do not follow probability theory when judging conjunctive probability. This paper shows that the conjunction fallacy can be explained by the standard probability theory equation for conjunction if we assume random variation in the constituent probabilities used in that equation. The mathematical structure of this equation is such that random variation will be most likely to produce the fallacy when one constituent has high probability and the other low, when there is positive conditional support between the constituents, when there are two rather than three constituents, and when people rank probabilities rather than give numerical estimates. The conjunction fallacy has been found to occur most frequently in exactly these situations. Copyright © 2008 John Wiley & Sons, Ltd.     

The consistency of inconsistency in serial homicide: patterns of behavioural change across series

Linkage analysis is a crucial part of the investigative process when faced with a possible series of related offences. Establishing behavioural consistency (i.e., offender's behaviours consistently present across the series) is at the core of linkage. Recent empirical studies have found little evidence of consistency looking at either individual or groups of behaviours in serial homicide. It is argued that behavioural changes are rooted in the changing cognitive strategies that offenders use to reach their ultimate goal (i.e. the commission of multiple homicides). Factors that could account for these changes include learning, situational factors, loss of control, and changes in the offender's fantasy. Patterns of behavioural change have been identified in serial crimes, such as rape. However, no empirical studies have looked at patterns of behavioural change in serial homicide. The present study examined patterns of consistency and change using a combination of thematic and behavioural subgroup approaches that use Multidimensional Scaling. Thematic differentiation indicative of behavioural manifestations of cognitive strategies was found in all three examined subgroups: planning, wounding, and offender–victim interaction, and patterns of change within these subgroups provided support for the above theories. Looking at behavioural patterns rather than individual behaviours, may be a more fruitful way of examining consistency in serial homicide, and could have significant implications for linkage analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Exploring the conjunction fallacy within a category learning framework

The literature presents two major theories on the cause of the conjunction fallacy. The first attributes the conjunction fallacy to the representativeness heuristic. The second suggests that the conjunction fallacy is caused by people combining p(A) and p(B) into p(A&B) in an inappropriate manner. These two theories were contrasted in two category‐learning experiments. As predicted by the latter theory, data showed that participants that could assess p(A&B) directly made fewer conjunction fallacies than participants who had to compute p(A) and p(B) separately and then combine them into p(A&B). Least conjunction fallacies were observed in the cases where the representativeness heuristic was applicable. Overall, data showed that an inability to appropriately combine probabilities is one of the key cognitive mechanisms behind the conjunction fallacy. Copyright © 2008 John Wiley & Sons, Ltd.     

The conjunction fallacy and interference effects

In the present article we consider the conjunction fallacy, a well known cognitive fallacy experimentally tested in cognitive science, which occurs for intuitive judgments. We show that the quantum formalism can be used to describe, in a very simple way, this fallacy, in terms of interference effect. We provide evidence that the quantum formalism leads quite naturally to violations of the joint probability rule of classic probability when considering the estimated probability of the conjunction of two events. By defining the concept of maximal conjunction error, we find a good agreement with experimental results. Thus we suggest that in cognitive science the formalism of quantum mechanics can be used to describe a quantum regime, where the cognitive fallacies and heuristics are valid.     

The degree of epistemic justification and the conjunction fallacy

This paper describes a formal measure of epistemic justification motivated by the dual goal of cognition, which is to increase true beliefs and reduce false beliefs. From this perspective the degree of epistemic justification should not be the conditional probability of the proposition given the evidence, as it is commonly thought. It should be determined instead by the combination of the conditional probability and the prior probability. This is also true of the degree of incremental confirmation, and I argue that any measure of epistemic justification is also a measure of incremental confirmation. However, the degree of epistemic justification must meet an additional condition, and all known measures of incremental confirmation fail to meet it. I describe this additional condition as well as a measure that meets it. The paper then applies the measure to the conjunction fallacy and proposes an explanation of the fallacy.     

The conjunction fallacy and the many meanings of and

According to the conjunction rule, the probability of A and B cannot exceed the probability of either single event. This rule reads and in terms of the logical operator wedge, interpreting A and B as an intersection of two events. As linguists have long argued, in natural language "and" can convey a wide range of relationships between conjuncts such as temporal order ("I went to the store and bought some whisky"), causal relationships ("Smile and the world smiles with you"), and can indicate a collection of sets rather than their intersection (as in "He invited friends and colleagues to the party"). When "and" is used in word problems researching the conjunction fallacy, the conjunction rule, which assumes the logical operator wedge, therefore cannot be mechanically invoked as a norm. Across several studies, we used different methods of probing people's understanding of and-conjunctions, and found evidence that many of those respondents who violated the conjunction rule in their probability or frequency judgments inferred a meaning of and that differs from the logical operator wedge. We argue that these findings have implications for whether judgments involving ambiguous and-conjunctions that violate the conjunction rule should be considered manifestations of fallacious reasoning or of reasonable pragmatic and semantic inferences.     

The conjunction fallacy: A test of averaging hypotheses

The conjunction fallacy, in which individuals report that the conjunction of two events is more rather than less likely to occur than one of the events alone, is a robust phenomenon. We assessed the possibility that an analysis in terms of functional measurement methodology might be consistent with occurrence of the fallacy. A 3 × 3 design in which we varied the judged likelihood of the two components constituting the conjunction permitted us to assess the possibility that subjects judge the likelihood of conjunctions by averaging the likelihood of their component parts. The results were consistent with this possibility, and this interpretation was supported by analysis of the results in terms of functional measurement methodology.     

On the nature of the conjunction fallacy

In a seminal work, Tversky and Kahneman showed that in some contexts people tend to believe that a conjunction of events (e.g., Linda is a bank teller and is active in the feminist movement) is more likely to occur than one of the conjuncts (e.g., Linda is a bank teller). This belief violates the conjunction rule in probability theory. Tversky and Kahneman called this phenomenon the “conjunction fallacy”. Since the discovery of the phenomenon in 1983, researchers in psychology and philosophy have engaged in important controversies around the conjunction fallacy. The goal of this paper is to explore the most important of these controversies, namely, the controversy about the nature of the conjunction fallacy. Is the conjunction fallacy mainly due to a misunderstanding of the problem by participants (misunderstanding hypothesis) or is it mainly due to a genuine reasoning bias (reasoning bias hypothesis)? A substantial portion of research on the topic has been directed to test the misunderstanding hypothesis. I review this literature and argue that a stronger case can be made against the misunderstanding hypothesis. Thus, I indirectly provide support for the reasoning bias hypothesis.     

On the reality of the conjunction fallacy

Attributing higher "probability" to a sentence of form p-and-q, relative to p, is a reasoning fallacy only if (1) the word probability carries its modern, technical meaning and (2) the sentence p is interpreted as a conjunct of the conjunction p-and-q. Legitimate doubts arise about both conditions in classic demonstrations of the conjunction fallacy. We used betting paradigms and unambiguously conjunctive statements to reduce these sources of ambiguity about conjunctive reasoning. Despite the precautions, conjunction fallacies were as frequent under betting instructions as under standard probability instructions.     

The dependence of the conjunction fallacy on subtle linguistic factors

Summary The three experiments reported in this article are concerned with moderating conditions of the so-called conjunction fallacy. Although the conjunction of two events (A&B) is necessarily less probable than one event alone, intelligent people's judgments sometimes violate this logical principle when it is easier to think about or imagine the conjunction A&B than the component event A. It was hypothesized that the fallacy might be reduced either by a priming treatment (directing the participants' attention to the logical rule prior to the conjunctions tasks) or by a linguistic manipulation (replacing the ambiguous term probability by the term frequency. Experiment 1 shows that the bias is drastically reduced by the subtle linguistic manipulation and suggests that the fallacy is partly due to a common misunderstanding of the probability concept. The lack of a priming effect seems to imply that cueing or activation of logical schemata is not a critical factor. Experiment 2 replicates the linguistic influence under slightly modified task conditions, and Experiment 3 corroborates the ineffectiveness of the priming factor using a stronger priming treatment.     

Lax monitoring versus logical intuition: The determinants of confidence in conjunction fallacy

The general assumption that people fail to notice discrepancy between their answer and the normative answer in the conjunction fallacy task has been challenged by the theory of Logical Intuition. This theory suggests that people can detect the conflict between the heuristic and normative answers even if they do not always manage to inhibit their intuitive choice. This theory gained support from the finding that people report lower levels of confidence in their choice after they commit the conjunction fallacy compared to when their answer is not in conflict with logic. In four experiments we asked the participants to give probability estimations to the options of the conflict and no-conflict versions of the tasks in the original set-up of the experiment or in a three-option design. We found that participants perceive probabilities for the options of the conflict version less similar than for the no-conflict version. As people are less confident when choosing between more similar options, this similarity difference is proposed to serve as a mediator in the task in a way that the conflict and no-conflict conditions have their effects on confidence ratings through manipulating the similarity of the answer options.     

A unified account of the conjunction fallacy by coherence

We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms (the M–A paradigm and the A–B paradigm). We compare our account with a recent proposal by Tentori et al. (J Exp Psychol Gen 142(1): 235–255, 2013) that attempts to generalize earlier confirmation accounts. Their model works better than its predecessors in some respects, but it exhibits only a shallow form of generality and is unsatisfactory in other ways as well: it is strained, complex, and untestable as it stands. Our coherence account inherits the strength of the confirmation account, but in addition to being applicable to both paradigms, it is natural, simple, and readily testable. It thus constitutes the next natural step for Bayesian theorizing about the conjunction fallacy.

     

Testing boundary conditions for the conjunction fallacy: effects of response mode, conceptual focus, and problem type

Two experiments used within-subject designs to examine how conjunction errors depend on the use of (1) choice versus estimation tasks, (2) probability versus frequency language, and (3) conjunctions of two likely events versus conjunctions of likely and unlikely events. All problems included a three-option format verified to minimize misinterpretation of the base event. In both experiments, conjunction errors were reduced when likely events were conjoined. Conjunction errors were also reduced for estimations compared with choices, with this reduction greater for likely conjuncts, an interaction effect. Shifting conceptual focus from probabilities to frequencies did not affect conjunction error rates. Analyses of numerical estimates for a subset of the problems provided support for the use of three general models by participants for generating estimates. Strikingly, the order in which the two tasks were carried out did not affect the pattern of results, supporting the idea that the mode of responding strongly determines the mode of thinking about conjunctions and hence the occurrence of the conjunction fallacy. These findings were evaluated in terms of implications for rationality of human judgment and reasoning.     

Suppressing natural heuristics by formal instruction: The case of the conjunction fallacy

A basic principle of probability is the conjunction rule, p(B) p(A&B). People violate this rule often, particularly when judgments of probability are based on intensional heuristics such as representativeness and availability. Though other probabilistic rules are obeyed with increasing frequency as people's levels of mathematical talent and training increase, the conjunction rule generally does not show such a correlation. We argue that this recalcitrance is not due to inescapable “natural assessments”; rather, it stems from the absence of generally useful problem-solving designs that bring extensional principles to bear on this class of problem. We predict that when helpful extensional strategies are made available, they should compete well with intensional heuristics. Two experiments were conducted, using as subjects adult women with little mathematical background. In Experiment 1, brief training on concepts of algebra of sets, with examples of their use in solving problems, reduced conjunction-rule violations substantially, compared with a control group. Evidence from similarity judgments suggested that use of the representativeness heuristic was reduced by the training. Experiment 2 confirmed these training effects and also tested the hypothesis that conjunction-rule violations are due to misunderstanding of “B” as “B and not A.” Changes in detailed wording of the propositions to be ranked produced substantial effects on judgment, but the pattern of these effects supported the hypothesis that, for the type of problem used here, most conjunction errors are due to use of representativeness or availability. We conclude that such intensional heuristics can be suppressed when alternative strategies are taught.