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.     

Paranormal belief and the conjunction fallacy: Controlling for temporal relatedness and potential surprise differentials in component events

Recent research suggests paranormal believers are especially prone to the ‘conjunction fallacy’. The current study extends this work by presenting believers and non‐believers with eight paranormal plus eight non‐paranormal scenarios. Participants were given either a paranormal or virtually identical non‐paranormal version of each scenario. Of these, half incorporated component events which were (virtually) co‐occurring with half including components which were temporally disjointed. Analysis of Covariance (ANCOVA; controlling for gender and maths/stats/psychology qualifications) found believers made more conjunction errors than non‐believers. Neither event type (paranormal vs. non‐paranormal) nor components' temporal relationship (co‐occurring vs. disjointed) had a significant effect on conjunction biases. Believers' tendency to produce larger conjunctive estimates was unrelated to group differences in component probability estimates (surprise values) and further, could not be attributed to group differences in the perceived functional relationship between component and conjunctive events. Possible explanations are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

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.

     

A conjunction in closure spaces

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of ω-conjunctive closure spaces (X is ω-conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:

1. For every closed and proper subset of an ω-conjunctive closure space its interior is empty (i.e. it is a boundary set).
2. If X is an ω-conjunctive closure space which satisfies the ω-compactness theorem and \(\hat P\) [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice.
3. A closure space is linear iff it is an ω-conjunctive and topological space.
4. Every continuous function preserves all conjunctions.

     

Inefficient conjunction search made efficient by concurrent spoken delivery of target identity

Visual search based on a conjunction of two features typically elicits reaction times that increase linearly as a function of the number of distractors, whereas search based on a single feature is essentially unaffected by set size. These and related findings have often been interpreted as evidence of a serial search stage that follows a parallel search stage. However, a wide range of studies has been showing a form of blending of these two processes. For example, when a spoken instruction identifies the conjunction target concurrently with the visual display, the effect of set size is significantly reduced, suggesting that incremental linguistic processing of the first feature adjective and then the second feature adjective may facilitate something approximating a parallel extraction of objects during search for the target. Here, we extend these results to a variety of experimental designs. First, we replicate the result with a mixed-trials design (ruling out potential strategies associated with the blocked design of the original study). Second, in a mixed-trials experiment, the order of adjective types in the spoken query varies randomly across conditions. In a third experiment, we extend the effect to a triple-conjunction search task. A fourth (control) experiment demonstrates that these effects are not due to an efficient odd-one-out search that ignores the linguistic input. This series of experiments, along with attractor-network simulations of the phenomena, provide further evidence toward understanding linguistically mediated influences in real-time visual search processing.     

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.     

Mathematical instrumentalism meets the conjunction objection

Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instrumentalism escapes the conjunction objection unscathed.     

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.     

Quantum probabilities and the conjunction principle

A recent argument by Hawthorne and Lasonen-Aarnio purports to show that we can uphold the principle that competently forming conjunctions is a knowledge-preserving operation only at the cost of a rampant skepticism about the future. A key premise of their argument is that, in light of quantum-mechanical considerations, future contingents never quite have chance 1 of being true. We argue, by drawing attention to the order of magnitude of the relevant quantum probabilities, that the skeptical threat of Hawthorne and Lasonen-Aarnio’s argument is illusory.     

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.     

Disjunction and conjunction fallacies in episodic memory

It has recently been found that episodic memory displays analogues of the well-known disjunction and conjunction fallacies of probability judgement. The aim of the present research was, for the first time, to study these memory fallacies together under the same conditions, and test theoretical predictions about the reasons for each. The focus was on predictions about the influence of semantic gist, target versus context recollection, and proactive versus retroactive interference. Disjunction and conjunction fallacies increased in conditions in which subjects were able to form semantic connections among list words. In addition, disjunction fallacies were increased by manipulations that minimised proactive interference, whereas conjunction fallacies were increased by manipulations that minimised retroactive interference. That pattern suggests that disjunction fallacies are more dependent on target recollection, whereas conjunction fallacies are more dependent on context recollection.     

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.     

Algebraic logic for classical conjunction and disjunction

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties.Dedicated to our master, Francesc d'A. Sales, on his 75th birthday     

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.     

Perceptual learning in visual conjunction search.

Perceptual learning in colour/orientation visual conjunction search was examined in five experiments. Good transfer occurred to other conjunction arrays when only one element of the conjunction (either colour or orientation) was changed. When both elements (colour and orientation) were changed, but the same feature spaces were used (i.e. other colours and orientations) or when a new dimension was introduced to the transfer task (shapes instead of orientation), transfer was poor. The results suggest that perceptual learning of visual conjunction search is constrained mainly by stimulus parameters rather than by changes in cognitive strategies which are common to all search tasks. Contrary to other reports we found little evidence of long-term retention of learning.     

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.