The conjunction fallacy is the well-documented empirical finding that subjects sometimes rate a conjunction A&B as more probable than one of its conjuncts, A. Most explanations appeal in some way to the fact that B has a high probability. But Tentori et al. (2013) have recently challenged such approaches, reporting experiments which find that (1) when B is confirmed by relevant evidence despite having low probability, the fallacy is common, and (2) when B has a high (...) probability but has not been confirmed by relevant evidence, the fallacy is less common. They conclude that degree of confirmation, rather than probability, is the central determinant of the conjunction fallacy. In this paper, we address a confound in these experiments: Tentori et al. (2013) failed to control for the fact that their (1)-situations make B conversationally relevant, while their (2)-situations do not. Hence their results are consistent with the hypothesis that con- versationally relevant high probability is an important driver of the conjunction fallacy. Inspired by recent theoretical work that appeals to conversational relevance to explain the conjunction fallacy, we report on two experiments that control for this issue by making B relevant without changing its degree of probability or confirmation. We find that doing so increases the rate of the fallacy in (2)-situations, and leads to comparable fallacy-rates as (1)-situations. This suggests that (non-probabilistic) conversational relevance indeed plays a role in the conjunction fallacy, and paves the way toward further work on the interplay between relevance and confirmation. (shrink)

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. © 2004 Cognitive Science Society, Inc. All rights reserved.

The conjunction fallacy has been a key topic in debates on the rationality of human reasoning and its limitations. Despite extensive inquiry, however, the attempt to provide a satisfactory account of the phenomenon has proved challenging. Here we elaborate the suggestion (first discussed by Sides, Osherson, Bonini, & Viale, 2002) that in standard conjunction problems the fallacious probability judgements observed experimentally are typically guided by sound assessments of _confirmation_ relations, meant in terms of contemporary Bayesian confirmation theory. (...) Our main formal result is a confirmation-theoretic account of the conjunction fallacy, which is proven _robust_ (i.e., not depending on various alternative ways of measuring degrees of confirmation). The proposed analysis is shown distinct from contentions that the conjunction effect is in fact not a fallacy, and is compared with major competing explanations of the phenomenon, including earlier references to a confirmation-theoretic account. (shrink)

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.

If people believe that some property is true of all members of a class such as sofas, then they should also believe that the same property is true of all members of a conjunctively defined subset of that class such as uncomfortable handmade sofas. A series of experiments demonstrated a failure to observe this constraint, leading to what is termed the inverse conjunction fallacy. Not only did people often express a belief in the more general statement but not (...) in the more specific, but also when they accepted both beliefs, they were inclined to give greater confidence to the more general. It is argued that this effect underlies a number of other demonstrations of fallacious reasoning, particularly in category-based induction. Alternative accounts of the phenomenon are evaluated, and it is concluded that the effect is best interpreted in terms of intensional reasoning [Tversky, A., & Kahneman, D.. Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment. Psychological Review, 90, 293-315.]. (shrink)

Crupi et al. (2008) offer a confirmation-theoretic, Bayesian account of the conjunction fallacy—an error in reasoning that occurs when subjects judge that Pr( h 1 & h 2 | e ) > Pr( h 1 | e ). They introduce three formal conditions that are satisfied by classical conjunction fallacy cases, and they show that these same conditions imply that h 1 & h 2 is confirmed by e to a greater extent than is h 1 (...) alone. Consequently, they suggest that people are tracking this confirmation relation when they commit conjunction fallacies. I offer three experiments testing the merits of Crupi et al.’s account specifically and confirmation-theoretic accounts of the conjunction fallacy more generally. The results of Experiment 1 show that, although Crupi et al.’s conditions do seem to be causally linked to the conjunction fallacy, they are not necessary for it; there exist cases that do not meet their three conditions in which subjects still tend to commit the fallacy. The results of Experiments 2 and 3 show that Crupi et al.’s conditions, and those offered by other confirmation-theoretic accounts of the fallacy, are not sufficient for the fallacy either; there exist cases that meet all three of CFT’s conditions in which subjects do not tend to commit the fallacy. Additionally, these latter experiments show that such confirmation-theoretic conditions are at best only weakly causally relevant to the presence of the conjunction fallacy. Given these findings, CFT’s account specifically, and any general confirmation-theoretic account more broadly, falls short of offering a satisfying explanation of the presence of the conjunction fallacy. (shrink)

Crupi et al. (Think Reason 14:182–199, 2008) have recently advocated and partially worked out an account of the conjunction fallacy phenomenon based on the Bayesian notion of confirmation. In response, Schupbach (2009) presented a critical discussion as following from some novel experimental results. After providing a brief restatement and clarification of the meaning and scope of our original proposal, we will outline Schupbach’s results and discuss his interpretation thereof arguing that they do not actually undermine our point of (...) view if properly construed. Finally, we will foster such a claim by means of some novel data. (shrink)

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. (shrink)

The conjunction fallacy has been a key topic in debates on the rationality of human reasoning and its limitations. Despite extensive inquiry, however, the attempt of providing a satisfactory account of the phenomenon has proven challenging. Here, we elaborate the suggestion (first discussed by Sides et al., 2001) that in standard conjunction problems the fallacious probability judgments experimentally observed are typically guided by sound assessments of confirmation relations, meant in terms of contemporary Bayesian confirmation theory. Our main (...) formal result is a confirmation-theoretic account of the conjuntion fallacy which is proven robust (i.e., not depending on various alternative ways of measuring degrees of confirmation). The proposed analysis is shown distinct from contentions that the conjunction effect is in fact not a fallacy and is compared with major competing explanations of the phenomenon, including earlier references to a confirmation-theoretic account. (shrink)

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.

Because the conjunction pandq implies p, the value of a bet on pandq cannot exceed the value of a bet on p at the same stakes. We tested recognition of this principle in a betting paradigm that (a) discouraged misreading p as pandnotq, and (b) encouraged genuinely conjunctive reading of pandq. Frequent violations were nonetheless observed. The findings appear to discredit the idea that most people spontaneously integrate the logic of conjunction into their assessments of chance.

In a famous experiment by Tversky and Kahneman (Psychol Rev 90:293–315, 1983), featuring Linda the bank teller, the participants assign a higher probability to a conjunction of propositions than to one of the conjuncts, thereby seemingly committing a probabilistic fallacy. In this paper, we discuss a slightly different example featuring someone named Walter, who also happens to work at a bank, and argue that, in this example, it is rational to assign a higher probability to the conjunction (...) of suitably chosen propositions than to one of the conjuncts. By pointing out the similarities between Tversky and Kahneman’s experiment and our example, we argue that the participants in the experiment may assign probabilities to the propositions in question in such a way that it is also rational for them to give the conjunction a higher probability than one of the conjuncts. (shrink)

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. (shrink)

Brock and Glasgow recently introduced a new definition of paradox and argue that this conception of paradox itself leads to paradox, the so-called Paradox Paradox. I show that they beg the questions during the course of their argument, but, more importantly, do so in a philosophically interesting way: it reveals a counterexample to the equivalence between being a logical truth and having a probability of one. This has consequences regarding norms of rationality, undermining the grounds for the Conjunction (...) class='Hi'>Fallacy. (shrink)

Intuitive predictions and judgements under uncertainty are often mediated by judgemental heuristics that sometimes lead to biases. Our micro-developmental study suggests that a presumption of rationality is justified for adult subjects, in so far as their systematic judgemental biases appear to be due to a specific executive-inhibition failure in working memory, and not necessarily to a lack of understanding of the fundamental principles of probability. This hypothesis was tested using an experimental procedure in which 60 adult subjects were trained to (...) inhibit the classical conjunction bias on a frequency judgement task derived from Tversky and Kahneman's work. Pre- and post-test performance was assessed via a probability judgement task. The data indicated a training effect, suggesting that subjects traditionally labelled as "irrational" with respect to the classical rules of inductive reasoning are in fact "inefficient inhibitors". These findings are discussed in terms of a polymorphous view of rationality. (shrink)

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. (shrink)

ABSTRACTThe 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 (...) class='Hi'>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 simil.. (shrink)

Human agents happen to judge that a conjunction of two terms is more probable than one of the terms, in contradiction with the rules of classical probabilities—this is the conjunction fallacy. One of the most discussed accounts of this fallacy is currently the quantum-like explanation, which relies on models exploiting the mathematics of quantum mechanics. The aim of this paper is to investigate the empirical adequacy of major quantum-like models which represent beliefs with quantum states. We (...) first argue that they can be tested in three different ways, in a question order effect configuration which is different from the traditional conjunction fallacy experiment. We then carry out our proposed experiment, with varied methodologies from experimental economics. The experimental results we get are at odds with the predictions of the quantum-like models. This strongly suggests that this quantum-like account of the conjunction fallacy fails. Future possible research paths are discussed. (shrink)

This paper raises a principled objection against the idea that Bayesian confirmation theory can be used to explain the conjunction fallacy. The paper demonstrates that confirmation-based explanations are limited in scope and can only be applied to cases of the fallacy of a certain restricted kind. In particular; confirmation-based explanations cannot account for the inverse conjunction fallacy, a more recently discovered form of the conjunction fallacy. Once the problem has been set out, the (...) paper explores four different ways for the confirmation theorist to come to terms with the problem, and argues that none of them are successful. (shrink)

This paper takes issue with a recent proposal due to Shogenji (Synthese 184:29–48, 2012). In his paper, Shogenji introduces J, a normatively motivated formal measure of justification (and of confirmation), and then proceeds to recruit it descriptively in an explanation of the conjunction fallacy. We argue that this explanation is undermined by the fact that it cannot be extended in any natural way to the inverse conjunction fallacy, a more recently discovered, closely related fallacy. We (...) point out that since the inverse conjunction fallacy occurs in the absence of any form of explicit evidence, formal measures of confirmation such as J, are hard pressed to apply to it at all. We then proceed to argue that this problem with Shogenji’s measure is actually quite difficult to come to terms with; even if the situations where people commit the inverse conjunction fallacy are understood in such a way that the measure is applicable, it doesn’t seem possible to reasonably interpret these situations in such a way that the measure can explain the found response pattern. (shrink)

We here report the findings of our investigation into the validity of the conjunction fallacy (Tversky & Kahneman, 1983), bearing in mind the role of conversational rules. Our first experiment showed that subjects found a logically correct answer unacceptable when it implied a violation of the conversational rules. We argue that tautological questions, such as those which concern the relationship of inclusion between a class and its sub-class, violate conversational rules because they are not informative. In this sense, (...) it is not understood that the question in a Linda-type problem involves a comparison between an inclusive and included class, but presumed that a different type of comparison is intended. Tautological questions (and, consequently, also their answers) do not become a matter of discussion except under certain specific conditions. Our second experiment showed that, providing the context was adequately marked (such as in the case of a rhetorical question), the conjunction fallacy disappears. In two further experiments, the implications of our view were compared with those of the other critical approaches to the heuristic programme: the classical pragmatic view (which we call logical complementarity) and the frequentist view. (shrink)

Can some evidence confirm a conjunction of two hypotheses more than it confirms either of the hypotheses separately? We show that it can, moreover under conditions that are the same for nine different measures of confirmation. Further we demonstrate that it is even possible for the conjunction of two disconfirmed hypotheses to be confirmed by the same evidence.

We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms. We compare our account with a recent proposal by Tentori et al. : 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. (shrink)

We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms. We compare our account with a recent proposal by Tentori et al. : 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. (shrink)

In the first edition of LFP, Carnap [2] undertakes a precise probabilistic explication of the concept of confirmation. This is where modern confirmation theory was born (in sin). Carnap was interested mainly in quantitative confirmation (which he took to be fundamental). But, he also gave (derivative) qualitative and comparative explications: • Qualitative. E inductively supports H. • Comparative. E supports H more strongly than E supports H . • Quantitative. E inductively supports H to degree r . Carnap begins by (...) clarifying the explicandum (the informal “inductive support” concept) in various ways, including. (shrink)

My purpose in this paper is to examine a pitfall in empiricistic analysis which has not been widely discussed, perhaps because it lies implicit in what may seem a harmless facet of such analysis. The kind of analysis I have in mind is analysis of any variety which seeks to reduce understanding of any object, concept event, institution, or whatever, and its appropriate properties, to understanding of discrete elements and their properties out of which the analytically reduced can be constructed. (...) Let us call the entity analysed into elements the molar object, and restrict elements hereafter to some set of discrete elements of some molar object which have been obtained by an analysis of the kind to be discussed. What I will have to say will apply most directly to the reductionist analyses of an older logical empiricism, but I hope to make it clear that more recent linguistic analyses encounter the same difficulty. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy called the conjunction fallacy. It discusses a case of Linda who was deeply concerned with issues of discrimination and social justice and also participated in antinuclear demonstrations. The chapter applies the concept the present fallacy to the Linda Problem by using a Venn diagram. When one needs to determine the relative probability of different scenarios, one way to prevent himself/herself from making the conjunction (...) class='Hi'>fallacy is to determine whether any of the options is a subset of any other. If so, he/she can conclude that the subset option is not more probable. This rule works because a specific event can never be more likely than a general event that encompasses the specific event. The chapter explains why people are so frequently drawn in by the conjunction fallacy. (shrink)

We provide a 'verisimilitudinarian' analysis of the well-known Linda paradox or conjunction fallacy, i.e., the fact that most people judge the probability of the conjunctive statement "Linda is a bank teller and is active in the feminist movement" (B & F) as more probable than the isolated statement "Linda is a bank teller" (B), contrary to an uncontroversial principle of probability theory. The basic idea is that experimental participants may judge B & F a better hypothesis about Linda (...) as compared to B because they evaluate B & F as more verisimilar than B. In fact, the hypothesis "feminist bank teller", while less likely to be true than "bank teller", may well be a better approximation to the truth about Linda. (shrink)

The claim that we have a moral obligation, where a choice can be made, to bring to birth the 'best' child possible, has been highly controversial for a number of decades. More recently Savulescu has labelled this claim the Principle of Procreative Beneficence. It has been argued that this Principle is problematic in both its reasoning and its implications, most notably in that it places lower moral value on the disabled. Relentless criticism of this proposed moral obligation, however, has been (...) unable, thus far, to discredit this Principle convincingly and as a result its influence shows no sign of abating. I will argue that while criticisms of the implications and detail of the reasoning behind it are well founded, they are unlikely to produce an argument that will ultimately discredit the obligation that the Principle of Procreative Beneficence represents. I believe that what is needed finally and convincingly to reveal the fallacy of this Principle is a critique of its ultimate theoretical foundation, the notion of impersonal harm. In this paper I argue that while the notion of impersonal harm is intuitively very appealing, its plausibility is based entirely on this intuitive appeal and not on sound moral reasoning. I show that there is another plausible explanation for our intuitive response and I believe that this, in conjunction with the other theoretical criticisms that I and others have levelled at this Principle, shows that the Principle of Procreative Beneficence should be rejected. (shrink)

ABSTRACTProbability judgements entail a conjunction fallacy if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies. A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 (...) uses data where BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs. IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based prob... (shrink)

As Russell's paradox of "the set of all sets that do not contain themselves" indicated long ago, matters go seriously amiss if one operates an ontology of unrestricted totalization. Some sort of restriction must be placed on such items as "the set of all sets that have the feature F' or "the conjunction of all truths that have the feature G." But generally, logicians here introduce such formalized and complex devices as the theory of types or the doctrine of (...) impredictivity. The present paper argues for the informal and elementary idea that the items invoked in a proper identification have themselves already been identified. Even as an explanation is not satisfactory that proceeds in terms of items that themselves require prior explanation, so the same holds with identification. And heed of this elementary idea suffices to sideline those otherwise paradoxical perplexities. (shrink)