Memory & Cognition 2002, 30 (2), 191-198 On the reality of the conjunction fallacy ASHLEY SIDES, DANIEL OSHERSON, NICOLAO BONINI, and RICCARDO VIALE Rice University, Houston, Texas 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 demon- strations of the conjunction fallacy. We used betting paradigms and unambiguously conjunctive state- ments to reduce these sources of ambiguity about conjunctive reasoning. Despite the precautions, con- junction fallacieswere as frequent under betting instructions as under standard probability instructions. The Conjunction Fallacy Mars before 2050) and is governed by principles familiar Here is the famous Linda story, to be labeled E (for ev- from discussionsof Bayesianism (as in Earman, 1992; Hor- idence) in what follows: wich, 1982; and Howson & Urbach, 1993). All the events (E) Linda is 31 years old, single,outspoken,and very bright. that figured in our experiments were singularin character, She majored in philosophy.As a student,she was deeply resisting placement in classes of similar cases that allow concerned with issues of discrimination and social jus- for a meaningful frequency count. tice, and also participatedin antinucleardemonstrations. Interpreting the Word Probability The task is to rank various statements “by their probabil- As documented in Hertwig and Gigerenzer (1999), ity,” including these two: probability is polysemous in the general population. It (B) Linda is a bank teller. has often been noted, moreover, that, through much of its (B ` F ) Linda is a bank teller and is active in the feminist premodern history, the term probable carried a connota- movement. tionof “approvableopinion”(see Hacking,1975, chap. 3). A majority of respondentsacross a variety of studiesranked Appeal to authority was one way that an opinion was ap- B ` F as more probable than B (see Hertwig & Chase, provable,but anotherwas via evidentialsupport.Thus, John 1998, for a review of findings;the original report is Tver- Locke (1671) defined probable propositions as those sky & Kahneman, 1983). This judgmentis in apparent vio- “for which there be arguments or proofs to make it pass lation of the conjunctionlaw Pr(X ` Y | Z) # Pr(X | Z)for or be received for true” (cited in Krause & Clark, 1993, any statements X, Y, Z, with strict inequality for nontriv- p. 71). A respondent working with the latter interpreta- ial cases such as the present example. tion of probabilitywould attempt to determine whether E The law is not violated,however, if participants in these providesmore supportfor B or for B ` F. In what follows, studies understand the word probability in a sense differ- we formalize support in a familiar way and observe that ent from the one assigned to it by modern probabilitythe- it justifies the intuition that E provides greater support ory. There is similarly no violation if B is interpreted to for B ` F than for B. Several alternative formalizations mean B ` ¬F or is interpreted in any way other than as a would serve our purposes just as well, but we do not at- conjunct of B ` F. The need for clarity about these issues tempt a survey of possibilities. Our point is that at least is discussed in the remainder of the present section. We then one plausible reading of probable exculpates reasoners describe experiments in which we attempted to provide a from the conjunction fallacy. sharper test of the thesis that naive conjunctivereasoning Many authors agree that a statement X supports a state- can be led into fallacy. ment Y to the extent that Pr(Y | X ) exceeds Pr(Y ) (see the Let us first note that we do not attempt to defend naive references cited in Fitelson, 1999, in which the term con- reasoning by denying the defective character of the judg- firmation is used in place of support). A simple way to ment Pr(X ` Y | Z) . Pr(X | Z) (if such a judgment is ever quantify this relation is via the quotient Pr(Y | X )/Pr(Y ) made). In particular, we believe the concept of probability (the difference works just as well). Here Pr denotes prob- can be sensibly applied to single events (like man reaching ability in the modern, technical sense, and the quotient Pr(Y | X ) / Pr(Y ) translatesthe support concept into mod- ern terms. According to the definition,E supports B ` F The authors thank Andrea Cerroni, Karin Dudziak, Denise Wu, An- more than E supports B if and only if drea Pozzali, and Zhihua Tang for assistance in performing Experiment 1. Correspondence should be addressed to D. Osherson, Department of Psy- Pr(BFE` ) Pr(BE) chology, MS-25, P. O. Box 1892, Rice University, Houston, TX 77251- | > | . 1892 (e-mail: [email protected]). Pr(BF` ) Pr(B ) 191 Copyright 2002 Psychonomic Society, Inc. 192 SIDES, OSHERSON, BONINI, AND VIALE An application of Bayes’s theorem reveals that the in- of B and B ` F they preferred to bet on. There was some equality above holds if and only if Pr(E | B ` F) . Pr(E | decline in the rate of conjunctionviolation,but it nonethe- B). And the latter inequality is entirely reasonable since less characterized a majority of responses. it asserts that Linda is more likely to be single, outspoken, The betting version of the Linda problem strikes us as and so on, on the assumption that she is a feminist bank inconclusive,however, in lightof the hypotheticalcharac- teller than on the mere assumption that she is a bank ter of the question. Linda is not a real person, and no bets teller.Hence, one interpretationof majoritychoices in the will be paid. Respondents may consequently be inclined conjunction problem is that (1) most respondents have a to interpret the query as a disguised probability question, supportinterpretation of probability, (2) their conception leadingto the same ambiguityas before. Similar concerns of the support that statement X provides for Y can be for- beset the bettingquestionsposed in Wolford et al. (1990). malized as the ratio of Pr(Y | X )toPr(Y ), and (3) they ac- Genuine bets were made (and paid) in Bar-Hillel and curately perceive E to provide more support for B ` F Neter’s(1993) careful study of violations of the disjunc- than for B, in the foregoing sense of support. tion law (according to which the probabilityof a given event The hypothesisthat the word probabilityis interpreted cannot be higher than the probabilityof any event that in- as support is not the same as claiming that the standard cludes it). But bets in their experiments concerned people probability of B given E is the probability of E given B, with whom the participantswere unfamiliarand outcomes and similarly for B ` F. The latter claim is explored in already known to the experimenter. This is not the usual Wolford, Taylor, and Beck (1990) and aptly criticized by kind of betting context. Wagers on sporting events, for Bar-Hillel (1991; see also Wolford, 1991). On the other example, bear on familiar teams and as-yet-undetermined hand, the support hypothesis is close to an analysis ad- outcomes. In the betting experiments reported below, we vanced in Hertwig and Chase (1998). The latter discus- therefore prepared questionsinvolvingfuture events about sion is complicated, however, by a relatively indirect mea- which our participants possessed background knowledge sure of support (due to Nozick, 1981). (the latter plays the role of evidence E). If responses to the Linda problem are often due to a sup- Bettingparadigmsalso discourageinterpretingthe prob- port interpretation of probability, we would expect more ability of a sentence as its expected informational value. conformity to the conjunctionlaw when the wording of the To explain, suppose that people judge B ` F to be much problem discouragesthis interpretation.One such wording more informative than B, but only slightly less probable. is due to Fiedler (1988), who asked participantsto estimate Then the product of probability and informativeness (the “how many out of 100 people who are like Linda” satisfy expected informativeness) might be higher for the con- B and B ` F. Relative to the original problem, it seems junction than for its conjunct. In response to a question harder to construe Fiedler’s frequency question as involv- about probability,participantsmight choose the alternative ing support. In fact, conjunction violations were substan- with higher expected informativeness on conversational tially depressed in response to the frequency wording.1 grounds (since one goal of polite conversation is to be Improvement with frequency formats, however, leaves maximally informative). This possibilitywas recognized open the possibility of irrational judgment when con- by Tversky and Kahneman (1983) and was also discussed fronted with single events.To assess conformity to the con- by Bar-Hillel and Neter (1993). Since conversationalgoals junction law in these circumstances, it seems necessary are clearly irrelevant to gambling, betting paradigms are to frame questions that avoid the word probability and well suited to minimize the impact of expected informa- its cognates. The obviousstrategy is to offer a choice be- tiveness on respondents’ choices. tween betting on a conjunctiveproposition versus one of its conjuncts. Such was the idea behind Tversky and Ambiguous Logical Form Kahneman’s(1983) problem involvinga regularsix-sided Answers to the Linda problem constitutea conjunction die with four green faces and two red faces. Participants fallacy only if the options labeled B ` F and B are inter- were asked to select one sequence, from a set of three, with preted as a conjunction and one of its conjuncts. It has a $25 prize if the chosen sequence appeared embedded been widely observed that, in the presence of the alterna- within 20 rolls of the die.