Conjunction and the Conjunction Fallacy

Abstract

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. Conjunction and the Conjunction Fallacy 3 Conjunction and the Conjunction Fallacy Fallacy or no fallacy? Despite extensive debate lasting several decades, there is still no consensus among cognitive scientists on a fundamental question about human reasoning. Are many Western college students disposed to elementary errors when evaluating the chances of simple events? In the present note, we attempt to settle the matter with respect to the conjunction fallacy. Let X and Y be the following English sentences (appearing as stimuli in Sides, Osherson, Bonini & Viale, 2002). (1) X: The cigarette tax in Texas will increase by $1.00 per pack by September 1, 1999. Y : The percentage of adolescent smokers in Texas will decrease at least 15% from current levels by September 1, 1999. Let X-and-Y denote the result of writing X followed by the word “and,” followed by Y . Sides et al. (2002) observed a majority of subjects preferring to bet on X-and-Y compared to Y . Likewise, a majority of a different set of subjects judged X-and-Y to have higher probability than Y . Similar results were obtained for other items in several replications, which included measures to prevent Y from being misinterpreted as Y -and-not-X. (The possibility of such misinterpretation is raised in Tversky & Kahneman, 1983; Morier & Borgida, 1984; Macdonald & Gilhooly, 1986; Politzer & Noveck 1991; and Dulany & Hilton, 1991). Concordant findings emerge from follow-up work reported in Bonini, Tentori & Osherson (2003) using a different betting paradigm. We invited subjects to choose bets among all three of X-and-Y , X, and X-and-not-Y . The presence of X-and-not-Y as an explicit alternative makes it pragmatically impossible to interpret X as X-and-not-Y since it would be uncooperative to needlessly repeat one of the options in altered form. Different choices for X and Y appeared in different items. For example, one item read as follows. Because of the Italian Rail’s new policies aimed at encouraging voyages longer than 100 km, the number of passengers . . . will decline by 5% on commuter trains and increase by 10% on long distance trains. [X-and-Y ] Conjunction and the Conjunction Fallacy 4 will decline by 5% on commuter trains and will not increase by 10% on long distance trains. [X-and-not-Y ] will decline by 5% on commuter trains. [X] Eighty percent of sixty Italian college students chose to bet some money on X-and-Y ; in fact, the average bet on X-and-Y was significantly greater than the average bet on X. Of course, if the goal is to maximize expected winnings, all bets should have been placed on X. Other items yielded similar preferences. The bets were genuine, and subjects were informed that they could limit their bets to a single statement. Sixty additional students performed in the same way even when the conjunctive statements were accompanied by the parenthetical remark: “both events must happen for you to win the money placed on this bet.” Thus (as originally reported by Tversky & Kahneman, 1983), for some choices of X and Y , many people endorse the inequality: (2) Prob(X-and-Y ) > Prob(Y ) Moreover, it seems justified to understand the function Prob(·) in (2) as reflecting reasoner’s judgments about chance, rather than some other concept that might be encoded by the locution “probability.” For, no appreciable differences were found in the Sides et al. (2002) experiments between the conditions in which subjects placed bets on statements versus comparing their probabilities; likewise, the experiments reported in Bonini et al. (2003) involved betting only. The experiments thus conform to the recommendations of several commentators (e.g., Hertwig & Gigerenzer, 1999, p. 293) who advocate betting instructions as a means of circumventing ambiguity in the word “probability.” The procedures followed in Sides et al. (2002) and Bonini et al. (2003) also make it unlikely that reasoners tended to construe Prob(X) as Prob(X-and-not-Y ) [and similarly for Prob(Y )]. It therefore seems safe to assume that participants endorse (2) literally, namely, as the attribution of greater probability to a conjunction compared to one of its conjuncts. (See Bonini & Savadori, 2003, for additional experimental evidence in support of this claim.) There remains the question: Is (2) a fallacy? From the theory of subjective probability (as presented, for example, in Skyrms, 2000, Chapter VI), one derives the following principle. (3) For all statements A, B, Prob(A ∧ B) ≤ Prob(A), Prob(B). But (3) does not, by itself, contradict (2) inasmuch as X ∧ Y fails to appear in (2); only X-and-Y appears. Thus is born the idea that (2) sustains no charge of fallacy except Conjunction and the Conjunction Fallacy 5 through illicit conflation of logical conjunction (∧) with natural language conjunctions like “and” (e.g., Gigerenzer, 2001, pp. 95-96). The conflation is illicit because “and” possesses semantic and pragmatic properties that are foreign to ∧. For example, uttering X-and-Y often conveys temporal succession or even causation, as in “Bill ate garlic ice cream and died.” (For discussion, see Carston, 1993; Sweetser, 1990.) Let us make the case that (2) is nonetheless a genuine conjunction fallacy. Suppose again that X and Y are as specified in (1). The reader will likely concur that: (4) (a) It is impossible for X to be false given the truth of X-and-Y . (b) It is impossible for Y to be false given the truth of X-and-Y . Indeed, we think that virtually every native speaker of English would agree with (4) even if she interprets X-and-Y as “X and then Y ” or as “Y because of X.” Now, obviously: (5) It is impossible for X ∧ Y to be false given the truth of the two statements X, Y . Immediately from (4) and (5), it follows that: (6) It is impossible for X ∧ Y to be false given the truth of X-and-Y . Using standard terminology, we can therefore say that the argument from X-and-Y to X ∧ Y is valid. Similarly, (4) shows that the arguments from X-and-Y to X, and from X-and-Y to Y are valid. Using |= to signify validity, we abbreviate these facts to: (7) (a) X-and-Y |= X. (b) X-and-Y |= Y . (c) X-and-Y |= X ∧ Y . Well-known analyses of the betting rates set by rational players lead to the (extremely reasonable) principle: (8) For any statements A, B, Prob(A) ≤ Prob(B) whenever A |= B. In view of (7)b and (8), a probability fallacy therefore results from believing (2). The only way to escape this conclusion is to deny that people believe (7)b. Such denial strikes us as a desperate defense of human judgment, and is also contrary to the experimental results reported below. Conjunction and the Conjunction Fallacy 6 We conclude that (2) is indeed a probabilistic fallacy. But does (2) qualify as a conjunction fallacy? Answering this question seems to involve little more than terminology; the important empirical issue is whether human judgment violates elementary principles of personal probability, and this is demonstrated by the combination of beliefs (7)b and (2). The terminological issue may nonetheless be addressed as follows. Let us first observe that the “official” conjunction fallacy (9) Prob(X ∧ Y ) > Prob(Y ) is easily derived from (2) via (7)c, (8), and the transitivity of numerical inequality. So (2) is either equivalent to (9) (if “and” means ∧) or else strictly stronger than (9) (if “and” means more than just ∧). Anyone endorsing (2) has therefore committed at least the official conjunction fallacy. Thus, the only reason to deny (2) the status of true conjunction fallacy is to insist that conjunction fallacies must endorse exactly (9) and nothing more. To perceive the peculiarity of such insistence, recall how easily (7)c is derived from the self-evident facts in (4). The validity of the argument from X-and-Y to X ∧ Y , along with its simple demonstration, make it natural to affirm that X ∧ Y is part of the meaning of X-and-Y , hence that X-and-Y expresses X ∧ Y . Such an affirmation does not require X ∧ Y to exhaust the meaning of X-and-Y , since (7)c does not require that X-and-Y be logically equivalent to X ∧ Y (i.e., that also X ∧ Y |= X-and-Y ). Analogously, it is accurate to say that the Declaration of Independence expressed the desire of the colonies to be free of England; it is not thereby asserted that nothing more was expressed. That ∧ does not represent all linguistic uses of “and” as a sentential connective is familiar and uncontroversial; it has been observed in logic texts for half a century. This truism should not prevent us from recognizing the equally evident fact that X-and-Y expresses the logical conjunction X ∧ Y , whatever else X-and-Y happens to express. For example, given our choices of X and Y , it may be that X-and-Y expresses (or pragmatically implicates) that X is a cause of Y . Even so, X-and-Y also expresses X ∧ Y , as shown by (7)c. Consequently, it is natural to assert that (2) expresses the (formal) conjunction fallacy (9). It remains to show experimentally that many people endorse statements of form (2) at the same time that they agree to (7)a,b. Conjunction and the Conjunction Fallacy 7

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 0 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.