Elsevier

Cognition

Volume 108, Issue 3, September 2008, Pages 740-753
Cognition

The conjunction fallacy and the many meanings of and

https://doi.org/10.1016/j.cognition.2008.06.008Get rights and content

Abstract

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 ∧, 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 ∧, 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 ∧. 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.

Section snippets

The and in research on the Linda task: Logical operator or natural language conjuction?

In their seminal article on the conjunction fallacy, Tversky and Kahneman (1983) distinguished between two experimental paradigms: the M → A and the A → B paradigms. The word problem that became the pivotal implementation of the M → A paradigm is the Linda task. Here, respondents read a brief personality sketch of a person, Linda. The sketch described her as an educated, outspoken, social activist. Based on this information, respondents were asked to rank eight statements about Linda according to

Method

One hundred and nineteen participants were presented with a frequency representation of the Linda task and asked to provide an estimate for a single target item. The majority of respondents were students of two universities in Berlin, and about a third of respondents were professionals from different fields (e.g., architecture, medical care). The study was conducted at the Max Planck Institute for Human Development, Berlin. Respondents were randomly assigned to one of three conditions, and were

Experiment 2

It would be plainly wrong to claim that recent research on violations of the conjunction rule has paid no attention to the ambiguity of and. Bonini et al. (2004) and Tentori, Bonini, and Osherson (2004), for instance, discussed this issue extensively. Notwithstanding their prudent treatment of semantic and pragmatic ambiguities in general they found numerous judgments that violated the conjunction rule. In what follows, we consider Tentori et al.’s (2004) Experiment II in more detail.

One of the

The and in causal conjunctions

As pointed out before, Tversky and Kahneman (1983) distinguished between two task structures, the M → A paradigm and the A → B paradigm. Originally, it was predominantly the first paradigm that attracted researchers’ imagination, and the Linda task evolved as its archetypical representative. Recently, however, the A → B paradigm has also been investigated (e.g., Sides et al., 2002, Bonini et al., 2004). According to Tversky and Kahneman, violations of the conjunction rule in this paradigm occur

Experiment 3

Experiment 3 is designed to investigate how people read the cigarette-tax task constructed by Sides et al.’s (2002). To do this we make use of Levinson’s (1983) analysis of and-conjunctions. He predicted that people read (9):

  • (9)

    He turned on the switch and the motor started

in a way that “is as ‘strong’ (informationally rich) as the world allows – and read in the following relations between two conjoined clauses wherever possible:

Given p and q, try interpreting it as:

  • (i)

    ‘p and then q’; if successful

Experiment 4

In Experiment 4, we investigate estimates of both conditional and conjunctive probability, elicited separately in two groups of respondents. If indeed, as Experiment 3 suggests, (some) people understand the tobacco-tax task as a cause–effect statement, and consequently, estimate the conditional probability, participants in the conditional probability group and the conjunctive-probability group will arrive at more or less identical estimates.

General discussion

We have used three different methods to probe people’s understanding of the and connective. In Experiment 1, we asked participants to shade the area in a Venn diagram that corresponded to the quantity that they had just estimated. In Experiment 3, we asked participants to imagine themselves in the role of a reader who relates a newspaper statement to colleagues, and to report how they would have reproduced the original statement. In Experiment 4, we asked participants in one group to estimate

The ambiguity of and and semantic or pragmatic inferences

Numerous linguists have emphasized that and-conjunctions can communicate a wide range of relationships between the states of affairs described by their conjuncts. Some linguists have taken the position that the natural language and should be analyzed independently of the logical operator ∧ (e.g., Sweetser, 1990); other linguists have proposed that and has a minimal truth-functional semantics (e.g., Blakemore & Carston, 2005). The key issue is on which levels do connotations arise that are not

The conjunction fallacy: A misunderstanding about conjunction?

In a series of studies, Noveck and Chevaux (2002) demonstrated that children appear less likely than adults to draw implicit meanings from the conjunction in statements that invert a series of events such as “Laurent went to the hospital and broke his ankle.” In their view, this and related findings (e.g., Noveck, 2001) show that “a child’s initial treatment of utterances … is consistent with logical interpretations and that linguistic–pragmatic interpretations evolve with age” (p. 6). Here is

Conclusion

The ability to make reasonable pragmatic inferences is a sine qua non of human intelligence, and people bring this intelligence into the laboratory. Any computer can mechanically apply the conjunction rule. In contrast, to winnow out the inappropriate readings of ambiguous words and statements, to infer the referents of pronouns and the meanings of polysemous words, and to fill in intended interpretations of what has been said is evidence of the remarkable linguistic abilities of humans. The

Acknowledgements

Ralph Hertwig was supported by Swiss National Science Foundation Grant 100014-118283/1. We thank Gregor Caregnato for help in data collection and Laura Wiles for editing the manuscript.

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