# Conjunction fallacy: Map

### Map showing all locations mentioned on Wikipedia article:

The conjunction fallacy is a logical fallacy that occurs when it is assumed that specific conditions are more probable than a single general one.

The most oft-cited example of this fallacy originated with Amos Tversky and Daniel Kahneman:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?
# Linda is a bank teller.
# Linda is a bank teller and is active in the feminist movement.

85% of those asked chose option 2. However the probability of two events occurring together (in "conjunction") is always less than or equal to the probability of either one occurring alone—formally, for two events A and B this inequality could be written as \Pr(A \and B) \leq \Pr(A), and \Pr(A \and B) \leq \Pr(B).

For example, even choosing a very low probability of Linda being a bank teller, say Pr(Linda is a bank teller) = .05 and a high probability that she would be a feminist, say Pr(Linda is a feminist) = .95, then, assuming independence, Pr(Linda is a bank teller AND Linda is a feminist) = .05 × .95 or .0475, lower than Pr(Linda is a bank teller).

Tversky and Kahneman argue that most people get this problem wrong because they use the representativeness heuristic to make this kind of judgment: Option 2 seems more "representative" of Linda based on the description of her, even though it is clearly mathematically less likely.

(As a side issue, some people may simply be confused by the difference between 'and' and 'or'. Such confusions are often seen in those who have not studied logic, and the probability of such sentences using 'or' instead of 'and' is completely different. They may infer sentence #1 assumes Linda is necessarily not active in the feminist movement.)

Many other demonstrations of this error have been studied. In another experiment, for instance, policy experts were asked to rate the probability that the Soviet Union would invade Poland, and the United States would break off diplomatic relations, all in the following year. They rated it on average as having a 4% probability of occurring. Another group of experts was asked to rate the probability simply that the United States would break off relations with the Soviet Union in the following year. They gave it an average probability of only 1%. Researchers argued that a detailed, specific scenario seemed more likely because of the representativeness heuristic, but each added detail would actually make the scenario less and less likely. In this way it could be similar to the misleading vividness or slippery slope fallacies, though it is possible that people underestimate the general possibility of an event occurring when not given a plausible scenario to ponder.

## A quantum probability explanation of the conjunction fallacy

In a recent work of Franco, the conjunction fallacy has been described with the mathematical formalism of quantum mechanics.In particular, it has been shown that each couple of mutually exclusive events (Linda is/isn't feminist, or Linda is/isn't bankteller) can be associated to a basis in a 2-dimensional vector space.Moreover, it is assumed that the subject's beliefs about such events is described by a vector (called opinion state |s>), which can be written as a superposition of the basis vectors:
|s>=s_0|a_0>+s_1|a_1>
where |a_0> and |a_1> are the basis vectors relevant to a particular couple of mutually exclusive events (for example, Linda is/isn't feminist). The subjective probability relevant to event a_1 (Linda IS feminist) is
P(a_1)=|s_1|^2
If we want to describe the subjective probability relevant to another couple of mutually exclusive events b_0, b_1 (Linda is/isn't a bankteller), the law of total probability is replaced in the quantum framework by the following law:
P(b_1)=P(a_1)P(b_1|a_1)+P(a_0)P(b_1|a_0)+Interference
where the interference term (with a precise mathematical form) has a very important role in the conjunction fallacy. In fact, the presence of strongly negative interference terms can makeP(b_1)

<>

math>, which is precisely the conjunction fallacy (the estimated probability that Linda is bankteller is lower than the estimated probability that Linda is feminist and bankteller).

An important fact is that in quantum mechanics it is impossible to measure simultaneously two non-commuting observable quantities. Thus the joint probability is replaced by the concept of consecutive probability P(a_1)P(b_1|a_1).

In other words, it has been shown that the use of quantum probability allows to describe in a natural way the conjunction fallacy.

## Notes

1. Tversky & Kahneman (1982, 1983)
2. Many variations of this experiment in wording and framing have been published. When Tversky and Kahneman (1983) changed the first option to "Linda is a bank teller whether or not she is active in the feminist movement" in the same experiment as described a majority of respondents still preferred the second option.
3. Tversky & Kahneman (1983)
4. Riccardo Franco (2007), The conjunction fallacy and interference effects, http://arxiv.org/abs/0708.3948v1

## References

• Tversky, A. and Kahneman, D. (1982) "Judgments of and by representativeness". In D. Kahneman, P. Slovic & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases. Cambridge, UK: Cambridge University Press.