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
- Tversky & Kahneman (1982, 1983)
- 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.
- Tversky & Kahneman (1983)
- 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.
External links