by Alan Walworth
(Draft 1.2, March 25, 2015)
In Extensional Versus Intuitive Reasoning (Psychological Review, 1983), Tversky and Kahneman reported finding that experimental subjects violated the conjunction rule, which says the probability of a conjunction cannot exceed the probability of either of its conjuncts; that is, P(A&B) <= P(A).
According to Tversky and Kahneman, people succumb to the conjunction fallacy (the “violation of the conjunction rule in a direct comparison of B to A&B”) because the conjunction is more “representative” and “available” than the conjunct:
- "A conjunction can be more representative than one of its constituents, and instances of a specific category can be easier to imagine or to retrieve than instances of a more inclusive category. The representativeness and availability heuristics therefore can make a conjunction appear more probable than one of its constituents."
Below I’ll offer a different interpretation of what may be their most notorious example, related to the likelihood that Linda, a philosophy major, has become a bank teller. (I have not thoroughly examined the related literature, so although I have not seen this interpretation elsewhere, it is conceivable it has been proposed earlier. Nor have I investigated carefully to what extent my interpretation can account for the results of other experiments alleged to demonstrate the conjunction fallacy.)
The question about Linda is in essence as follows:
- 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 likely?
- 1. Linda is a bank teller.
- 2. Linda is a bank teller and is active in the feminist movement.
A subject asked which is more likely might answer, “Since those are statements about a fictional character, they have no relation to reality; therefore it is not legitimate to attach probabilities to them. Asking how likely they are is like asking “What is the probability that Scarlett O’Hara had a birthmark on her back?” The author of Gone with the Wind could have said Scarlett had such a birthmark, or did not, whichever she liked, without being mistaken; but if the author said nothing about it there is no truth of the matter, so it makes no sense to talk of the probability that such a statement matches reality. In other words, it makes no sense to ask how likely such statements are to be true.”
The experimenter might chide such a subject for being uncooperative, similar to a subject who, when asked to say what a Rorschach image looks like, answers, “It looks like an inkblot.” “To participate properly in this experiment,” the experimenter might explain, “you need to play the implicit game. Imagine that the story about Linda is true, that you know she is 31 years old, single, outspoken, concerned about social justice, and so on. Then imagine that, knowing these facts about her, you are told that she is a bank teller, and ask yourself, How likely is that to be true? Next imagine that instead you are told that she is a bank teller and is active in the feminist movement, and ask yourself how likely that is to be true.”
Regardless of whether this explanation of how a subject should approach answering “Which is more likely?” is correct, it seems some subjects might approach the task this way, by considering the likelihood the statements would be true if made in the context described. If so, should that in any way affect our assessment of the conventional wisdom that the correct answer is that “Linda is a bank teller.” is more likely?
The logical case for 1 being more likely than 2 is simple and compelling. A conjunction can never be more probable than one of its conjuncts. Since 1 is entailed by 2, it is true whenever 2 is true, and in addition might be true when 2 is false, so it follows directly that 1 is more probable than 2, more likely to be true.
This logical analysis correctly assesses the relative probabilities of the propositions represented by 1 and 2. But what if the question is taken instead to relate to the statements 1 and 2 used to express those propositions? By a statement I mean an expression, a meaningful thing with physical reality, typically created by an act of speaking or writing. A reasonable way to think about the likelihood of a statement being true in specified (possibly hypothetical) circumstances is to consider what fraction of all instances of it in such circumstances are (or would be) true, much as one might assess the likelihood of a person having type O blood by determining what fraction of all people have that characteristic.
Since the propositions are presented by means of corresponding statements, it tends to be unclear whether the question “Which is more likely?” relates to the propositions or the statements. That this distinction can make a difference to the answer is evident from examples like this:
Which is more likely to be true?
- 3. Je parle seulement anglais.
- 4. I speak only English and I’m over five years old.
Logic argues that 3 is more likely than 4, since it is entailed by 4 and can be true when 4 is false. Common sense, however, says that in the real world 3 will almost certainly be false, whereas 4 may well be true; thus 4 is more likely to be true than 3. The conclusions differ not because the logical analysis is invalid, but because it concerns propositions, whereas common sense considers the statements.
Since in real life people are mostly concerned about whether statements are true, rather than about abstract considerations regarding the probability of propositions, it is natural to think of such questions about which is more likely as relating to statements. Consider the following scenarios:
- A) A bank in Bangalore is robbed and the robber escapes. A waiter from a restaurant across the street makes the following statement:
- 5. The robber escaped in a red pickup truck, a blue hybrid car, a green school bus, or a silver Sikorsky helicopter.
That statement is not very credible, because it indicates that the person making it, even if serious, has little if any idea what happened. Thus it is not very likely to be true. It might reasonably be thought its likelihood of being correct is less than 50%.
- B) A bank in Bengaluru is robbed and the robber escapes. A waiter from a restaurant across the street makes the following statement:
- 6. The robber escaped in a blue Toyota Prius.
This statement is fairly credible, for it appears to come from someone who knows what happened. In the absence of opposing evidence, it might reasonably be thought more likely correct than not.
Thus it seems statement 6 is more likely to be true than statement 5. In the long run, people in such situations who believe statements like 6 will be right more often than people in such situations who believe statements like 5. Thus the judgement that 5 is unlikely to be true and the judgement that 6 is likely to be true are both reasonable.
What if it turns out that Bengaluru is Bangalore, and the robberies are one and the same? If so, one can argue that 5 is more likely than 6. But it does not follow that the earlier judgements were irrational.
At the conclusion of a physics lecture, a student leaving the lecture hall makes one of the following statements. Which is more likely to be true?
- 7. Someone within 100 meters of here has a headache.
- 8. I have a headache.
The student saying 8 can be presumed to know the truth of the matter, and to most likely be telling the truth, so the probability is very high that 8 is true. The student who says 7, however, cannot be assumed to be basing the claim on her own headache, since in that case it would be extremely odd to say 7 instead of 8. She may instead be basing her statement on an estimate of the number of people in the surrounding area and the frequency of headaches. Since she cannot be presumed to be so well informed about the truth of the statement, we might reasonably conclude that the likelihood statement 7 is true is not so high as the likelihood 8 is true. The person whose policy is to believe statements like 8 while remaining skeptical about the truth of statements like 7 is likely to be right more often than the person whose policy is the reverse. In other words, statement 7 is less likely to be true than statement 8, even though proposition 7 is entailed by proposition 8.
Tversky and Kahneman’s subjects may have regarded the question “Which is more likely?” as being about the statements presented, imagined such statements being made in a real situation with the background described, and reasonably concluded that, in such a situation, since the statement “Linda is a bank teller and is active in the feminist movement.” provides evidence that the person making it is well-informed about Linda, it is more credible, and more likely to be true, than the alternative “Linda is a bank teller.”, which provides no such evidence. It is arguably reasonable to suppose that in such situations, where a statement like 2 is made, a person who believes what is said is likely to be right more often than a person who, in such situations where a statement like 1 is made, believes what is said. If so, statements like 2 are more likely to be true than statements like 1, and the responses of Tversky and Kahneman’s subjects may reflect their recognition of this fact, rather than demonstrating irrationality regarding the relative likelihood of related propositions.
Addendum (April 3, 2015):
It is customary for academic papers to include a review of relevant prior literature. Though there is much to be said for not ignoring prior work, there is also some risk of giving it too much attention, as when long ago scholars sought truth by focusing on what was said by the ancients rather than by looking at the world around them. Currently, paying too much attention to those who came before can lead to requiring students to memorize facts about the history of philosophy in place of having them do philosophy, which makes about as much sense as a physical education course in which the students study the history of tennis instead of playing the game. Paying respect to earlier academics can become a stultifying ritual, in which references to predecessors serve to acknowledge the validity of their work much as in certain countries one is required to hang a picture of the dictator on the wall to demonstrate acceptance of his greatness. Attending too much to what others have done earlier may restrict one's thinking to the paths they took, a limitation that may explain the conventional wisdom that the best new ideas often come from someone outside the mainstream of a field. There is much to be said for thinking through an issue on one's own, without first seeing what others have had to say about it, somewhat as it is good practice to work a math or physics problem on one's own, before consulting the answer in the back of the book.
It turns out there is a substantial—at least in quantity—body of literature about Tversky and Kahneman's experiments and speculations, and in particular about the "Linda" example. Some published articles make points related to the theme above, though without focusing on the distinction of propositions from statements. Jaako Hintikka's 2004 article in Synthese, A Fallacious Fallacy, may be the best example of this sort. After discussion of Bayesian methods, prior probability distributions and Carnapian assumptions, Hintikka makes the point that the likelihood an answer is true depends on the credibility of its source, and that an answer itself may affect the assessment of the source's credibility:
- Now we come to the crux of the conjunction “fallacy”. The fact that answers like T or (T & F) come from a different source of information does not change the objective evidence E on the basis of which their credibility has to be judged. But in the light of what has been said it may affect the prior probability distribution. For the sources of the information may affect the prior probabilities P1 (T) and P2 (T & F) which are the basis of the conditionalized probabilities P1 (T/E) and P2 ((T & F)/E). And even though P1 and P2 are known technically as “prior” probabilities, according to what was found they can be affected by evidence, though not be conditionalization. And part of that evidence are the two different messages that the two sources provide the evaluator with. For when we are evaluating an answer A from a source of information with an unknown but presumably constant propensity to give true answers, that particular answer A itself constitutes part of the data on the basis of which the reliability of its source and hence its own reliability must be judged. Hence the probabilities assigned in the course of an actual inquiry to two answers (messages) A, B from two different sources of information may have to be judged by reference to different “prior” probability distributions.
Isaac Levy, in a rejoinder to Hintikka in the same issue of Synthese, argues that Hintikka has failed to show the "Linda" experimental subjects are not irrational:
- If I have understood Hintikka’s suggestion correctly, he has not succeeded in saving the rationality of the experimental subject. In the first place, nothing in the scenario suggests that the hypotheses in question are or ever have been the testimony of any witnesses. But suppose we waive that point and accept Hintikka’s elaboration according to which the experimental subject takes one witness to have testified that T and another that T&F. So the total relevant evidence is now the background biography for Linda (E) and the testimony of both witnesses. If, as Hintikka, along with Kahneman and Tversky, seem to think, the experimental subjects are ranking the propositions on the list with respect to probability and these probabilities are posteriors conditional on the the total relevant information available, the comparison being required is a comparison using the same probability evaluation. To assign probability to T higher than the probability for T&F where these two posterior judgments are constituents of the agent’s probability judgment in a single context is simply incoherent. If it is claimed that the contexts are different, then the pair of probability judgements do not appear responsive to the question the experimental subjects are invited to answer. Hintikka’s experimental subjects seem to be as incoherent in their probability judgments as those of Kahneman and Tversky.
Levy then proceeds to give his own account of why the experimental subjects are not behaving irrationally, based on the notion (fairly popular among those defending the rationality of Kahneman's subjects) that they are judging "epistemic utility" rather than probability:
- I proposed a different measure of expected epistemic utility. It is represented by a function P(H/B&E) – qM(H/B&E) where the M-function is, like the P-function a probability measure formally. Its intended application is different. 1 – M(H/B&E) measures the value of the information added to B&E by H. q is an index of boldness and is restricted to values between 0 and 1.
- This measure does satisfy the condition on equivalence of hypotheses that the difference between posterior and prior does not. And it is readily seen to be a generalization of the defective measure in terms of the difference between posteriors and priors.
- I have no idea whether experimental subjects use my proposed measure of epistemic utility or not. I think they should use it or some improvement on it. But that is not my current concern. The measure I propose, like the measure favored by Hintikka and Pietarinen a long time ago, evaluates T&F over T. In general, there is an abundance of measures that do the trick.
- On this construal of the responses of the experimental subjects, there is no fallacy in the use of expectation determining probabilities.
Here we have an all too typical academic discussion, displaying erudition and technical sophistication but adding little to understanding of the world, more soporific than enlightening. It reminds me of a political science class, taught by a highly rated professor, that I sat in on with my daughter during a recent college visit. She was thinking of majoring in political science, so it seemed suitable to visit a class in that subject. Unfortunately, the discussion was so incomprehensible and boring that although she is still interested in politics, she has lost interest in studying political science. In retrospect, we would have done better to visit a class discussing something other than Hegel's Philosophy of Right, but the tendency toward obscurity in academic discourse is widespread.
As it happens, the professor leading that Hegel discussion wrote a letter to the New York Times, defending academics against the charge that they are insufficiently engaged with real world issues. Academic subjects, he pointed out, often do require complex technical considerations that make them inaccessible to ordinary people. That is true, but it is also true that the ivory tower has become largely divorced from the problems of the real world. It is all too easy to argue that intellectual exercises have value regardless of any evident practical application, and on that basis to enjoy one's privileged, tenured position, regardless of whether their technical exercises are truly worthwhile.
Addendum (April 16, 2015):
Here are some additional related references:
- The Conjunction Fallacy and the Debate on Human Rationality by Rodrigo Moro (2009) (email@example.com)
- Linda is Back: Controversies Around the Conjunction Fallacy in the Psychology of Reasoning by Rodrigo Moro (2007)
- On the Nature of the Conjunction Fallacy, Rodrigo Moro, Synthese, 171 (1): 1-24 (2009)