# Is it ever rational to stumble onto the conjunction fallacy in probability?

The conjunction fallacy is the phenomenon where many people believe that the probability of the event (A AND B) is strictly greater than the probability of the event A. It is usually thought of as an error in reasoning. However, I recall reading a paper in a philosophy journal where the author claimed that, at least in some cases, it is rational to believe that the probability of the event (A AND B) is strictly greater than the probability of the event A. Could that actually be the case? Note, this is separate from the issue of whether it is correct to believe it. Some things are rational to believe, but incorrect. (For example, the case of not drinking a bottle labeled "POISON", when in fact the bottle is perfectly fine, some prankster just labeled it "POISON"). My personal belief is that it is not only incorrect but also irrational to believe the conjunction fallacy, and that the author of that paper is simply trying to use bogus arguments to desperately avoid the conclusion that most people are irrational in believing the conjunction fallacy. But, have any reputable philosophers argued for the rationality of (sometimes) stumbling onto the conjunction fallacy?

• It is never rational to "stumble onto" a fallacy. Maybe you could reword the question. Commented May 3, 2023 at 21:36
• You're not actually talking about mathematical probabilities here, like the probability of a coin landing on heads. You're talking about subjective assessments of plausibility and reliability of evidence. "He was drunk and I saw him fall over" might be more believable than "He was drunk", but that doesn't make it more probable. Commented May 4, 2023 at 10:52
• "However, I recall reading a paper in a philosophy journal where the author claimed that, at least in some cases, it is rational to believe that the probability of the event (A AND B) is strictly greater than the probability of the event A." Give an example. Commented May 4, 2023 at 14:52
• Yes, please edit your question to include the details of this paper and quote the relevant part of it, otherwise it could be distorted by being paraphrased from memory. Commented May 4, 2023 at 23:01
• @kaya3 I don't remember exactly which article in which journal where I read this. Commented May 5, 2023 at 1:52

## 2 Answers

Well, the conjunction fallacy is a fallacy, so it's not rational to believe it. However...

Suppose Dave tells you he was just robbed and that's why he's were late to meet with you. You are skeptical, so you question him about the details of the robbery. He easily gives a lot of specific details.

Now, "Dave was just robbed AND it was on the corner of Elm and Woodbury AND the robber jumped off a motorcycle to steal Dave's phone AND the motorcycle license plate was 8LEM94 AND the robber was short, wearing a black motorcycle helmet and white sweater" is less likely than "Dave was just robbed"; the first is a conjunction of many events and the second is a single event.

However, it is nonetheless rational to believe Dave's story more, on the basis that he was easily able to give such details. This is not the conjunction fallacy, but it can be mistaken for it. P(Dave is telling the truth) < P(Dave is telling the truth | Dave is easily able to give many plausible details of the event) is perfectly possible and not fallacious. Dave's ability to give details (quickly and off-the-cuff) is increased if the event actually happened.

• Good example. In all generality, P(A | B) = P(A and B | B), so P(A) < P(A | B) can easily be mistaken for the conjunction fallacy.
– Stef
Commented May 4, 2023 at 7:59

Here is an example of how p(A&B) can be greater than p(A) or p(B): suppose Aunt Gertie dies under very mysterious circumstances. Someone proposes:

A Aunt Gertie was murdered.


You might know that Aunt Gertie had some skeletons in her close, but still, that seems unlikely, so you assign it a probability of p(A)=0.3.

Someone else tells you

B Aunt Gertie had a gambling problem and owed money to shady people.


Independently, you would assign p(B)=0.3. But now, given other knowledge, you realize that A&B, along with other knowledge forms a coherent pattern of events that is more plausible than A or B alone, so you assign p(A&B)=0.5.

The essential point is that p(A&B) is not being calculated from p(A) and p(B); it is being assigned its own probability based on the contents of the propositions themselves and the knowledge base.

This is arguably how science works. The individual propositions of a scientific theory are individually improbable, but they support each other in such a way that the theory as a whole is more probable than the individual propositions in it.

ADDENDUM

Due to carelessness in editing, I left out an important point which the commenters have rightly scolded me for. What I should have said is that it can be rational to assign a higher probability to p(A&B) than one originally assigned to p(A) or p(B), but as pointed out in the comments, if you retain all of the original assignments of probability, you are left with a set of assignments that violates the theory of probability. Rationally, then, you have to either revise your judgements of the probabilities of A and B or just drop them, asserting ignorance of the individual probabilities.

• Is this technically correct? It feels like you're actually assigning P(A | B) = 0.5, not P(A & B) = 0.5, where B is either "AG had a gambling problem and owed money" or "somebody said that AG had a gambling problem" Commented May 4, 2023 at 5:20
• or perhaps P(A & B | B*) where B is "AG had a gambling problem" and "B* is somebody believes AG had a gambling problem". I still upvoted this answer anyway Commented May 4, 2023 at 5:22
• @BugCatcherNakata, why would you think it's p(A|B) rather than p(B|A)? You are evaluating the two propositions together and each gives support to the other. Commented May 4, 2023 at 9:33
• I think the actual proper way of phrasing this is such: p(B) = 0.3 is only accurate prior to the events. It's intuitive to think it doesn't change upon getting the news of her death, because it only describes events that happened earlier in time, but actually you have the event D: Aunt Gertie has died under mysterious circumstances. So now you need new values p(A | D), and p(B | D), which apparently are larger than 0.5 so you arrive at p(A&B | D) = 0.5 without conjunction fallacy. Commented May 4, 2023 at 10:53
• This isn't "an example of how p(A&B) can be greater than p(A) or p(B)" at all. It's true that a person may estimate p(A) = 0.3 and p(B) = 0.3, and then, after thinking things over some more, estimate p(A&B) = 0.5. But now that this person has come to estimate that p(A&B) = 0.5, they must abandon their previous estimates of p(A) and p(B) and change to new estimates that are both greater than or equal to 0.5. Commented May 4, 2023 at 16:27