Thanks for the example. Notice that given the same evidence, you increased P (B,L) from 1/4 to 5/12. Notice that P (L) doesn’t increase. But now the problem comes in treating these as credences. Once you do that, the contradiction is clear. Given the same evidence, you become more confident in Linda being a Librarian and a Banker. You do not become more confident that Linda is a Librarian. But Linda being a librarian and a banker implies her being a librarian. This should imply an increased credence in the latter logically. But it doesn’t in bayesianism.

How do you define the difference between a probability and a "credence", what exactly do you mean by that term. In Bayesianism probabilities can be attached to the truth of a hypothesis. In Bayesianism a probability represents your "confidence", i.e. degree of belief/state of knowledge.

By credence I simply mean confidence/belief/degree of beliefs. Exactly what you stated. The point is that if you take probabilities as credences, you are updating them in a way that violates the logical implication of sentences. When I say that I am confident in X and Y, it implies that I am confident in X and I am confident in Y. I cannot be confident in X and Y while only being confident in X. Similarly, I cannot increase my confidence in X and Y while only increasing my confidence in X (from a logical perspective).

"Given the same evidence, you become more confident in Linda being a Librarian and a Banker. " and less confident that Linda is a librarian and not a banker?

Note by the way that your example does not violate the probability calculus. It violates logic. You could escape the logic violation by increasing your credence in Linda being a librarian. However, that will ultimately lead to a violation of probability axioms for the reasons mentioned in the OP. So either way, it leads to issues

No what I’m saying is that if you become more confident in Linda being a librarian and a banker, you must become more confident that Linda is a librarian. Because that is what “I am more confident that Linda is a banker and a librarian” as a matter of logic implies

We are back where we started. Bayesian probability is not logic, it is logic extended so that it can deal with reasoning under uncertainty. They are not the same thing, and the problem here is with your logic.

I am not saying that Bayesian probability is logic. I am saying that it violates logic. Now, you can adopt a system that violates our normal sense of logic l guess, but I don’t think most Bayesians would want to accept this. If you accept that then go ahead. But this is not a matter of my logic. “I am more confident in X and Y” implies “I am more confident in X and I am more confident in Y” as a matter of logic/meaning. This is not my subjective opinion. That’s the implication in English.

To see this with a simpler example: If I love one of my children more but not the other, I don’t say I love my children more. Every time someone says “I love my children more” they mean they love each more

Common sense tells me that Linda entering a bank tells me nothing about whether she is a librarian. So if logic tells you that it should make you more or less confident on that question, then your logic is wrong. What does your common sense tell you?

I agree completely. But that’s not what my logic said. Common sense also tells me that I am not going to be more confident in Linda being a librarian and a banker from seeing she goes to a bank. Notice that the Bayesian does become more confident that she is a banker and a librarian in your example. That’s where the issues come in. I do not

So again, I did not say you should increase your credence in her being a librarian just from the evidence so that’s a strawman. I’m saying that if you increase your credence in her being a librarian and a banker, you must also increase your credence in her being a librarian to prevent violating logic.

@Stella "Notice that the Bayesian does become more confident that she is a banker and a librarian in your example." yes, but in my example P(B,L|E) = P(B|E), it just tells us that we are more confident that she is a banker, it doesn't give us any reason to be more (or less) confident that she is a librarian. The increase in P(B,L|E) is solely because of the increase in P(B|E).

"I agree completely. But that’s not what my logic said" right, so your logic is inconsistent with your common sense. That should tell you something about either your logic or your common sense. IMHO the problem is obviously the former.

"So again, I did not say you should increase your credence in her being a librarian just from the evidence " what else do we have?

I know what the increase is because of. Your reasoning explains why it doesn’t violate the probability calculus. The probabilistic statements in and of themselves have no issues with them. The problems come with treating them as credences as I already explained. It’s a bit like having a subset of a set. You cannot increase the elements of the subset without increasing the set. The way to escape this logic violation is by simply not treating probabilities as credences which is what Deutsch recommends

And again, no, my logic (and it’s not mine, it’s logic in general) is not inconsistent. I did not say that I should increase my credence in Linda being a librarian. I said that if one increases the credence of linda being a banker and a librarian, then they should increase their credence in her being a librarian (for otherwise it violates logic)

How many times does it need to be said that as soon as you start reasoning under uncertainty, standard logic is inadequate. That is why we have probability.

No. Probability does not violate logic. Treating probabilities as credence is not part of probability theory. In general, if a reasoning scheme violates logic, the problem is not with logic, it’s with the reasoning scheme. For otherwise you are employing a system that is incoherent

Anyways, I don’t want to have a very long winded discussion about this here. If you’re interested, I recommend that you check out the video that I linked. He goes over why it doesn’t make sense. He’s about to finish a paper against Bayesianism soon as well. And not to fanboy, but he’s not your average thinker.

"I said that if one increases the credence of linda being a banker and a librarian, then they should increase their credence in her being a librarian " no you keep saying that, but it obviously isn't true. It is true in logic that if we know she is a banker and a librarian then that implies that she is a librarian (pretty obvious). As soon as you introduce credances, rather than logic, that is no longer the case. See the answer by bumble.

"He goes over why it doesn’t make sense." according to Bumbles answer, Carrol spotted his error and he had no answer. I may watch it, but it doesn't sound a promising use of my time. Academics have a tendency to "go emeritus" and ruin their good reputations with fringe theories outside their area of genuine expertise. I hope that isn't the case here.