She does.  She wasn't that awake anyways.

"So what I think #6 has to be pointing at is, okay, I'm going to just say this and hope your mind doesn't collapse because you're one of the Special Girls, it's pointing at how Asmodeans sort of go around trying to convince themselves really hard that sure they want to go to Hell and actually they don't and this is not a way a dath ilani would ever think.  Ever.  #6 is going to be the fragment of Law that prohibits that."

"It's going to mean a lot more than that, unless we've already lost our game to Keltham completely.  And if he was planning to convince everyone around him to turn traitor, if he was directly playing against the real Cheliax, then he wouldn't give us advance warning, he'd just hit everyone in the class with the full lecture."

"How does the Law of Probability tell people not to do what you just said?  It doesn't seem connected to any of the quantities we've been using before now.  If you understand the meaning of the words, tell me about the math, connect it to anything and I'll be able to unravel everything else from there and maybe in less than a minute."

Apparently 'Asmodeans are lying to themselves about wanting to go to Hell' is just a fine thing to say to Asmodia and she doesn't argue with it.

Ione really wishes she knew what was going on here.

"I think problems #6 and #7 are meant to be harder than the first five, probably require the first five to get.  I only got #2 on my own and I've been too distracted by watching you figure out the others.  I'm guessing you don't want me to take the time to figure them out on my own -"

"#1 is about how there's all the different things a hypothesis could predict, and all the probabilities it puts on them have to sum to 1, if there's like five different things that could happen, you can't say 40% for each, even if you come up with a really clever explanation for each of them, if all the explanations sound equally clever they get the same probability which is 20% each and that's just the same as going 'I don't know' about each of them."

"#2 you got, it sounds like, it just says that if something is half as likely as something else, you need evidence twice as likely as if not to start believing it, and if it's a thousandth the evidence has to be a thousand times as likely -"

"Write it down so I can think about it.  Include the formulas so my brain can connect this to the math."

"That's going to take me five fucking minutes!  I've only got twenty left on the hour!"

"I'm not going to be able to tell you how #6 is about the probability of anything conditional on anything until you spend those five minutes.  Maybe you want to go it alone."

"#6 is about - the probability of something, conditional on your trying to convince yourself of it.  It has to be zero."

"I'm trying to convince myself that I lack a million gold pieces.  Oh look, now I have a million -"

"Yes I know it's stupid I just thought if I said it maybe you'd correct me and your correction would actually be helpful."

"Gonna have to be less stupid first."

"If you see yourself trying to convince yourself of something, it can't be evidence, because you'd see that with the same probability in, the Ordinary world, the Conspiracy world, any world you're inside, you can still try to convince yourself of things."

"So just wanting to convince yourself of something isn't evidence by itself, sure, good job not getting smitten by Asmodeus there and aren't we all glad Security isn't torturing us for saying it.  What if you come up with, like, actual arguments?"

"You can come up with arguments in any world, so coming up with arguments for anything isn't evidence for it."

"I want to believe you about that, but unfortunately, it sounds like neither my wanting to believe you, nor the fact that you came up with any arguments for it, should ever convince me of anything.  By the way, can I have all your money?  And before you try to convince yourself otherwise, keep in mind that you should never be able to convince yourself of anything."

"No!  You can never expect to convince yourself of anything.  You can convince yourself of something, you just can't expect to.  The word 'expect' has to be the key because if we take it out the claim is wrong."

"Well, in the context of the Law of Probability, expect should mean predict, right?  You can't predict yourself convincing yourself of something?"

"So you can't predict seeing evidence that - where after you see it, you believe - good work Ione, shut up Ione, I think that's enough and I can take it from there -"

Asmodia is now completely ignoring Ione and scribbling something about P(observation1 ◁ hypothesis1) and P(hypothesis1 ◁ observation1) and P(observation1) and P(hypothesis1) and P(~observation1) and what appears, roughly, to be every possible combination of the symbols in Keltham's language, which, once exhausted, she starts multiplying and adding together in quantities like P(hypothesis1 ◁ observation1) * P(observation1) + P(hypothesis1 ◁ ~observation1) * P(~observation1) and now she's expanding the definitions and trying to prove things.

Right.  Ione will just try to work out some of the other problems on her own, then.

It does, in fact, take her only a minute.

(P(hypothesis1 ◁ observation1) * P(observation1)) + (P(hypothesis1 ◁ ~observation1) * P(~observation1)) = P(hypothesis1).

If you add up how much you'd believe in something, given a piece of evidence, times your chance of seeing that evidence, with how much you'd believe something, if you saw not that evidence, times the chance of seeing not that evidence, it's just the same probability you started with.  You can end up convinced of things, you just can't expect to be convinced of them.  Any time you go look for something, that you're hoping will convince you of something, there's a counterbalancing chance you won't find it and then the math requires you to end up believing less.  On net it all balances out.

"I've got #6, no time to explain it though, tell me about #7, not in math I have the math now just what it does it mean."

"I want that headband."

"Even I won't have it for very much longer at all so shut up and talk."

It couldn't literally be the Crown of Infernal Majesty, could it?  That's just flatly impossible even on Project Lawful.

"Well, if we're going full-out on heresy - and hoping that, for some reason, you and not just I survive that experience - then, the same way #6 is about how you shouldn't be able to convince yourself of things - #7 is about how you shouldn't believe the Church when they try to convince you about it."

"We've already been through this you can end up convinced of things you just can't expect to be convinced of them.  Why can't you expect the Church to convince you of anything?  They're allowed to know all sorts of things you don't, they could have all sorts of properly convincing evidence you haven't seen."