"Chaotic Good, huh."

"YES."

"Cookie of +4 Intelligence lasting 1 hour."

"Nope."

Pilar contemplates how this thought occurred to Keltham within seconds, and to her not at all over days.  It probably has something to do with dath ilanism, but, how would there be a Law for something like that?

"Then I suppose 'Project Lawful' will continue on, with snacks catered by Cayden Cailean.  Thank you, Cayden Cailean, the magnitude of your contribution there seems difficult to understate.  Though I do appreciate the Pilar save during the attack, to be clear."

"I'm going to take a brief break and then get back to math.  Fifteen minutes, say."  He's got to use the washroom, for one thing.

Actually, he should take a bite of the cookie, just in case it contains edible knowledge?  Nope.  It's a good cookie, though.

(Keltham departs.)

"Do you remember Elysium?" Meritxell asks Pilar. "Did they explain anything there?"

"Mostly they - I thought at the time - tried really hard to talk me into staying.  At the very end after I said no, they told me that the whole point was so that I'd be, certain of myself and my choices, I forget how they put it exactly.  Said I was going to be used for Lord Asmodeus's interests, not against Him, because Good would mostly rather not use people against the ones they're truly loyal to.  The Grand High Priestess thought that someone like them might maybe be telling the truth about that, to someone like me, but it doesn't mean Cayden is on Asmodeus's side, he could be plotting to destroy Cheliax and then I prevent a new Worldwound from opening in the center."

Some other things happened since then that do look more like Cayden Cailean cooperating.  Is Pilar supposed to say anything about those?

(These girls don't need to know the other things.)

Safe topics of conversation are hard to find, so the girls mostly review their notes while they wait for Keltham to return.

Keltham is now here!  Perhaps he was really always with you all along.

So, math, yeah.

When we'd previously seen our plucky heroines, they had just realized that everything, or at least, all the positive real numbers, can be seen as being made out of continuous quantities of 2s being multiplied together.  A 3 is a bit more than 19/12 of a 2, so 1/12 more than 1-and-a-half 2s.  So diminishing something via multiplying by 8/9, is taking a bit more than 1/6 of a 2 out of its bag.

Predictions chain together by multiplication.  If you spin a fair coin once, the probability of it coming up Abrogail is 1/2.  Knowing whether it came up Abrogail or Text doesn't change the probability on the next spin, so the chance of two sequential Abrogails is 1/4, the chance of three such is 1/8.

Each time you say 1/2, and the event happens, that's like taking another 2 out of the bag containing your total prediction over all the events.  Where, to be clear, your bag started with zero 2s in it, or probability 1.  After guessing 50/100 three times at three fair coinflips, your bag would contain -3 2s or a probability of 1/8.

What if you predicted Abrogail with 2/3 probability instead, on one spin?  Well, if the coin comes up Abrogail, good for you, you've only lost - how many 2s, roughly?  Raise your hand once you've got an estimate.

(Keltham may eventually notice that all his students assiduously say 'the Queen' even when repeating after him otherwise.)

How many twos in 2 - one. How many 2s in 3 - 19/12ths, ish. So 1 - 19/12ths = -7/12th. You've lost seven twelfths of a two. 

"I'm confused about what losing twos corresponds to. You lose them when you're right!"

"Hold that thought just a little longer."

"If you predict Queen with probability 2/3, then if you get Text instead, as happens half the time, you thereby predicted that with probability 1/3.  Though actually we'd say that it's a tiny bit less than 2/3 and 1/3 because maybe the coin could land on its edge or just mysteriously vanish, but leaving that aside for now.  If you predict Queen with 2/3 probability and the coin comes up Text, how many 2s do you lose?"

"...nineteen twelfths?" says Tonia.

"Yup.  So if you lose seven-twelfths 50 out of 100 times, and nineteen-twelfths 50 out of 100 times, how many 2s do you lose on average each round?"

"Thirteen twelfths."

"Better or worse than if you'd just predicted 1/2 every round?"

"That depends on whether losing twos is good or bad! But presumably it's meant to be bad since this is a dumber way to guess."

"You lose 12/12ths of a 2 each time you predict 1/2, so you lose 1/12th of a 2 less that way.  Which is better if losing 2s is bad."

"So would you agree that this scoring function..."

"Gives you more points, or rather, has you lose fewer 2s, the more probability you assign to whatever happened?"

"Gives you the same final number of points, or 2s lost, whether you're predicting two coinspins at once, or predicting them separately in different rounds?"

"And, at least in this particular example we checked, it wasn't possible to expect to score more average points, or lose fewer 2s, by giving an answer other than reality's answer for how often something happens?"

Yep! That does seem true!

Why does that still feel like a surprise even though she predicted it way earlier?  "And it's the only scoring function like that which can possibly exist," Asmodia states.

"Not exactly.  Counting lost 3s will also work.  Or counting lost 5s.  But that just scales the number of points you win.  There's around nineteen twelfths of a 2 in a 3, so if you know how many 3s you lost, you can convert to how many 2s you lost.  It's not so much that there's only one function, as that all the functions like that, are basically doing the same thing and have outputs that are trivial to convert back and forth."

"The Law, in this case, is not an exact function or an exact number of points, it's a structure such that every solution shares that structure and does almost exactly the same thing.  Like a simpler and clearer version of the way that lots of logics are ultimately equivalent to first-order logic in what they end up deriving."

"And you can - use this to figure out who's the best at predicting things?"

"If everyone is predicting the same questions using the same knowledge.  If your sole goal is to end with as many 2s as possible, and you get to pick whether or not to play the game, the only winning move is not to play, so you can end up with the same zero 2s you started with."

"Otherwise you start with nothing, and then lose more every time you try to predict anything that isn't absolutely certain, and the best you can do is losing the least 2s possible, which will always still involve losing some, it's just that if you don't match reality you do even worse.  So, yes, if that was a game with, like, actual penalties, and no other reward for playing it, nobody would play that game if they had a choice."