Mari glances at her textbook, turns to Ellen:

"I was looking ahead to different ways of casting the same spell that are logically the same, and it referred me back to something we covered Thursday when you weren't here. How can a negative statement be the same as a positive one?"

She points at a passage in the textbook.

Ellen leans over to look.

"That's the contrapositive, not the negative. If A implies B, then obviously not B implies not A and vice versa."

It is not obvious.

"A and B don't mean anything."

Mari looks around for help, sees Franklin two seats over.

Oh, he is supposed to contribute to discussion and not just read random parts of the textbook, yes. "Stuck on the contrapositive? It could help to try phrasing the implications as disjunctions, which makes it clear that they're equivalent. Or make up a concrete example, some people do that."

"A concrete example, please. Ellen's no help."

"I suppose I could give you an example. There is the one about crows, which gets into ... but the fact that A and B could be anything matters. That's the point of logic. It isn't about crows, it's about everything."

Ellen, unpersuaded, looks back at Franklin.

"Tell me about crows."

"Oh, yes, the crow joke. Consider the statement 'all crows are black'. I don't know if that's actually true or not but it doesn't matter. You can rephrase that statement as 'all nonblack things are not crows', and it's the same statement, because it's false if there's such a thing as a nonblack crow and true otherwise. Uh, and I called it a joke because you can say 'I'm going to do an experiment on whether all crows are black' and poke a random nonblack thing and say 'Oh look it isn't a crow, I made a correct prediction.'" 

"To put it formally, 'If something is a crow then it is black' — 'if A then B" — is equivalent to 'if something is not black then it is not a crow' — If not B then not A — its contrapositive. 

"What Franklin calls a joke isn't exactly, because the logic is right. A sheet of white paper is a not-black thing that is not a crow. A not-black thing that is a crow would be evidence — conclusive evidence — against the proposition that 'if something is not black then it is not a crow.' So the fact that sheet of paper isn't a crow is evidence for the proposition, and that's equivalent to the proposition that if something is a crow then it is black.

"So a sheet of white paper is evidence that all crows are black."

Mari does not look amused, or even less puzzled, so Ellen turns to Franklin.

"Isn't it?"

"Technically yes," he says cheerfully, "but it's really bad evidence. It's like claiming there are five thousand beans in a bucket and supporting it by pulling out ten beans and going 'Look, it's at least ten, and it can't be more than a billion, so we've ruled out a whole lot of numbers and not five thousand.'"

Ellen grins at him:

"Right. It's evidence, but very weak evidence. That's what I told Apa when he gave me that puzzle, back when I was little and he was teaching me logic. It goes asymptotically to zero as the number of things in the world goes to infinity, and for all we know the universe might have an infinite number of things in it."

She turns back to Mari:

"When I was little, Apa used cards to teach me about logic. Suppose we could represent everything in the world with twelve cards, each of which had 'black' or 'white' on the back and 'crow' or 'not-crow' on the face. The cards are lying face down on the table. We turn over all the white cards and all of them say 'not-crow.' Now we know that all the cards with "crow" must be black. If turning over all the white cards proves that, then the first white card you turn over is evidence for it.

"But the more white cards there are, the less evidence one card gives you, and there are a lot of not-crows in the world.  So it isn't just a logic puzzle, it's a pointer to how induction, the logical version not mathematical induction, works, and if you follow that out you get into probability theory."

She turns back to Franklin:

""I'm Ürömi Heléna, but everyone calls me Ellen."

"Pleased to meet you; I'm Franklin Leister. What language is your name from? I don't recognize it."

"Magyar. It isn't Germanic or Slavic or Romance or even Indo-European, so it isn't surprising that you didn't recognize it.

"'Ürömi' means that I'm from Üröm. It's a town outside Budapest. She's Budi Maria because she's from Buda. And I wasn't here first week and have to leave pretty soon because I'm actually in the other section. On my schedule this hour is maintenance but Mari got someone else to do it for me so I could attend the first half of the Tuesday class with her. That makes my section even more boring than it was but I can always spend the time doing the homework for one of my other classes or reading a book I found in the library about linear algebra because I couldn't fit that class into my schedule."

Those are excellent uses of time. Also she's an indie who's friends with an enclaver and this is the point in the conversation where Franklin is supposed to politely but subtly reveal that he's an enclaver but having been coached on doing that doesn't actually mean he can do it correctly. 

"That explains why I didn't see you last week. Linear algebra sounds interesting. I didn't get it this term either but I got trigonometry, mathematical models for artificing, and introduction to mechanics, so pretty good set."

"I have mathematical models for artificing too. Apa is an artificer, a jeweler. He made Narya for me — she helps me burn things." She holds up her right hand with the ring on it. "I've done a little, but only jewelry. I thought I might learn useful things here if I wanted to make more, maybe even things Apa doesn't know so I could teach him something instead of the other way around. But probably not — he's very good and he knows a lot more math than I do."

"Don't be too sure, the library is very big and there are probably things in it nobody knows about. That is a very nice ring, though. Is jewelry your affinity as well, or just his?" He's not kidding about the ring; that's some quality metalwork.

"Apa's affinity is fire, like mine; fire is useful for making jewelry. Anya is a weaver; they make barriers of woven fire to protect our house. I've been hoping to find a weaver here I could work with. I don't think the classes are going to teach me things about fire or artificing that Apa doesn't know but the library might, or I might get something useful from the void. Mari might be able to find me a book — she is good at finding things.

"My classes so far are not very good; I think I could write a better textbook than the one we have for incanting and I'm sure Apa and Anya could. But this class might be better, when it finishes with obvious things about logic and tells us how to use it for spellcasting. It never occurred to me that you could use logical equivalence to find ways of revising a spell to cast better."

"Yeah, I'm excited to get to the practical things in this class. I like incantations a lot less than artificing, but maybe I'll like them more with additional math."

Sándor is INTERESTED in this conversation. Most people do not have interesting conversations but Franklin and Ellen do have interesting conversations, so he's eavesdropping. He doesn't have anything clever to say, but he's happy to listen to them talk.

Ellen, of course, has never put the obvious question to their new acquaintance, and Mari can think of a lot of things more interesting to talk about than math. More useful, too.

"Ellen told you we were from Buda but she never asked you where you were from."

Oh good, being asked straight out makes it so much easier to avoid sounding like a person who takes every opportunity to bring it up. "I'm from Boston."

"I've heard good things about Boston. Kevin really knows what he is doing, unlike most of the freshmen maintenance people. When letters go out at the end of the year I'm planning to write my father suggesting that he should find someone in the enclave with a suitable affinity or someone good outside it who wants to train for maintenance.

"András takes good care of Buda but he's getting old, could use more help, and  won't live forever."

"Kevin's great! His affinity's plumbing and he's been following our maintenance lead around since he was six. Though I don't know how much of that is the plumbing affinity and how much is the mad genius." He considers asking what her father's role is, since it sounds like he's pretty high up, and decides the risk of being weird outweighs the value of asking a question in keeping the conversation going smoothly.

"I spent a lot of time in the last year with András. Mother says you can't supervise people unless you know how to do what they are doing and father says the Scholomance is a stress test for people and I want to make sure Buda ends up with another competent maintenance person, someone we can trust. My guess about what happened to Chicago is that they didn't."

"I don't know if we'll ever find out for sure what happened to Chicago, but finding people you can trust is definitely important, in here and outside. I bet Kevin will end up knowing who's maintenance track and good."

Apparently some enclaves understand the importance of trust better than others; he has heard some things about for example Shanghai that make him especially glad to be from Boston. Mari seems pretty sane so far about things that aren't math, but Marcy's a better judge of character than he is.

"If he is willing to talk to me about it, not now but in a couple of years when he will know more, I would be grateful."

She is not saying that one of the things he will know more about is which of the maintenance track kids are still alive.

The lecture has started but Mari thinks Ellen plus the textbook is probably a better way of learning the material than paying attention to it. She looks around the room to see who else might be worth talking with, thinks about what she can do to get closer to Boston; they feel more like her sort of people than the bigger enclaves, although London might be an exception.

Franklin had a lot of practice listening to lectures on things he's already read but that might unexpectedly get interesting (or where he might get called on and asked a question, though that's less of a thing in here); he listens with one ear and contemplates ways logic might be useful in spell design.

Ellen listens for a few minutes, concludes that the lecture is not saying anything she didn't know when she was eight. The book might be better, at least the later chapters, so she reads that.

She notices Mari is taking notes. Ellen has been told that tests are sometimes on things of no importance at all that are in the lecture and that's why you have to listen and take notes instead of just reading the text, which is much faster. If Mari takes notes and Ellen learns anything new from the book,  Ellen doesn't have to attend her section on Thursday and can attend Mari's Magical History class instead. It might be interesting and if any mals show up she will be there to protect Mari. Things will be more dangerous next semester; they have to figure out a way of getting all their classes together, or as  many as they can. If they meet as soon as they have their homeroom assignments and she gets the number of Mari's homeroom,  she can pick up her schedule in her homeroom, run over to Mari's, and ...

Franklin ends up with two columns of notes, one from the lecture and one of speculation and diagrams. Eventually they get the homework (all the even problems on page 68-70) and he writes that at the top and sticks the paper in his textbook at the appropriate page.

Ellen will have had to leave in the middle, but presumably Mari is still around?

The class ending, Mari catches Franklin's eye.

"I'll be meeting Ellen for lunch, if you want to continue your mathematical conversations. I don't mind — it's not a language the school will try to give me spells in."

"I'd like that; I was actually thinking of inviting you both to eat with us."

Because he wants to talk about math with Ellen and Marcy would like to meet Mari. His own impressions from eating with Buda would also be somewhat useful, but Marcy is good at that sort of thing and also somehow likes it. 

Two minds with but a single thought; Ellen and Franklin can talk about math while she and Marcy get to know each other.

"Thank you. Yes."