"Now there's something that might be true everywhere, which, you might think, would make it an important fact; and if it's important, then it's important to know exactly what it is, that's true everywhere.  So what do you mean, when you say that one equals one?"

"I mean, I'm not at all sure it's an important fact, it's mostly just saying that we defined equals, and the way we defined equals, the things on both sides of it are the same, and things are the same as themselves. But it does seem like it'd be true everywhere."

"It's something of a mischievous question, but mischief is also important in learning, so I'll ask.  One common way to ask what something means, is to ask what you experience when that proposition is true.  If you say 'water is liquid', for example, and I ask you what that means, you might tell me that 'water' describes the clear stuff inside a glass you hold up, and that 'liquid' means that a substance tries to cling to itself but has no set shape, and so conforms itself to the shape of its container; and when I see you pour the water from the glass, onto the floor, I should expect to see it spread out across the floor, while still locally clinging to itself and staying in contiguous puddles.  Now, what do you see when one equals one?"

This is SO STRESSFUL. 

"If you use a spell to duplicate something it'll have all the same properties as the original."

"You don't see anything, it's just a definition."

"Things ...exist at all? ...that'd imply it's not true in the Maelstrom, though -"

"If you try to do math and you assume it, your math will keep making sense."

"Positive reinforcement for continuing to be wrong instead of quiet!  Now, really I only told you half of a proverb, just then.  The real proverb says that to ask what a proposition means, we ask what you should see that's different, depending on whether the proposition is true or false.  Yesterday, water was liquid; tomorrow, water won't be liquid.  How are yesterday and tomorrow different?  Well, yesterday, when I poured water from the cup, it spread out over the floor, in puddles where it clung to itself.  So if tomorrow, I pour out water, and it stays in the same shape as when it left the cup, then tomorrow, 'water is liquid' is false.  Yesterday, you used a spell to duplicate something - let's say a small flower, a dandelion - and the duplicate dandelion seemed just the same as the original.  Tomorrow, you use a spell to duplicate a dandelion, and the resulting flower is blue instead of yellow.  Is one no longer equal to one, tomorrow?  Yesterday, one equaled one; tomorrow, it won't.  What will you see tomorrow that's different from yesterday?"

AAAAAAAHHHHHHHH

"I...don't think tomorrow sustains conscious life that's observing things."

"That's a cop-out, whatever, you're scrying the place where this is true."

"I still think - you try to do math, and your math doesn't work anymore."

"'Tomorrow, it won't' can't be true."

"Can't be true?  Well, if it can't be true that something is false, that would make it a necessary truth, I suppose.  Dath ilan might imagine that it'd managed to deduce what was true in all planes, if it couldn't be false.  But if for that reason you can't tell me what you expect to see, what will happen to you, as a consequence, does your necessary truth really mean anything?  After all, if it meant only some things could happen to you, but not others, it would cease to be true if you traveled to a plane where other things happened to you instead.  So whatever is true no matter what happens to you, never helps you figure out what will happen to you; and, therefore, is absolutely useless.  Now I have just proven to you that all necessary truths are absolutely useless.  And some of you have suggested that math is made of necessary truths.  So have you just proved that math is absolutely useless, since, whatever could happen to you, that wouldn't make math false, and therefore math can never say anything about what will happen?"

Otolmens is watching this classroom SO HARD right now.  The mortal had BETTER not be going anywhere weird with this.  Physics disasters are BAD but math disasters are SO MUCH WORSE.

 "You can use math to derive how to move a spell, and then the spell works or it doesn't."

"And target a catapult."

"And build a bridge."

"If I have one hat and one head, one equalling one means that after I have put the hat on the head there won't be any spare hats or any spare heads. It seems - possible to imagine observing instead that if you have one of something and one of another thing it doesn't mean they match up to each other with none going spare."

The group is divided on whether this is in fact possible to imagine. 

"Just to check, Carissa-Sevar, can you describe to me in additional detail what you'd imagine it to be like to observe that?"

Keltham has had a pretty strange couple of days and is, in fact, less sure of some things than he used to be.

"I mean if it happened I'd assume someone was messing with my head, or I was dreaming, but - well, imagine instead we have five weapons and five spots on a weapons rack, it's not hard to imagine that you put a weapon in each slot but then there's still one slot left over, and you go back and count and there are five slots, one of them empty, and you count the weapons and there are five, all in a rack. It's harder to imagine with one because in dreams sometimes counting to five doesn't quite work but counting to one still does."

"Saying those words out loud is one thing; could you create a detailed illusion of it happening?"

"Not a motionless one. I bet I could - do one that took advantage of how people can't look at a whole landscape at the same time and changed where they weren't looking at it. You'd just be tricking them, though, even if you did it perfectly, you wouldn't have changed what one equalled."

"If it's not possible to create an illusion of something being false, you might not need to travel to other planes to guess it would be true there.  But I offer the same mischievous objection as before:  To say that you can't make an illusion of something, doesn't narrow down what kind of future follows from the past - we can make an illusion of a plane where jumping up puts you at the bottom of an ocean, instead of off the ground.  Even if in all previous history, jumping just lifted you off the ground a bit, we can make a detailed illusion of a world where that happens the first trillion times, and on the trillion-and-first time, jumping teleports you under the ocean instead.  So if math is about truths we can't make even an illusion deny - then why is math any good for building bridges?  We can make an illusion of a bridge falling down."

They are so confused and varying degrees of distressed about it.

"Actual bridges fall down more if you did the math wrong."

"Making an illusion of casting a spell isn't - the same thing as actually casting the spell - sometimes the way to pass the test is to be able to actually do it, not just to make it look like you can -"

(Keltham does not have the faintest chance of noticing that somebody who did well in a Chelish academy is leaking tiny signs of distress past their routine concealment thereof.)

"Well, I think I've created enough explicit confusion that you'll notice learning something that makes you feel less confused," Keltham says, and then makes a brief sad face about how this snappy statement sounds so ridiculously long in Taldane.  What kind of language makes confusion a three-syllable word, anyways?  One that has no idea what its nearly neural-level cognitive primitives are, presumably.

Keltham goes to the improvised whiteboard, and starts drawing squares and triangles, red and green, large and small, inside some bigger blue circles.

"Consider each of these blue circles and their contents as depicting - we would say in Baseline - possible worlds.  By possible, I don't mean it's especially likely that you'll find yourselves in them; these possible worlds I'm depicting are much too tiny to support intelligent life.  They've only got a few squares and triangles inside.  By 'possible' I do mean that one could make a fully detailed illusion of the world, given the ability to cast arbitrarily large illusions; my using markers to draw a world in complete detail similarly shows that world to be 'possible'.  Now, consider these propositions -"

Keltham writes, in black marker:

Z.  All triangular things are red.
H.  All red things are large.
Q.  All triangular things are large.

(Dath ilan has some different conventions for symbols to use in equations, for example, all the symbols should be as topologically and typographically distinct as possible.)

"As you can see, I have shown worlds where Z is true, and worlds where Z is false.  I have shown worlds where H is true, and worlds where H is false.  I have shown worlds where Q is true, and worlds where Q is false.  None of Z, H, and Q, then, are necessary truths, nor necessary falsehoods; for they are all true in some illusionable worlds, and false in others.  Then is there anything useful here for math, logic, and necessity to say?"

It takes a couple of minutes of muttering and frowning and guessing "no?" and "there are triangular things in all the world- oh, no, not that one -" before - "well, if Z and H are true, then Q is, you can't have any with Z and H but not Q."

That took them longer than Keltham expected.  He frankly would not have expected that all the exercises he had to do as a kid were, like, required for getting that point instantaneously as an adult.  Not to mention, they know topology but not predicate logic?  Right, because you need topology for spells, but not, apparently, predicate logic.  If he'd realized he sure would've told them to learn that in yesterday evening's afterhours instead of calculus.  Oh, well, he'll plunge on and see how far he gets.

Keltham goes to the whiteboard and draws some conscious observers inside his blue circle-worlds.  Much as some other world might indicate observers with smiley-faces, dath ilani convention calls for Keltham to draw a number of glaring eyes inside his worlds, creating a tableau that somebody from a differently-troped world might regard as eldritch.

"Well, now I've put some conscious observers inside these worlds!  Not that my tiny drawings embody real experiences, of course, they're not detailed enough drawings for that; so now these pictures are no longer being drawn in full detail, which is something we might need to watch out for if this was a more complicated debate about conscious experiences."

"Some of these observers, in the worlds where Z is actually true, might see twenty triangles being red, and zero triangles being green, and hypothesize a general law: all triangles are red.  They might be able to deduce, without having to actually scry into other planes, that Z was not a necessary truth; they might be able to cast illusions, draw on walls, or just use their imaginations to see that.  So they would not be certain that all triangles are red.  For all they know, the world might up and present them one day with a green triangle.  But the next time they saw a triangle, even if their world made them slower to see colors than shapes, they could guess even in advance of observing; they would guess the triangle was red."

"Let's also suppose that you can tell whether an object is small or large, but it's an expensive measurement; an observer has to actually wander over close to the object, to determine its size; because if they're looking at the object from a distance, they're not sure if it's nearby and small, or large and far away.  These observers have only one eye, as you can see; no binocular vision for tracking distances.  Let's say they have to pay one labor... one silver piece each time they want to move to an object."

"In worlds where H is true, observers who pay to measure a few red things will find, that of all the red things they have measured, every one of those red things was large."

"Now let me ask again, in case anyone has seen the point before I speak it:  How can knowing necessary truths save you money?"

"Well, if you know that triangles are red, and that red things are large, then you don't have to go check the size of triangles."

"To state it precisely, some observers may have guessed the unnecessary truth that all triangles are red, observing the redness after the delay.  They may have separately guessed the unnecessary truth that all red things are large, after paying to measure some red things.  Maybe they've never measured any of the red things that were triangles! we can suppose for the sake of clarity.  Then the necessary truth, 'Q is true in all worlds where Z and H is true', can allow them to guess the unnecessary truth 'All triangles are large', which necessarily follows from other unnecessary truths they've guessed.  And even if they've never measured the size of a single triangle before, they can guess - though not know for certain - that every triangle they've seen was large, and that the next triangle they see will be large.  If it's the kind of knowledge that matters, but not enough that you need to be very sure of it, they could use that guess in place of paying a silver piece to measure it."

"Of course, it isn't a necessary truth that the observers are capable of figuring this all out - that they can operate the necessity, 'Z and H yield Q'.  We could draw an illusion of a world where the observers totally fail to figure that out.  It would still be true across all planes and all illusions that could ever be drawn in full detail, but the people in that illusion wouldn't know it."

"It isn't necessary that entities successfully operate universal necessities in order to see which new guesses must follow from old guesses, which means that some entities do better or worse at this than others.  This is true when considering all possible worlds as a whole, and also happens to be true within my homeworld, and almost certainly in this one."

"So now we shall turn to the question: suppose you were constructing a new entity from scratch.  How would you go about embedding in them an internal reflection of the interuniversal Law, the ability to operate necessary truths correctly...  No, sorry, that's probably too much of a leap to ask in one go.  Strike that, restart.  Suppose you were comparing two entities: how would you say that one was doing better or worse than the other at being Lawful in this exact sense?"

- Keltham hasn't noticed but his teaching style clearly has half the class extremely panicked. They are concealing it very well.

...it really seems bizarre, that you could teach Law this way, with trick questions and guessing games and strange rules about how you're supposed to volunteer wrong answers if you aren't sure you know the right one. It seems like the habits of mind that would teach are - well, does she actually think that it'd teach unLawful habits of mind, or just horrendously ill-advised ones, there is a difference -

- if you built a military out of Kelthams it would not be a very good military, which is a perfectly serviceable definition of Law, the discipline and coordination required to win wars. The Kelthams -- and, plausibly, the people taught like Keltham - would be wrong, a lot, out loud and cheerfully, they'd consider everything their business, they'd ask questions they shouldn't ask -

- he did behave differently with Contessa Lliratha, maybe there's a kind of distinction the mind can successfully maintain, irreverent in most contexts but deferential where it actually matters - but it seems like it would be hard to tell if someone will be deferential when it actually matters, if they've spent their entire life in contexts where it doesn't, not being sufficiently deferential at all -

"You could look at ...how good they were at making those guesses? How often when they guessed they were right, how often they missed a pattern..."

"Measuring how good people are at guessing final conclusions in reality - whether, when they say 'I assign 90% probability this triangle is large', the triangle is actually large 9 times out of 10 - sure is a metric of how much Law people contain and are using correctly!  But there's more than one kind of Law you need to build an agent, and the piece of Law we're trying to isolate is the one that's about using necessary truths correctly.  One way of looking at that part is that it's about which conclusions follow from which premises.  To demonstrate -"

Keltham has seen one or two fragments of algebra in his reading, enough that he has some idea of what Chelish algebra conventions look like.  Though it's a bit weird that they teach algebra without, like, teaching people what algebra means.  Hopefully it's not a piece of knowledge that's infohazardous here but not in dath ilan.

He sketches a series of equations:

Otolmens is now in EMERGENCY PANIC OVERDRIVE, which you would be able to distinguish from her usual state of being if you looked carefully.  This particular proof of an inconsistency in first-order arithmetic is safely flawed, but if the foreign mortal is plotting to produce a valid proof of inconsistency - why won't they move the mortal somewhere prophecy still works?

She can't trust Abadar anymore, fellow Lawful Neutral god or not.  Abadar might not be useful in this emergency even if she could trust Him; He's scarcely better at decoding mortal minds than Herself. 

Otolmens sends a message reading simply HELP, tagged with a location.

"Now I'm not so much asking 'What is the flaw in this proof?'," Keltham is saying, now that he's given the classroom the few required seconds to look over his derivations, "as asking, 'How would you go about finding the flaw, if you couldn't spot it at a glance or on your first try at looking?'"

Irori has never once received an emergency summons from Otolmens that was actually important.

He nonetheless maintains a habit of responding with alacrity, just in case.  The concept of 'anthropic selection' is not lost on him, and zero urgent summonses from Otolmens is not quite as reassuring as a mortal might think.

Yes?

You USED to be a MORTAL.  I request you to read this mortal's mind and inform me whether it is plotting to write down a series of VALID proof steps proving an inconsistency in first-order arithmetic.

Otolmens isn't sure, for obvious reasons of resulting inconsistency, but She suspects that She internally uses ordinal induction up to epsilon-zero.  They'd have to boot up Metatolmens to fix Her!

Ex-mortal or not, from where Irori truly stands far above Golarion and other places, it isn't easy for Him to look inside the mind of a mortal not pledged to Himself and praying.  Otolmens only needs to pay attention to relatively few things going on, inside the multiverse, and then She is a relatively materially-focused entity on top of that, designed to be able to check all the electrons in a room to make sure none of them have the wrong mass.  Irori, if He hasn't formed an avatar and sent it into the room, cannot read the writing on the whiteboard the way Otolmens can; He can barely tell that these souls are in a library surrounded by books.  He definitely can't hear the sounds, the pressure patterns transmitted through the air as vibrations.

Still, it is Otolmens who calls, and the mortal is more Lawful Neutral than usual even for those that register Lawful Neutral.

From the mortal's general spiritual posture, Irori can already guess what He'll see.  But just in case, Irori expends the energy to take a very brief look at the surface of the mortal's mind.  It's not as difficult as it would be at other times of this mortal's life, given his current endeavors.

...he's not planning to destroy mathematics.  He only intends to teach of his Way to others.

Irori shifts most of His delegated attention back to other aspects of His businesses, leaving only a tiny fragment to look at the Chelish place a bit longer.

...Irori shifts somewhat more of His attention back to that location.

Carissa feels that she could grasp what Keltham is pointing at a lot faster if she were reading his mind but that's disallowed, now, he's a fourth-circle caster and reasonably likely to notice. She can't even ask him whether it'd be all right if she read his mind because they haven't acknowledged mindreading to be a thing that magic can do.

It remains bizarre, to think that Law has anything to do with formal mathematical logic. You don't need to understand the gods to be Lawful, you just need to obey them. But - but Keltham's world is more Lawful than hers, and -

- so there's nothing heretical about the claim that humans are using a mediocre approximation of Law, which is a god-concept that doesn't mean quite what humans understand it to mean. And there's nothing heretical about the idea that humans ought to use the real thing, except that they're too stupid and limited to understand it, so they have to settle for their wrong approximations. And there's...nothing very heretical about the claim that, actually, there's a way to teach humans the real thing, despite their stupidity and limitations -- at least, to teach smart humans, to teach humans in Keltham's world with a median INT of 16 or 17, and the people in this room have a median INT of 16 or 17, so the people in this room can learn it. 

And the true structure of Law would be mathematical, because it's about - regularities, consistencies, treaties among the gods aren't promises so much as fundamental changes, becoming the kind of structure of which the promise is true, and there is, actually, an obvious parallel to math there, even if she can't properly articulate it. The way the gods are is inevitable; in many ways they vary much less than humans, because there is only one way to be right and many many ways to be wrong. 

And the gods wouldn't be very suited to figure out what math, specifically, to teach to humans, especially if it requires obnoxiously counterintuitive tactics like making everyone limp their way through the lesson guessing - and perhaps, too, this wouldn't even have been worth trying anywhere in the world until quite recently, you need a bunch of smart people in a room and Cheliax is the first society in recorded history to look for all their smart children and teach them math -

"Well, you'd know there has to be an error somewhere, since you got it wrong."

"You could - check each line and see where the error showed up first -"

"Check each line to see where the error showed up first?  How would you check a line for error?"

" - well, there's obviously a problem in the eighth line, where if you substitute in '1' for X and Y you've got the error already. And there's...not a problem in the seventh line, because that one comes out to 2*0 = 1*0. Which is true."

Keltham takes a quick look at the nametag of whoever that was.  Why the Chelians collectively aced this problem but not the predicate-logic one... presumably it's just down to more actual practice with algebra?

"Precisely.  If we substitute in 1 for x and y, and evaluate the left-hand sides and right-hand sides of each equation, we get the following assertions:"

"The tactics of algebra - like being allowed to add 3 to both sides of an equation - are meant to preserve truth, not create it from scratch.  If an equation starts out true, a tactic in algebra should not produce a false equation from that true equation."

"This way of thinking holds even if the elements of the equation refer to things in the outside world.  Let x be the number of people sitting in the brown chair, 2 as it happens, and let y be the number of people sitting in the red chair, currently 3.  It is then an unnecessary truth, not a necessary truth, that x + 1 = y, as I have defined those terms to refer to the outside world.  In our world, x + 1 = y evaluates to 3=3, which happens to be true; but if you cast an illusion showing two people sitting in the brown chair and two people sitting in the red chair, the equation in that world would evaluate to 3 = 2, which is false.  And if I said x + 10 = y, that would be an unnecessary falsehood; in our world it evaluates to the false statement 12 = 3."

"Now apply the rules of algebra, add 2 to both sides, and transform the first equation x + 1 = y to the new equation x + 3 = y + 2.  In our world, this evaluates to 5 = 5, which is again true.  If we apply the same tactic to x + 10 = y, it yields x + 12 = y + 2, which evaluates to 14=5, again false."

"We term a step of inference valid when it is truth-preserving; when it transforms true statements into only other true statements.  It doesn't have to preserve falsehood; multiplying both sides of an equation by zero will produce truth even where it didn't previously exist."

"What makes the tactic of adding 2 to both sides of an equation, allowed in math, is not that some Watcher or representative from Governance told you it was allowed."  This part got hammered into Keltham and his agemates a lot as a kid, so it was probably determined to be important in practice to emphasize??  "What makes it an allowed step is that, if you have two weights balanced on either side of a scales, and you add two identical rocks to both the left side and the right side, the scales will still balance after that."

"If you look back at the original flawed proof that 2=1, it goes from a true statement in step [7], to a false statement in step [8].  Then between [7] and [8] we must have applied some operation of inference which is not 'valid', which has the ability to take in a true statement and spit out a false statement.  This tactic was canceling the multiplication by (x - y) from both sides, which is to say, dividing both sides by x - y.  Dividing both sides of an equation by 2 is valid; if you have a scales in balance, and remove half the weight from each sides of a scale, it will still be in balance.  Here, we see that division by 0 is not valid, because it can produce falsehood from truth.  What makes division by 0 unlawful is not that your Watcher told you not to do it while doing algebra; it is that division by 0 is not generally truth-preserving.  We can find some equations that will still be true after dividing both sides by a term equal to 0, but it is not a safe step in general."

"Sorry if that part about Watchers seems overly obvious, by the way.  It's just that apparently human brains by default try to reuse the part of ourselves that learns from adults not to steal cookies outside of mealtimes or we'll get slapped on the wrist, in order to relate to the rules for manipulating necessary truths that existed outside the start of Time.  And these are actually quite different topics; like, rules change sometimes, when Legislators vote on them, but algebra doesn't.  So you want to be explicitly aware of the difference, and not go bugging adults to let you divide by zero just this once."

"So the argument is that part of Law is - the habits of mind so you only reason in truth-preserving ways?" Meritxell, who was also fastest on the algebra, says. 

"I am still not entirely sure what the word 'Lawful' means to y'all.  Multiple different words in my native language all come out as 'Lawful' in Taldane and I'm mostly running with those.  Cheliax is supposedly a 'Lawful' country, but the books are written with what look to me like appalling jumps of reasoning, and somebody seems to have taught y'all algebra without teaching you what math is or why it works.  But Lrilatha-whose-job-title-I-already-forgot is supposed to be more innately Lawful, and she did not talk with those appalling jumps in her reasoning.  Which suggests to me that the word 'Lawful' is translating to me mostly correctly, or that the concept I hear is at least a real part of what 'Lawfulness' is; and the humans here simply are not being taught about that part of Lawfulness, or how to flow along with it on purpose instead of by accident."

"That said, not being taught something is not the same as having none of it inside you.  Your eyes can see without you being taught how the - part of the mind that handles vision - is doing the work it does.  And if you could never see the implications of other guesses you'd already made, you wouldn't get far enough in life to reproduce.  Everyone here has bits and pieces of them that imperfectly echo the shard of Law about which conclusions follow from which premises.  I also happen to have studied that Law explicitly and went through standard training for not being quite as messy about it.  That's part of the process that dath ilan went through to put together aeroplanes that could fly across oceans.  We aren't perfect at it, to be clear, just better than whoever wrote the so-called books in this library.  I really want to see what happens if we match up Lrilatha against a Keeper - one of the people from my world who are actually specialized in being more perfect reflections of Law - but I doubt we'll ever get a chance to try."

"....you think that in a Lawful country all the books should only use truth-preserving arguments?" someone says, somewhat dumbfounded.

It makes sense, though. Mortals didn't have free will. Now they do, and it displeases Asmodeus, but no one has a complete account of what free will is, because they're not gods, and don't understand what exactly displeases Asmodeus. But that might just be it. Gods, innately, reason in truth-preserving ways. Of course they would. Lying to yourself for self-preservation is a thing you only have to do if you have wrong beliefs and can't argue yourself out of them because you don't know the counterarguments, and so you have to stop thinking about them. That is not a problem gods have. Gods just reason correctly. And in Keltham's world - there's still the concept of infohazards, things you're not supposed to learn, presumably because you're only human and can't properly have the kind of mind that entertains that fact in a way that allows for continued useful functioning -

- something about that frame isn't quite right but despite that she feels like everything is coming together.

Minds should reason in truth-preserving ways. Someone, a long time ago, robbed humans of that, and Asmodeus is angry. Carissa is angry! That was her birthright, and she wants it back. And Asmodeus thought, until Keltham arrived, that the scars they'd wrought on human souls could only be corrected in Hell - or at least could most cheaply for Asmodeus be corrected in Hell - but in Keltham's world, where humans do not magically reason in truth-preserving ways, they figured out, possibly over many thousands of years of careful experiments, how to teach it. And Asmodeus saw that and immediately told them not to hurt Keltham, because -

- okay, that line of thought she's going to tuck away for later, it seems maybe ill-advised. Sufficient that Keltham got Asmodeus's endorsement immediately.

Minds should reason in truth-preserving ways. The books ought to have good arguments. Devils are masters of propaganda, but aren't convinced by it. Carissa - doesn't think of herself as convinced by it, the books are really presenting their conclusions not their arguments, but - but that's because the books think humans aren't doing reasoning well enough to be persuaded by argument, and humans can learn that. At least smart ones. And if they knew it, then you could just argue everyone out of all the heresies, their minds wouldn't possess the weaknesses that make that strategy doomed, that make it necessary to present them with conclusions they won't be able to understand. Or at least - less of it. Keltham did have the concept of things he was not meant to learn. 

(More things that suddenly make sense: what the Starstone does to you, why it changes some people more than others. Godhood, even more than devilhood, would preserve you to the extent that you are worth preserving - to the extent that you have learned the processes of reasoning - Irori ascended just by becoming perfect, and everyone writes that off as a strange one-off that only Irori could do but in dath ilan they teach it -)

It has to be done all at once, she realizes. There's a terrible middle ground where you are trying to reason things out, but you are incompetent to do it, and so you run right into all the heresies that you could have been protected from by not trying to reason. You would absolutely fail a loyalty check, in the middle of trying to learn how to think. But at the end of it - Asmodeus arrived at His beliefs through reason. And He hates it, that humans were changed, so they can't, and He wants them changed back.

She rereads everything on the board, though there's not much written on the board. The new thing she's learned here isn't that there are necessary truths and empirical truths, or that you shouldn't divide by zero, it's that it is possible for humans to learn how to reason well enough they're better off trying it.

"If you found yourself in an unfamiliar country and you opened up a book and it was like, 'The sky is green.  How do we know this?  Because teddy bears are cute!  My dad once bought me a cookie!' would you suspect you were in a Chaotic country or a Lawful one?  Now, I admit this example is unrealistic; generalizing from my reading experiences, a Chelish author would never explicitly ask 'How do we know this?'  And yes, I'm sure places outside of Cheliax are even sillier but your book authors are still all very silly and if Lrilatha had infinite free time I would lock all of them in a room with her until they learned better."

"That's kind of what Hell is," someone offers. The other people who were totally thinking that but not sure if they were allowed to say it giggle. 

"The Worldwound isn't in Hell, it's here.  And I don't know why you can't have people train in Lawfulness in the whole post-life thing for a few years, and then resurrect them here, if that's a thing in the first place; or why Lrilatha hasn't been able to train teachers who could train teachers who could train you.  But the Worldwound isn't in Hell, it's here, and it's this world that needs to become saner and wealthier and better at repelling demons, or die." 

Those questions don't...sound like they're meant to answer them? Instead, they nod vigorously. 

- no, actually, she thinks they're meant to answer that. Or she thinks they ought to, regardless of whether they're meant to. "Becoming a devil in Hell takes centuries," she says. "You can't be resurrected after that long. It's been widely assumed there just wasn't any way to make a useful amount of progress on - being Lawful the way devils are - in a human lifetime. Or in time to close the Worldwound. But it seems to me that the reason Asmodeus intervened directly to tell us to make this a priority is that - the way you know is a lot faster."

"Asmodeus would also bet significant resources on that even if he only estimated a small probability of it working, so let's not get overconfident.  But yeah.  I don't know how long dath ilan took to get where we did, starting from scratch and baseline - we had to screen off our history, for reasons that are apparently also infohazardous to know about.  But the pieces all fit together, and you should be able to complete the whole thing once you have enough hints from me.  Even if there's no spell to give me perfect recollection of all the training I went through, I'm hoping it should be possible to get, like, 80% of the benefit from going off my memory of, hopefully, the most critical parts.  Not to mention, you're not all 8 years old and that should count for something when it comes to learning this part a little faster."

Keltham turns back toward the whiteboard, completely unconscious of any effect the declaration about 8-year-olds might've had on the rest of his audience, who are all concealing their reactions anyways.

"When it comes to algebra over continuous quantities," Keltham says, gesturing at the tactics written between the steps of the equations, "we have rules like being allowed to multiply both sides by the same quantity, or divide both sides by the same quantity so long as it isn't zero.  If you imagine building a mind to reason inside a universe that was full of hidden order that could be described by algebra - if it was an observer surrounded by, like, piles of fruit containing twice as many cherries as apples, that sort of thing, it was just how that world worked - then you could imagine building that mind with rules like, 'If I believe an equation, I should also believe that equation with both sides multiplied by the same quantity' or 'If I believe an equation, I can believe that equation with both sides divided by the same quantity, so long as I already believe that quantity isn't zero.'  I say this to introduce a new topic: the concept of hidden order within the rules of reasoning themselves.  There are hidden patterns and deep explanations to be found in this subject matter, as, in my world, there was a reason why snowflakes had sixfold symmetry."

"As a very simple example, the rule 'You can divide by nonzero quantities' can be seen as a pure special case of 'You can multiply by any quantity.'  To say you can divide both sides by 2 is the same as saying you can multiply both sides by 1/2.  The reason you can't divide both sides by zero is that zero is the only continuous quantity which lacks an inverse.  Once you see things from that angle, in fact, you might say that it's a simpler viewpoint to say that there's just one rule to use there, about valid inference in algebra: the rule that you can multiply both sides by any quantity.  Say just that, and you don't need that darned rule with the extra complication about 'Oh well you can divide by anything unless it might be zero.'  You just have the rule that you can multiply by anything, and the rule that everything except zero has an inverse.  You could also add the rule about division, nothing invalid would happen to you if you did, but it would be redundant; the mind you were constructing could reach the same conclusions either way.  Through perceiving hidden order in the rules of reasoning, you would be able to simplify the mind's thought processes and arrive to the same ends - though it might also take longer to reason that way, it might take extra steps if you eliminated the extra rule."

"But meanwhile, back in the real world, we deal more with the equivalent of triangles and red things than the equivalent of numbers and addition.  I mean, this world has both, but still, let's go back to shapes and colors and sizes.  What sort of truth-preserving rules analogous to 'you can multiply both sides by any quantity' in algebra, might we use to combine beliefs like these?"

Z.  All triangular things are red.
H.  All red things are large.

"All triangular things are large."

Why are they so inconsistently math??

"That's the conclusion you want, yes; what rules did you follow and what road did you walk to get there?  If you were making a child from scratch, and you stood too far back of the child's future situation to know exactly what situations they would encounter or what conclusions they would need, how would you make the child to reason to Q from Z and H?"

This question is somehow really confusing to them!!

"...well, if all triangular things are red and all red things are large, then - you can't have a triangular thing that isn't large, that'd mean something was triangular and not red, or red and not large."

"Ah, well, that is a very persuasive argument, I am totally persuaded.  But what rule are you using to find this persuasive, what shard of structure embedded within me leads me to find it persuasive?  Is it the sort of rule that has some important exception we need to know about, like not being able to divide by zero?  Does it only work sometimes and sometimes give wrong results?  Is it maybe a bit of complete nonsense that somehow got embedded into both of us, causing us to both arrive at the same wrong conclusions?  If we don't even know what rules we're following, how could we begin to tell?  Imagine getting to Hell and being locked in a room with Lrilatha and now she has to explain everything you're doing wrong, only you don't know what you're doing at all and she has trouble empathizing because, I'm guessing, all the nonsense in our heads is contrary to her own nature.  Think of how much of her valuable time you could save her - not to mention your own time locked in the room - if you actually knew which rules were operating inside you, to cause you to be persuaded by arguments like that one.  So what renders persuasive 'Z and H implies Q', or your own statement 'for there to be a non-large triangle implies either a non-red triangle or a non-large red thing' - how would you construct an entity from scratch to be persuaded by a statement like that?"

These people are stunningly motivated to skip through as much as is possible of the being locked in a room with a frustrated devil once they die! They are very aware that it will suck and they are so eager to get to do less of it!!!! They....do not understand Keltham's question at all. 

"An ...entity that wasn't doing that kind of reasoning would be really bad at inference and waste a lot of time."

"Kids will just naturally pick it up, they actually tend to overgeneralize - I have a kid sister who'd say things like 'all boys have long hair' after she'd seen three -"

"I think it'd have an exception for like - cases where we're using the words differently in different contexts, like, if we say 'all criminals are punished' and 'all punishments are painful' that doesn't mean 'all criminals are painful' -"

Even Keltham has managed to pick up on the rise in energy levels in the room!  He's not sure why this math-marketing tactic is so much more effective than other marketing tactics in Cheliax but he's willing to roll with it!  Though he should probably also be careful not to overuse it, whatever the ass it is he's doing, especially when he has no idea why it's working.  He sets aside a question about what kind of game theory criminals use here, and what sort of bizarre equilibrium results, to an enormous ill-organized heap of similar plaintive questions.

Keltham goes over to one of the few remaining empty spaces on the wall-whiteboard; he'd rather not have it laundry-magicked clean just yet.

Z':  All male objects have long hair.
H':  All long-haired objects wear shirts.

"When you're confused, one of the macro reasoning strategies is to find the smallest, simplest problems that still contain your confusion.  Can you state a general rule like 'It's okay to add 2 to both sides of any equation' that covers how to combine Z' and H', which also says how to combine Z and H, without explicitly mentioning Z and H?  Like stating a rule for adding 2 to both sides of an equation, which doesn't mention the particular equation you're using.  That takes on some of the challenge of creating an agent who'll reason in the world, when you don't know which particular equations or statements that agent will encounter."

"You mean like, change the sentences to... 'all somethings have a trait' 'all things with a trait have a second trait'..."

"Well, yes!  You don't have to work out the entire hidden order all at once, in order to make progress on it a piece at a time, speaking of macro reasoning strategies!  Before you've worked out that it's okay to add any quantity to a balanced equation, it's fine to start by noticing just that it's okay to add 2 specifically to any balanced equation.  That's a legitimate step towards starting to put the pieces together for yourself."

Require (Z-generalized):  All objects with trait-1 have trait-2.
Require (H-generalized):  All objects with trait-2 have trait-3.
Conclude (Q-generalized):  All objects with trait-1 have trait-3.

"When you build an entity with a rule in its mind that looks for a case where it believes any instance of Z-generalized and H-generalized, and concludes Q-generalized, you're building an entity that's operating a much broader necessary truth than the very narrow universal truth that connects 'If all triangles are red and all red things are large, then all triangles are large.'  You might be able to build a few dozen fairly general rules like that into a mind, whose outputs feed into each other as inputs, and have thereby given it a noticeably-sized shard of the Law that connects premises and conclusions, instead of just a very narrow guideline about shapes and sizes in particular."

"Does anyone want to try naming another candidate for a belief-manipulating rule like that?"

"....there's the opposite, like, no objects with trait 1 have trait 2. Or, uh, I guess you'd want - no objects with trait 1 have trait 2. All objects with trait 2 have trait 3. No objects with trait one - no, that doesn't actually hold -"

"No objects with trait 1 have trait 2. All objects with trait 3 have trait 2. No objects with trait 1 have trait 3," another girl says, a little too competitively for this to sound like helpfully supplementing the first one's train of thought.

"Well, I'm starting to run out of room on this wall, so forgive me if I write that down in dath ilani shorthand," says Keltham.

    \ z. t1(z) -> ~t2(z)
    \ h. t3(h) -> t2(h)
__________________

    \ q. t1(q) -> ~t3(q)

"Now this is a valid reasoning rule to be sure," says Keltham, "but just like dividing over a balanced equation can be seen as multiplying by an inverse, I think we don't need to add this whole rule to our entity.  The form of this rule looks really quite similar, in some ways, to that earlier rule about Z-generalized, H-generalized, and Q-generalized.  I think we can add a smaller new rule to our entity, which already has that previous rule, and get this rule back out as a special case - like adding the inverse operation to an algebra that already has the rule about multiplying over a balanced equation, and automatically getting out the power to divide over a balanced equation."

"I don't predict, based on your past performance, that you can derive the missing rule on your own; but beliefs like that ought to be tested rather than just assumed.  Wanna surprise me?"

They're so upset not to get it! They're - not getting it, though. They're distracted by trying to follow the dath ilan notation and they're not quite generalizing far enough, proposing variants on the rule that aren't actually simpler. 

It's encouraging that his students aren't showing any visible sign of emotional disturbance at the prediction or at failing to overcome it; they have some traces of dath ilani dignity, at least.  Keltham was wondering whether a lack of training in dignity would require him to back off a little on challenges like those, but his students' dignity is unperturbed so far as he can see.

     \ h. t3(h) -> t2(h)
___________________

   \ h. ~t2(h) -> ~t3(h)

"So long as we have this reasoning tactic in our tactical repertoire - go ahead and take a moment to convince yourself that you couldn't cast an illusion violating it - we can combine it with our previous rule to get the combined rule we wanted:"

[1]    \ z. t1(z) -> ~t2(z)                (Premise)
[2]    \ h. t3(h) -> t2(h)                 (Premise)
_______________________

[3]    \ h. ~t2(h) -> ~t3(h)            (one person's modus ponens is another person's modus tollens [2])
_______________________

[4]    \ q. t1(q) -> ~t3(q)                (syllogism [1], [3])

"Anyone want to propose yet another universal rule?  Here's some shorthand language to help you express yourself:"

blue(k) \/ red(k)             "k is blue or k is red"
blue(k) /\ ~red(k)           "k is blue and k is not red"
\k. ~(blue(k) /\ red(k))    "for every k, it is not the case that (k is blue and k is red)"
blue(k) -> small(k)         "if k is blue, then k is small"
~~~blue(k)                     "it isn't wrong that k is not blue"

They take a while just to figure out how the symbols work and then they're full of ideas.

\k, blue(k) V ~blue(k)

blue(k) -> ~~blue(k)

~blue(k) -> ~blue(k) "That doesn't count!" "Yes it does, it's like the 1 = 1 thing!"

"Except we're not really using 'blue' to mean anything, right, we can just write those with t, like dath ilan does it-"

Now they're thinking with average intelligence!  While they're doing that, Keltham will helpfully write down some statements for them to decide on as valid or not valid.

((p -> q) -> p) -> p

"If p then q, ...if it's true that if p then q, then p...if it's true that if p then q then p, then p. Uh, I think that's...not true? Like, if p isn't true, then -"

"It's basically just saying, is p being true required from the fact that if it's true that - okay, (p -> q) -> p is not necessarily true, it could be, like, say p is 'men are immortal' and q is 'they will all become ninth-circle wizards', so obviously you can have p-> q but p is false -"

"That's not what it's asking. It's saying, if p-> q does imply p, then does that mean p is always true."

"- nooo? Like, okay, what's something where p-> q implies p? I'm just not sure that's a thing at all!"

"I think I see the problem.  The Taldane word 'implies' probably means all sorts of vague things besides... anyways.  Let's use 'material implication' to narrowly denote the particular kind of 'implies' I used here.  Now, we're going to have to erase this wall soon, but let's look back at the blue circles.  In particular, let's look at this blue circle containing a large red triangle, a large blue square, a small blue square, and a large red square.  The way I define material implication, we can take the statement 'For all z, z being triangular, materially implies z being red, and say that it's true of every object z, including the ones that aren't triangles.  We could look at this small blue square, and say of it truthfully, 'if a small blue square is triangular, then a small blue square is red' - the way we're defining material implication, that symbol I wrote like this," Keltham points to a -> symbol, "that would be a true thing to say.  Why define it that way?  So that the statement over here," Keltham points to \ h. red(h) -> large(h), "can be true when we evaluate it at every object h could refer to, including the objects that aren't red at all.  If we said that 'red h materially implies large h' was false whenever h wasn't red, putting a blue square in the world would mean we could not say of it, 'for every object in the world, the redness of that object materially implies its largeness'."

"Now, wanna take another shot at 'if p materially implying q materially implies p, then p'?  True across all possible worlds, or false in some of them?"

"So p->q implies p if there aren't any p."

"Well, p isn't quantified here - it's not ranging over possible objects.  p is here some proposition that could be either true or false, not an object with a property like redness.  So it's that p materially implies q whenever p is false, whether or not q is true."

"That seems -"

"No, that makes sense, that's like - I read a theological argument like that once -"

It's very hard for Keltham at this point to predict what Chelish practical-topologists will get instantly versus not at all.  Maybe once he's had longer than a day to experiment and figure it out.

He'll give them another couple of half-minutes on ((p -> q) -> p) -> p, but if they haven't gotten it by then, he'll leave coming to a definite decision about that as a homework problem, and tell them to get back to inventing other logical rules.

"(p -> q) if p is false, and also occasionally if p is true and the world happens to be that way. so (p -> q) -> p if the reason p implies q isn't that p is false?"

"Well, if p is false, then p->q doesn't imply p - it can't, since p is false. So if p->q does somehow imply p, then that would be...because p is true?" 

"No, it'd be not because p is false but that doesn't mean p is definitely true, we just don't know."

They're still all, to external appearances without a lot of experience reading Chelish people, very calm and unbothered by this!

"I'll leave that one as an exercise to try to solve afterwards - come back tomorrow with your own best guess, even if you haven't proven it, about whether it's necessarily true, necessarily false, or neither."

"Now, let me present you with a different puzzle, one that starts to lead into a higher lesson.  I was constructing an agent but, oops, I forgot to give it the 'or' concept," Keltham points to where \/ was written.  "It's got all the other concepts here like forall, and, not, implies, but darn it, I just forgot to give it the 'or' concept.  Can you form a statement that's equivalent to 'for every object h, h is red or h is blue' out of the concepts I did remember to put in?  So I can explain that important fact to my poor confused entity?"

\ h. red(h) \/ blue(h)  =  ???

"Sorry for making you clean up my mess, there," Keltham adds, "but the entity's already created and I can't redesign its mind now."

Giggles. 

"For every object h, h not red implies h is blue," someone calls out almost instantly.

Why can they - but not - nevermind.  Keltham glances at that nametag.

"Correct!  Wait, oops, I forgot to give them the 'implies' symbol too - anything you can do now?"

That was Asmodia.

"A implies B is the same as....for all h where A is true, B is true - if I try to write that out I use the implies symbol, though -"

"Kill them and start over?"

"Sorry, I screwed up even more, they're already sapient and Governance would take a dim view of killing them.  Or it's Golarion and they just end up in an afterlife anyways, and Hell will be annoyed if you made extra work for them."

"A implies B is the same as...not B implies not A - that doesn't help -"

"Construct a C, where C is everything that is in both A and B. for all h in A, h is in C," says Meritxell.

     "Where are you getting a both-A-and-B."

"- I haven't sketched out how I'd do it yet but I'm sure I could, it's obviously the sort of thing that's not hard to specify -"

     "Without 'implies', though?"

"x is in C if x is in A and x is in B. No implies."

"You do have the 'and' symbol.  And the 'forall' symbol, and the 'not' symbol, and the parentheses.  And the object variable symbols, of course, and the 'red' and 'blue' function symbols.  That's all you've got, though, you can't bring in Taldane language for describing things beyond that."  Keltham taps again where the whiteboard now shows, with its last gasp of open space: \ h. red(h) \/ blue(h)  =  ???

" \h, ~ (red(h) ^ blue(h)), ~(~red(h))^(~blue(h))?" Patxi ventures?

Carissa is being a bad student. This is, in part, because she is no longer in school and no longer feels with aching intensity that the entirety of her being as a person is her perfomance in school, and being lashed for inattentiveness doesn't hold the soul-consuming horror it once did either. It is in part because her mind keeps running ahead - she can't always see the answers to the specific questions, and probably she should focus her attention on them at some point, to crystalize the skill of turning all her thoughts into the crisp precise symbolic bounded versions of them, but she can see the broad outlines of what the questions let you do. Everything, maybe, if you're a god. If you're a human - 

How would you express 'the best outcome a human can reasonably get is to live such that when they die and go to Hell, they are useful?' For all humans - but no, she's not really making a claim about all humans, she's really only interested in the implications of this question for one human, and the other ones are relevant because she knows exactly how exceptional she is - there exists a human, such that, in the space of all eternities for that human, ordered by how strongly preferred they are, the most preferred is - well, no, it wouldn't be Hell, because of all possible eternities there are certainly some better ones -

This is of course not an argument against Hell, it's not like she could formulate any other important claims about the world either. It is an argument against sucking at thinking. It is an argument for - if there were a book that tried to convince you, what would it say -

"Indeed, or rather, we just need the second part - a red object counts as 'red or not-blue', we don't demand that only one side be true."

In that last bit of improvised whiteboard, Keltham extends his last equation, and then writes down one more on the edge of the wall below:

\h. blue(h) \/ red(h)   =   ???   =   \h. ~red(h) -> blue(h)   =    \h. ~(~blue(h) /\ ~red(h))
\h. blue(h) /\ red(h)   =   \h. ~(~red(h) \/ ~blue(h))   =   ~(red(h) -> ~blue(h))

"Now, given that - if you have 'not' - you can make 'and' out of 'or', or make 'or' out of 'and', or make either one out of 'materially implies' - why not just design an entity that thinks in terms of implication?  Why bother making an entity that tends to think in terms of 'P is true or R is true', instead of 'if P is false then R is true'?  This is not a theoretical question: if your mind works anything like mine does, your mind sometimes thinks in terms of 'or' and not just 'implies'.  You've probably thought using 'and' too.  Why is a human mind - which includes your mind - designed so... inelegantly?"

Nervous glances.

"- because humans were given free will and it was done very haphazardly and made us worse at reasoning like the gods," says Tonia, when no one else has said anything for a moment.

"Actually, there's something of a questionable assumption I've been making, which is that your biology is a possibly-modified version of biology that got copied off of a... branch of time, I don't think Taldane has a word for it... that's very close in branching time to dath ilan.  I think dath ilan can't see your world, can't be affected by it; but I did manage to show up in this world at all, even if that's a very rare phenomenon.  So your world can see my world, be causally affected by it, even if my materializing like this very rarely happens.  And your bodies look a lot like mine, and more importantly, I can eat your food without immediately falling over dead, which implies a lot of shared hidden order between our biology, which wouldn't exist without common ancestry.  If it's possible for me and somebody from this world to have kids, which is mostly what I'd expect, that would absolutely prove the point."

"Where the point is that while some stuff may have modified you relative to where a dath ilani starts, and dath ilan may have developed and diverged some from whenever your biology was copied from our cousin or ancestral world - remind me of how old human life on Golarion is, again? - human biology on Golarion is, I would strongly guess, basically a copy of dath ilani biology.  Some of my distant ancestors or cousins got materialized here and had kids, maybe.  Or some god read the - heredity code - for one of us and materialized some entities like that."

"If all of that is true, then the reason your underlying mind design looks like it was slapped together by monkeys on drugs, is the same reason our baseline mind design looks like it was slapped together by monkeys on drugs.  I wasn't born like this, we have to give people extensive training to get them to work at all correctly, instead of them just working correctly straight out of the womb, the way we would if we were designed by sane designers instead of... well, the thing that actually made us.  A weird pseudo-nonentity that had literally no idea what the ass it was doing.  Frankly it's sort of a big topic here, though it sure is a fundamental one so I'll probably get to the details at some time.  The point is, I fully expect that by the time we're done in class here, you will be looking over your mind design and thinking that you could accidentally sneeze a better mind design than that.  I'm not quite sure what the 'given free will' thing was about, the Taldane term 'free will' doesn't translate well into Baseline so we may not have whatever you were given, but trust me, your species's mind design was horrible crap even before then.  You can tell this because I had to go through lessons similar to what you're going through now.  Though, if 'free will' makes you even worse at sanity, which sure is plausible given this total mess of a planet, I probably need to have that explained to me at some point... I don't suppose it's easy to describe?"

Horrified silence.

She does not want to interact with this but she has the twin qualifications of being particularly unlikely to be executed for misstepping, it'd be conspicuous, Keltham can definitely tell her apart from everyone else, and having spent the last half hour dwelling on it.

"I don't think I have ever encountered the theory that the gods were copying," she says, "but it does seem odd, for there to be a world with a longer history and humans that came about some other way. I think that these lessons have helped me make more sense of the free will thing, actually. It used to be that humans didn't make mistakes of reasoning, but also that they didn't have their own goals, just the goals of the gods they served. It sounds like....you think maybe those necessarily went together, that it wasn't possible, for humans to stop making mistakes of reasoning while - being more than automata -"

"Yeah, that'd make its own kind of sense.  The event your history has down as 'humans suddenly acquired free will' could've been a magical template superposed on human biology, producing agents working for gods, and then somehow that magical template stopped working and suddenly you had the original humans again.  I do not know nearly enough of your history to guess what parts of the template versus original human nature were locked together, I am guessing at a lot here.  I'd ask if the magical template made people - nonconscious, nonexperiencing - but I wouldn't expect you to have any way of knowing that, given the general fuzziness of your prehistory.  That whole scenario would actually be a pretty optimistic result, from my standpoint?  It means you don't have additional features making you crazier, and dath ilani training should still work on people here with high baseline intelligence."

"The scenario you described matches all our histories, but we don't know details of the - magical template - aside from that the gods were divided over the change that made it stop working."

"Yeah, I'm not going to say details like that are unimportant, they're obviously hugely important and at some point I want to know everything that's known about it, but they're not obviously urgent details, especially compared to the general project of me transferring knowledge Golarion will need for industrialization and scaling up to fight the Worldwound."

"So back to where your mind design actually comes from.  I'll endeavor to be brief because this lesson is mainly about Validity, but now we're talking about how shards and reflections of Validity even got into human minds at all, and soon we're going to ask whether there's maybe something better than the version of Validity we have; and I'm not sure how you could reason well about those topics if you had no idea where your mind design came from in the first place."

"This part is actually a pretty simple idea.  If anything you should be careful not to overthink it.  You know how a pair of tall parents will probably, though not always, have a kid who's taller than average?  And a pair of short parents will probably, though not always, have a kid who's shorter than average?  It may help for the sake of concreteness to know that inside you there are extremely tiny, extremely long spirals of... stuff Taldane doesn't have a name for, but capable of encoding information.  Like, imagine there's four kinds of tiny parts that can make up each bit of spiral, labeled 0, 1, 2, and 3; so a section of the spiral might read 1032, that is, it'd be the second kind of bit, connected to the first kind of bit, connected to the fourth kind of bit, connected to the third kind of bit.  Each spiral is around three billion of those units long, but the parts are so tiny that even three billion of them curled up in spirals are still too tiny to see.  Your body is full of identical copies of your version, and it carries the information that told your body how to develop fingers and toes and a liver and so on, when you were forming in your mother's womb.  Variations in that code, between individuals, might cause some to grow up taller and some to grow up shorter.  You got half of your spiral sections - they're broken up into twenty-three pairs of sections - from your father, and half from your mother, which is why a pair of taller parents will tend to have taller kids."

"Now suppose that taller parents tend to have more kids than shorter parents.  Then the next generation will end up taller than the previous generation; the variations in codes that tell bodies to construct taller bodies will be more common among the next generation's inner spirals."

"Pile on one change after another, after another, after another, that contributes to some couples having more kids than another.  Even though each change is a single alteration, if you iterate that process thousands of times, millions of times, it can build whole complicated parts.  But it builds them without foresight, without planning.  Every part of your body is made up of a cumulation of changes that started as copying errors in the tiny spirals; they're mistakes that happened to work.  That's also where your mind design comes from - from the copying errors, and from some of those copying errors leading parents to have fewer kids and those errors dying out of the population, and a few copying errors accidentally constructing people who had more kids and those variations spreading throughout the population.  If I was actually focusing on this topic properly, I'd sketch the design of an eyeball on the wall, and show how it can develop in tiny changes starting from a single light-sensitive spot on the forehead of some tiny crawling creature a hundred million years ago."

"For now, the key thing to know - going back to our actual current subject, Validity - is that your mind design accreted on the ability to think using 'and', and the ability to think using 'or', and the ability to think about stuff implying other stuff, and the ability to imagine facts being true about all the objects inside a collection.  It's not all quite as redundant as it looks - the human native ability to reason about 'or' isn't quite the 'or' that appears in very simple logic, we're more likely to say an object is 'red or blue' meaning that it's either one or the other but not both, and less likely to say that this table is 'brown or not green', considering that in fact it is both brown and not green.  We are, in teaching ourselves to reason using the sharper simpler forms of logic, repurposing bits of our mind away from their original contexts, and stripping off real functionality along the way.  But that's part of the story of why we have such redundant facilities for thinking logically, 'and' and 'or' and 'implies' all at the same time."

"So would you like to guess, now, as to whether I'm about to tell you about some new connectors that would let your mind expand to even more powerful ideas - represent ideas that native human concepts can't represent at all?"

Is he going to do that. That would be so cool.

"When I was a bit younger and learning this stuff for the first time, I went straight to the Watcher - the adult who was there to make sure the older kids weren't teaching us anything too wrong - and demanded that I immediately be taught the most powerful kind of logic there was.  The Watcher told me that the logic I was learning was the most powerful kind of logic on offer - that it was, in fact, the most powerful kind of logic that could exist.  I didn't see how anyone could possibly know that even if it was true, so I figured this was another of the lies-they-tell-children, or maybe that the best kind of logic was probably being kept secret by the Keepers.  Those being the people who would learn a more powerful kind of logic, if it existed, and was too dangerous for everybody to have.  I wanted that for myself, so I tried inventing other kinds of logic with more powerful symbols in it, symbols that could connect three or even four propositions together, instead of just the one-or-two symbol connectors the older kids were telling me about."

"But before I tell you about the results of that particular journey of thinking, and whether or not it did turn out to be a lie-they-tell-children in the end, let me pause and ask another question first.  In algebra we have rules for producing new equations from old equations, or combining old equations.  Here we have rules for producing new statements from old statements, if those statements are written in a particular language.  Both algebra and the statement-rules obey the higher principle of Validity - we have ways of comparing equations and statements to worlds, to see if they're true or false; and if an equation or statement is true in a world, the rules for manipulating it should produce only more true equations or true statements.  In the world of statements, we managed to reduce 'or' to 'and' and 'not'.  In the world of algebra, we reduced the rule 'divide both sides by a nonzero quantity' to 'multiply both sides by an inverse'.  Can we in some way combine the rules of algebra, and the rules of statements, since they are both born of the same truth-preserving principle?  Can we reduce algebra-rules to statement-rules, or reduce statement-rules to algebra-rules, and so simplify our mastery of truth-perservation?"

"This one's actually quite hard to solve from scratch at our intelligence level - I didn't get it as a kid and wouldn't expect myself to get it now, if I didn't already know it.  But it is important to know your own emptiness before trying to fill yourself, so go and speak aloud any really bad wrong answers you come up with here."

"I mean, you could write the rules of algebra in statement logic- is that what you mean? Like, a + b = c if and then a bunch of stuff that correctly defines what 'plus' is - I don't know what stuff but I think there'd be stuff -" Merixtell says.

"Show me your shot at it?  I've been wrong once or twice guessing what you all can't do."

"Uh, okay. a plus b = c if, uh - oh, I think I actually only know what I'd do if a and b and c were all whole numbers -"

"I'll take it."

"If they're whole numbers, they're made of ones. a plus b = c if, uh, the process of taking ones from each side gets you zero on both-" She bites her lip. "- but then you still have to define taking one, I guess."

"Go ahead and define it then!  Don't worry too much about doing it wrong the first time, this one is hard and I'm impressed you're even trying.  Actually, I'm wondering if you've encountered something reflecting the correct answer from somewhere else in Golarion mathematics, because if you're literally doing this part from absolute scratch it's seriously impressive."

She beams at him. "Minus one is ...

...maybe you could do something with, a contains one more thing than b if for every thing in b, there's a thing in a, and for every thing in a, there's a thing in ...b plus one - no, now I've just needed to invent plus. ...maybe I can do that. a is b plus - no, sorry, I don't know -"

"If you don't know the right answer, make up a wrong one!  Maybe you'll be able to see why it's wrong and correct it, so long as you think it out loud!  And saying things out loud is a straightforward way to learn to think them out loud."

"I don't even know a wrong one!"