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!"