"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?"