I know it's just an example, but concerning

I find it hard to do something I consider worthwhile while on a spring break

maybe you have learned to be lazy on spring break? I mean, the theory that it's a habit seems more prosaic to me than being tired or something about "activasion energy".

A conclusion which is true in any model where the axioms are true, which we know because we went through a series of transformations-of-belief, each step being licensed by some rule which guarantees that such steps never generate a false statement from a true statement.

I want to add that this idea justifies material implication ("if 2x2 = 4, then sky is blue") and other counter-intuitive properties of formal logic, like "you can prove anything, if you assume a contradiction/false statement".

Usual way to show the latter goes like this:

1) Assume that "A and not-A" are true

2) Then "A or «pigs can fly»" are true, since A is true

3) But we know that not-A is true! Therefore, the only way for "A or «pigs can fly»" to be true is to make «pigs can fly» true.

4) Therefore, pigs can fly.

The steps are clear, but this seems like cheating. Even more, this feels like a strange, alien inference. It's like putting your keys in a pocket, popping yourself on the head to induce short-term memory loss and then using your inability to remember keys' whereabouts to win a political debate. That isn't how humans usually reason about things.

But the thing is, formal logic isn't about reasoning about things. Formal logic is about preserving the truth; and if you assumed "A and not-A", then there is nothing left to preserve.

How Wikipedia puts it:

An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false.

I can't find it by search, but haven't you stated that you've written hundreds of KLOC?

Took the survey and reminded my fellow Russians to participate too.

A sorcerer has two ways to manipulate people:

1) Move things around in the world.

2) Directly influence people's minds.

I'm not going to talk about option 2 because it stops people from being perfect reasoners. (If there's a subset of option 2 that still lets people be perfect reasoners, I'd love to hear it - that might be the most interesting part of the puzzle). That leaves option 1.

Here's a simple model of option 1. Nature shuffles a deck of cards randomly, then a sorcerer (if one exists) has a chance to rearrange the cards somehow, then the deck is shown to an observer, who uses it as Bayesian evidence for or against the sorcerer's existence. We will adopt the usual "Nash equilibrium" assumption that the observer knows the sorcerer's strategy in advance. This seems like a fair idealization of "moving things around in the world". What would the different types of sorcerers do?

Note that if both Bright and Dark might exist, the game becomes unpleasant to analyze, because Dark can try to convince the observer that Bright exists, which would mean Dark doesn't exist. To simplify the game, we will let the observer know which type of sorcerer they might be playing against, so they only need to determine if the sorcerer exists.

A (non-unique) best strategy for Bright is to rearrange the cards in perfect order, so the observer can confidently say "either Bright exists or I just saw a very improbable coincidence". A (non-unique) best strategy for Dark is to leave the deck alone, regardless of the observer's prior. Invisible has the same set of best strategies as Dark. I won't spell out the proofs here, anyone sufficiently interested should be able to work them out.

To summarize: if sorcerers can only move things around in the world and cannot influence people's minds directly, then Bright does as much as possible, Invisible and Dark do as little as possible, and the observer only looks at things in the world and doesn't do anything like "updating on the strength of their own beliefs". The latter is only possible if sorcerers can directly influence minds, which stops people from being perfect reasoners and is probably harder to model and analyze.

Overall it seems like your post can generate several interesting math problems, depending on how you look at it. Good work!

No one makes the wrong decisions for reasons they think are wrong. The more clever the man, as the Nroni were fond of saying, the more apt he was to make a fool of himself. We all argue ourselves into our mistakes.

Scott R. Bakker, The White-Luck Warrior

If you find yourself taken unawares by someone you thought you knew, recall that the character revealed is as much your own as otherwise. When it comes to Men and their myriad, mercenary natures, revelation always comes in twos.

– Managoras, Ode to the Long-Lived Fool

Scott R. Bakker, The White-Luck Warrior

This was what made the fall of Iothiah so disastrous. <...> Strategically, the loss of Iothiah was little more than a nuisance.

Symbolically, however…

The crisis she faced was a crisis in confidence, nothing more, nothing less. The less her subjects believed in the Empire, the less some would sacrifice, the more others would resist. It was almost arithmetic. The balance was wobbling, and all the world watched to see which way the sand would spill. She had made a resolution to act as if she believed to spite all those who doubted her as much as anything else, and paradoxically, they had all started believing with her. It was a lesson Kellhus had drummed into her countless times and one she resolved never to forget again.

To know is to have power over the world; to believe is to have power over men.

Scott R. Bakker, The White-Luck Warrior

If you find yourself taken unawares by someone you thought you knew, recall that the character revealed is as much your own as otherwise. When it comes to Men and their myriad, mercenary natures, revelation always comes in twos.

– Managoras, Ode to the Long-Lived Fool

Scott R. Bakker, The White-Luck Warrior

(potential spoilers removed, so if this dialogue doesn't make sense, be assured that it makes sense in context)

"Just wait," Zsoronga said. "Something auspicious will happen. Some twist will keep you here, where you can discharge your fate! Wait and see."

"And what if they know?" Sorweel finally asked, voicing the one alternative they had passed over in silence.

"They don't know."

"But wh-"

"They don't know."

Zsoronga, Sorweel was beginning to realize, possessed the enviable ability to yoke his conviction to his need - to believe, absolutely, whatever his heart required. For Sorweel, belief and want always seemed like ropes too short to bind together, forcing him to play the knot as a result.

Scott R. Bakker, The White-Luck Warrior