Don’t forget you’re contradicting, dear
If you do formal logic wrong even once it blows up. This is called the principle of explosion. If you’ve allowed a single contradiction into your system, you can use it to prove any incorrect statement you like - or the opposite of that statement! You’re left with a bombed-out wreckage of a formalism, completely incapable of distinguishing any truths whatsoever.
Knowing that contradictions are so dangerous, it may surprise you to learn that one of the most powerful techniques in mathematics is the “proof by contradiction”, dancing right on the razor’s edge of explosion. It works like this: start by assuming something is true with no evidence. Execute steps one after the other that have all been previously proven valid until you reach a contradictory conclusion. Something must be wrong for a contradiction to be let in, and there’s only one possible option: the assumption you made at the start. So that assumption must be false.
You can see that if you’re doing a proof by contradiction, it’s awfully important to remember what it is you assumed without evidence. If you went through so many steps that you’ve forgotten the initial assumption, you’ll get to the contradiction and think: “Wow, this goes against normal mathematics, but all of my steps were valid, so it must actually be right!” And once that happens you have explosion, gleefully chasing other findings that seem to “follow” from your contradiction without realizing that all statements are equally true or false in your exploded formal system.
In Being incoherent is Lindy, Crispy Chicken makes the point that the “classic” philosophical paradoxes are often just unhelpful conceptualizations obscuring the point (because well-formulated questions get answers). Let’s think about these contradictory statements as evidence that we’re in an exploded logic, and hunt for the assumption that the unhelpful conceptualization is hiding.
Take perennial TIS punching bag The Ship of Theseus paradox. The ship was rebuilt piece by piece, and a scavenger from behind made a ship from the discarded parts. Is one or the other the “real” ship of Theseus? How does one ship become two? We could play around with this exploded logic, or we could just notice: we assumed there was such a thing as “real” identity indepdent from a use case. We reached a contradiction. Ergo, our initial assumption was wrong: there must not be such a thing as a metaphysical “real ship”.
The sorties paradox: taking away or adding one leaf doesn’t meaningfully change a leaf pile into “not a leaf pile”, so how do we get leaf piles? Well, we’re assuming that a threshold of when leaves are a “pile” is a strict function of leaf count, but we seem to have reached a contradiction, so it’s probably wrong. If you see a leaf pile and are about to jump in, but then someone walks past you with a bloody forehead and says “That’s a single layer of leaves on top of a rock to trick people, don’t do it”, it sure doesn’t feel like a leaf pile anymore, does it?
What about the extremely-annoying-in-rationalist-circles Newcomb’s paradox? There’s a reliable predictor that’s changing the reward in a box based on it’s prediction of your behavior, so do you make the choice that “cooperates” with the future prediction, or are you safe to maximize since whatever it’s deciding has been decided? You could try to make your decision by going down a Wikipedia page that has separate headings for “causality and free will”, “conscioussness and simulation”, and “fatalism”. Or you could say, wait, this all seems extremely goofy and totally disjoint to how anti-inductive dynamics actually work in practice. We started this by assuming a reliable predictor that’s reliable even when the agent in question knows the options the predictor is predicting between and has no cost to change their mind, and now we seem to be in an exploded formalism, so probably our assumption was wrong and we’ve just accomplished a proof by contradiction against this kind of reliable predictor.
Remember the hallmark of a formal logic that’s been struck by the principle of explosion: anything is true, because you can prove half of everything with one version of the contradictory statement and the other half with the other version. If you’re in a situation where a statement and it’s opposite both seem to have valid proofs, take a step back and ask yourself: what concept am I assuming exists? You may have just been contradicting so hard you forgot you were doing it, and you haven’t actually proven the statement OR it’s opposite, but instead disproven the assumption that got you there in the first place.