I know enough logic to go through the steps to prove that P must be true, but boy am I struggling to get an intuition for why this statement makes P true.
To a non-logician, “P → Q” means something like “when the current state of the world contains ‘P is true’, I may then deduce Q”. To a logician it means something like “when I add ‘P is true’ to the current state of the world, I may then deduce Q”. Only the logician’s version allows you to conclude that a false proposition implies any proposition.
Your intuition is right; it doesn’t. It’s true whenever P is true/logically equivalent to just asserting P, which I think is the joke, but it’s not valid in general.
As an example “If ‘if it’s raining the streets are wet’ then it’s raining” is clearly false.
(Sorry if you are also participating in the joke in a way that is going over my head)
Logicians: (P → Q) → P
LLMs: It’s not just Q. It’s P.
GK Chesterton: It is not because of P that Q, but precisely because not P.
I know enough logic to go through the steps to prove that P must be true, but boy am I struggling to get an intuition for why this statement makes P true.
To a non-logician, “P → Q” means something like “when the current state of the world contains ‘P is true’, I may then deduce Q”. To a logician it means something like “when I add ‘P is true’ to the current state of the world, I may then deduce Q”. Only the logician’s version allows you to conclude that a false proposition implies any proposition.
This made it click for me. My interpretation is:
(P → Q) could be true while P is false, or while P is true (and Q is true).
The assertion (P → Q) → P is saying “if (P → Q), then it’s the ‘P is true’ version.”
It can’t be the ‘P is true’ version if P is false. Therefore P is true.
(This requires grokking the fact that not-P instantly makes (P → Q) true, regardless of the value of Q.)
Your intuition is right; it doesn’t. It’s true whenever P is true/logically equivalent to just asserting P, which I think is the joke, but it’s not valid in general.
As an example “If ‘if it’s raining the streets are wet’ then it’s raining” is clearly false.
(Sorry if you are also participating in the joke in a way that is going over my head)
“(P → Q) → P” is intended as an argument for P.
((P → Q) → P) → P
Suppose (P → Q) → P. If P is false it implies anything, so P → Q, so by our assumption P is true. This is a contradiction, so P is true.