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.
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.)