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Will_Sawin comments on Rationality Quotes: January 2011 - Less Wrong
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What's specifically going on here is that (P=>Q) => (Q=> P) is false whenever P is false and Q is true.
Adding it as an axiom schema to propositional calculus results in a contradiction. It cannot be added as a single axiom to first-order logic.
Yes, you are correct. I was confused in a very stupid way.