Peirce's law

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Proper noun

1. (logic) The classically valid but intuitionistically non-valid formula ((P \to Q) \to P) \to P of propositional calculus, which can be used as a substitute for the law of excluded middle in implicational propositional calculus.

   Consider Peirce's law, ((P \to Q) \to P) \to P) . If Q is true, then P \to Q is also true so the law reads "If truth implies P then deduce P" which certainly makes sense. If Q is false, then (P \to Q) \to P \equiv (P \to \bot) \to P \equiv \neg P \to P \equiv \neg P \to P \and \neg P \equiv \neg P \to \bot \equiv \neg \neg P so the law reads \neg \neg P \to P , which is intuitionistically false but equivalent to the classical axiom \neg P \vee P .

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