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Definition 2024

Peirce's law ‎(uncountable)

(logic) The classically valid but intuitionistically non-valid formula $((P\to Q)\to P)\to P$ of propositional calculus, which can be used as an 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 $(P\to Q)\to P\equiv (P\to \bot )\to P\equiv \neg P\to P\equiv \neg P\to P\land \neg P\equiv \neg P\to \bot \equiv \neg \neg P$ so the law reads $\neg \neg P\to P$ , which is intuitionistically false but equivalent to the classical axiom $\neg P\vee P$ .