# Peirce's law

## English

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### Etymology

Named after the logician and philosopher Charles Sanders Peirce.

### Noun

Peirce's law (uncountable)

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 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 $(P \to Q) \to P \equiv (P \to \bot) \to P \equiv \neg P \to P \equiv \neg P \to P \and \neg P \equiv \neg P \to \bot \equiv \neg \neg P$ so the law reads $\neg \neg P \to P$, which is intuitionistically false but equivalent to the classical axiom $\neg P \vee P$.