Peirce's law

English

Etymology

Named after the logician and philosopher Charles Sanders Peirce.

Proper noun

Peirce's law

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

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