Peirce's law

Named after the logician and philosopher Charles Sanders Peirce.

Peirce's law

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

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

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