Peirce's law

English

Etymology

Named after the logician and philosopher Template:W.

Proper noun

1. Template:Lb 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 $(P\to Q)\to P\equiv (P\to \mathrm{\perp })\to P\equiv \mathrm{\neg }P\to P\equiv \mathrm{\neg }P\to P\wedge \mathrm{\neg }P\equiv \mathrm{\neg }P\to \mathrm{\perp }\equiv \mathrm{\neg }\mathrm{\neg }P$ so the law reads $\mathrm{\neg }\mathrm{\neg }P\to P$, which is intuitionistically false but equivalent to the classical axiom $\mathrm{\neg }P\vee P$.

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