Peirce's Law

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication.

In propositional calculus, Peirce's law says that ((PQ)→P)→P. Written out, this means that P must be true if there is a proposition Q such that the truth of P follows from the truth of "if P then Q". In particular, when Q is taken to be a false formula, the law says that if P must be true whenever it implies the false, then P is true. In this way Peirce's law implies the law of excluded middle.

Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone.

Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/cc in Scheme.

Read more about Peirce's LawHistory, Other Proofs of Peirce's Law, Using Peirce's Law With The Deduction Theorem, Completeness of The Implicational Propositional Calculus

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