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Peirce's law in logic is named after the philosopher and logician
Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. The axiom can be used as an alternative to the excluded middle.In propositional calculus, Peirce's law says that ((P→Q)→P)→P. Written out, this says that if you can show that P implying Q forces P to be true, then P must be true.
Peirce's law does not hold in intuitionistic logic or intermediate logics.
Under the Curry-Howard isomorphism, Peirce's law is the type of continuation operators.
Showing Peirce's Law applies does not mean that P→Q is true, we have that P is true but only (P→Q)→P, not P→(P→Q) (see affirming the consequent).
The fastest proof of Peirce's Law is to prove the contrapositiveIn predicate logic, the contrapositive (or transposition of the statement p implies q is "not q implies not p''. A statement and its contrapositive are always logically equivalent, unlike a statement's inverse or its converse. One can informally convince ¬P→¬((P→Q)→P) thus:
But then we have (P→Q) and ¬P, so ((P→Q)→P) is false.
Thus ¬((P→Q)→P) is true, Q.E.D.For other meanings of the abbreviation "QED", see QED. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, "that which was to be demonstrated"). This is a translation of the Greek oper edei deixai which was used by many early mathem
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