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.
In propositional calculus, Peirce's law says that ((P→Q)→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...
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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.
In propositional calculus, Peirce's law says that ((P→Q)→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.
Here is Peirce's own statement of the law:
Peirce goes on to point out an immediate application of the law:
Showing Peirce's Law applies does not mean that P→Q or Q is true, we have that P is...
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