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double negation principle

Principle that, for any proposition P, P logically implies not-not-P, and not-not-P logically implies P.

Classical logic accepts both these halves of the principle, but intuitionist logic accepts only the first half, and not the second. This is because it accepts the law of contradiction (and so, given P, cannot allow not-P), but rejects the law of excluded middle (and so, given not-not-P, does not consider itself forced to accept P).

Source:
G T Kneebone, Mathematical Logic and the Foundations of Mathematics (1963), 243-50; elementary account of intuitionism

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