We saw in an earlier video that, on Classical Logic, every well formed statement is either true or false, and so (p V ~p) will always hold. This is sometimes called the Law of ExcludedMiddle.

But there are famous objections to this claim, one of which is made by Intuitionistic Logic. Intuitionists read a sentence like (p) as making an assertion that we have a PROOF that (p) holds. But if you read (p) that way, then (p V ~p) looks a bit weird, since it says we have a PROOF that (p V ~p). But there are many claims for which we do not have a proof that-(p) or a proof that NOT-(p).

And so Intuitionism gives up excluded middle.

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