TITLE: Alternatives to the law of the excluded middle
QUESTION [32 upvotes]: As a sentential logic, intuitionistic logic plus the law of the excluded middle gives classical logic.
Is there a logical law that is consistent with intuitionistic logic but inconsistent with classical logic?

REPLY [21 votes]: In first-order logic, the sentence
$$\neg\forall x,y(\neg\neg x=y \to x=y)$$
is consistent with intuitionist logic but not with classical logic.
One might call this "the fuzziness of identity". In synthetic differential geometry, as axiomatized in Models for smooth infinitesimal analysis by Moerdijk and Reyes, this is actually a theorem about the real numbers.