  We present a general method for deciding whether a Grothendieck topos
satisfies De Morgan's law (resp. the law of excluded middle) or not;
applications to the theory of classifying toposes follow. Specifically, we
obtain a syntactic characterization of the class of geometric theories whose
classifying toposes satisfy De Morgan's law (resp. are Boolean), as well as
model-theoretic criteria for theories whose classifying toposes arise as
localizations of a given presheaf topos.
