linguistic:translation
In intuitionistic logic

Three out of the four implications of de Morgan's laws hold in intuitionistic logic. Specifically, we have

and

The converse of the last implication does not hold in pure intuitionistic logic. That is, the failure of the joint proposition P ∧ Q {\displaystyle P\land Q} cannot necessarily be resolved to the failure of either of the two conjuncts. For example, from knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow who did not show up. The latter principle is equivalent to the principle of the weak excluded middle W P E M {\displaystyle {\mathrm {WPEM} }},

This weak form can be used as a foundation for an intermediate logic. For a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience L L P O {\displaystyle {\mathrm {LLPO} }}, which however is different from W L P O {\displaystyle {\mathrm {WLPO} }}.

The validity of the other three De Morgan's laws remains true if negation ¬ P {\displaystyle \neg P} is replaced by implication P → C {\displaystyle P\to C} for some arbitrary constant predicate C, meaning that the above laws are still true in minimal logic.

Similarly to the above, the quantifier laws:

and

are tautologies even in minimal logic with negation replaced with implying a fixed Q {\displaystyle Q}, while the converse of the last law does not have to be true in general.

Further, one still has

but their inversion implies excluded middle, P E M {\displaystyle {\mathrm {PEM} }}.