I might be working up a rant about the elegant symmetry of De Morgan's laws, and how unfortunately it's wrong.
There are four laws:
1. ¬A ∧ ¬B → ¬(A ∨ B)
2. ¬(A ∨ B) → ¬A ∧ ¬B
3. ¬A ∨ ¬B → ¬(A ∧ B)
4. ¬(A ∧ B) → ¬A ∨ ¬B
Intuitionistic logic accepts 1–3, but not 4. #4 claims that if there's no proof of A and B simultaneously, it must be for one of two reasons: there are no proofs of A at all, or there are no proofs of B at all.
But that's obviously not true except in extremely limited situations. It's like saying that if a restaurant never serves meat and fish on the same day, it's because it never serves meat or because it never serves fish.
(In contrast, the converse is obviously true: a meatless restaurant certainly never serves meat on the same day as fish. And the inverse: if the restaurant never serves meat, _and_ it never serves fish, then certainly it never serves either meat or fish, and vice versa.)
The symmetry is elegant, but wrong in general.
IL has a similar asymmetry in the duality between ∃ and ∀, of which ∨ and ∧ are special cases.

André Polykanine
in reply to Elena Brescacin • • •