nLab
de Morgan duality

In logic, de Morgan duality is a duality between intuitionistic logic and paraconsistent logic (in its dual-intuitionistic form). In classical logic and linear logic, it is a self-duality mediated by negation. Although it goes back to Aristotle (at least), its discovery is generally attributed to Augustus de Morgan.

More explicitly, this is the duality between logical constants and operators as shown in the table below:

Intuitionistic operator		Dual-intuitionistic operator
$\top$ (truth)		$\perp$ (falsehood)
$\wedge$ (conjunction)		$\vee$ (disjunction)
$\Rightarrow$ (conditional)		$\setminus$ (without?)
$\iff$ (biconditional?)		$+$ (exclusive disjunction)
$\neg$ ($p\Rightarrow \perp$)		$-$ ($\top \setminus p$)
$\forall$ (universal quantification)		$\exists$ (existential quantification)
$\square$ (necessity?)		$◊$ (possibility?)

Note that the first two operators in each column exist in both intuitionistic and dual-intuitionistic propositional logic and the last two in each column exist in both forms of predicate logic and modal logic (respectively), but they are still dual as shown. All of these exist in classical logic (although some of the paraconsistent operators are not widely used), and the two forms of negation ($\neg$ and $-$) are the same there.

In linear logic, this extends to a duality between conjunctive and disjunctive operators:

Conjunctive operator		Disjunctive operator
$\top$		$0$
$1$		$\perp$
$\&$		$\oplus$
$\otimes$		$⅋$
${}^{\perp }$		${}^{\perp }$
$\bigwedge$		$\bigvee$
$!$		$?$

As with classical negation, linear negation is self dual.

The first two rows of the intuitionistic/paraconsistent/classical duality generalise to arbitrary lattices, including subobject lattices in coherent categories, and from there to the duality between limits and colimits in category theory:

Limit		Colimit
top		bottom
meet		join
intersection		union
terminal object		initial object
⋮		⋮

So in a way, all duality in category theory is a generalisation of de Morgan duality.

The de Morgan laws are the statements, valid in various forms of logic, that de Morgan duality holds. In the foundations of constructive mathematics, de Morgan's Law usually means the statement $\neg (p\wedge q)⊢\neg p\vee \neg q$, since the other de Morgan laws of intuitionistic propositional logic (the converse of this one, as well as $\neg (p\vee q)\equiv \neg p\wedge \neg q$ and the nullary versions $\neg \top \equiv \perp$ and $\neg \perp \equiv \top$) are already constructively valid.