nLab De Morgan duality

Idea

In logic, De Morgan duality is a duality between intuitionistic logic and dual-intuitionistic paraconsistent logic. 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.

The dualities

More explicitly, De Morgan duality is the duality between logical operators as shown in the table below:

Intuitionistic operator		Dual-intuitionistic operator
$\top$ (truth)		$\bot$ (falsehood)
$\wedge$ (conjunction)		$\vee$ (disjunction)
$\Rightarrow$ (conditional)		$\setminus$ (subtraction)
$\Leftrightarrow$ (biconditional)		$+$ (exclusive disjunction)
$\neg$ ($p \Rightarrow \bot$)		$-$ ($\top \setminus p$)
$\forall$ (universal quantification)		$\exists$ (existential quantification)
$\Box$ (necessity)		$\lozenge$ (possibility)

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
$\top$		$0$
$1$		$\bot$
$\&$		$\oplus$
$\otimes$		$\parr$
$^\bot$		$^\bot$
$\bigwedge$		$\bigvee$
$!$		$?$

As with classical negation, linear negation is self-dual. (For the categorical semantics of this see at dualizing object and at Wirthmüller context – Comparison of push-forwards.)

The first two rows of the intuitionistic/dual-intuitionistic/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.

Related concepts

• De Morgan laws