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.

The De Morgan laws

The De Morgan laws are the statements, valid in various forms of logic, that De Morgan duality is mediated by negation. For example, using the second line of the first table, we have

$$\array { \neg(p \wedge q) \equiv \neg{p} \vee \neg{q} ,\\ \neg(p \vee q) \equiv \neg{p} \wedge \neg{q} . }$$

Traditionally, the term is reserved for this line.

In constructive mathematics

In the foundations of constructive mathematics, De Morgan's Law usually means the statement

$$\neg(p \wedge q) \vdash \neg{p} \vee \neg{q} ,$$

since every other aspect of the first two lines is already constructively valid, the claim that negation mediates the De Morgan self-duality already has a name (the double negation law, equivalent to the principle of excluded middle), and no other line involves only operators that appear in intuitionstic propositional calculus. This de Morgan’s law is equivalent to the law of weak excluded middle.

In homotopy type theory

In the context of homotopy type theory, there are two versions of the constructive De Morgan’s law, depending on whether the “or” in the law is interpreted as a propositionally truncated sum type

$$\prod_{A,B} \neg(A \wedge B) \to \Vert \neg A + \neg B \Vert$$

or as an untruncated sum type:

$$\prod_{A,B} \neg(A \wedge B) \to (\neg A + \neg B).$$

However, Martin Escardo proved that the truncated and untruncated versions of De Morgan’s law are the same:

Lemma

Truncated De Morgan’s law implies weak excluded middle:

$$\prod_{A} \neg A + \neg\neg A.$$

Note that truncation or its absence is irrelevant in weak excluded middle, since $\neg A$ and $\neg\neg A$ are mutually exclusive so that $\neg A + \neg\neg A$ is always a proposition.

Proof

Let $B=\neg A$ in the truncated De Morgan’s law, and notice that $\neg(A\wedge \neg A)$ always holds.

Lemma

Weak excluded middle implies that binary sums of negations have split support:

$$\prod_{A,B} \Vert \neg A + \neg B \Vert \to (\neg A + \neg B).$$

Proof

By weak excluded middle, either $\neg A$ or $\neg\neg A$. In the first case, $\neg A + \neg B$ is just true. In the second case, either $\neg B$ or $\neg\neg B$. In the first subcase, $\neg A + \neg B$ is again just true. In the second subcase, we have $\neg\neg A$ and $\neg\neg B$, whence $(\neg A + \neg B) = 0$ and in particular is a proposition.

Theorem

Truncated De Morgan’s law implies untruncated De Morgan’s law,

$$\prod_{A,B} \neg(A \wedge B) \to (\neg A + \neg B)$$

Proof

Combine the two lemmas.

In the theory of pseudolattice ordered abelian groups

Peter Freyd’s definition of a pseudolattice ordered abelian group in Freyd 08 is equational, and so is a Lawvere theory and could be defined in a category with finite products and a generic object $A$ where every object is equivalent to a finite product of $A$.

Freyd then showed that the de Morgan laws are satisfied in any pseudolattice ordered abelian group:

$$-(p \vee q) = (-p) \wedge (-q)$$

$$-(p \wedge q) = (-p) \vee (-q)$$

with the join and meet being the pseudolattice operations and negation being the negation of the abelian group.

In particular, the de Morgan laws are valid for the integers, the rational numbers, and the Dedekind real numbers, with the join being the maximum function and the meet being the minimum function.

Related concepts

• De Morgan Heyting algebra

• De Morgan Heyting category

• De Morgan algebra

• De Morgan topos

• excluded middle

References

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)