nLab De Morgan laws

Idea

In logic, the De Morgan laws refer to the the following laws:

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

More generally, the De Morgan laws are the statements, valid in various forms of logic, that De Morgan duality is mediated by negation.

In constructive mathematics

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

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

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.

Equivalent statements

This de Morgan’s law is equivalent to the following:

• weak excluded middle

• That the negation of every equivalence relation is an apartness relation

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 lattice-ordered abelian groups

Peter Freyd’s definition of a lattice-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 lattice-ordered abelian group:

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

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

with the join and meet being the lattice 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

• De Morgan duality

References

Named after Augustus De Morgan.

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

See also:

• Wikipedia, De Morgan’s laws