# De Morgan's Laws (Predicate Logic)

This proof is about De Morgan's Laws in the context of predicate logic. For other uses, see De Morgan's Laws.

## Theorem

These are extensions of De Morgan's laws of propositional logic.

They are used to connect the universal quantifier $\forall$ with the existential quantifier $\exists$.

They can be stated as:

### Assertion of Universality

$\forall x: \map P x \dashv \vdash \neg \exists x: \neg \map P x$
If everything is, there exists nothing that is not.

### Denial of Existence

$\forall x: \neg P \paren x \dashv \vdash \neg \exists x: P \paren x$
If everything is not, there exists nothing that is.

### Denial of Universality

$\neg \forall x: P \paren x \dashv \vdash \exists x: \neg P \paren x$
If not everything is, there exists something that is not.

### Assertion of Existence

$\neg \forall x: \neg \map P x \dashv \vdash \exists x: \map P x$
If not everything is not, there exists something that is.

## Source of Name

This entry was named for Augustus De Morgan.