# 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.*

## Contents

## 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

- $\forall x: P \left({x}\right) \dashv \vdash \neg \exists x: \neg P \left({x}\right)$

*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

- $\neg \forall x: \neg P \left({x}\right) \dashv \vdash \exists x: P \left({x}\right)$

*If not everything***is not**, there exists something that**is**.

## Source of Name

This entry was named for Augustus De Morgan.