# De Morgan's Laws (Predicate Logic)/Denial of Universality

## Theorem

Let $\forall$ and $\exists$ denote the universal quantifier and existential quantifier respectively.

Then:

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

## Proof

 $[\forall x.P \paren x]^1$ $[P \paren y]^2$ $\neg P \paren y$ $\bot$ $_1$ $\neg \forall x.P \paren x$ $\exists x. \neg P \paren x _2$ $\neg \forall x.P \paren x$

So $\exists x. \neg P \paren x \vdash \neg \forall x.P \paren x$

$\blacksquare$

## Examples

### Example: $\forall x \in S: x \le 3$

Let $S \subseteq \R$ be a subset of the real numbers.

Let $P$ be the statement:

$\forall x \in S: x \le 3$

The negation of $P$ is the statement written in its simplest form as:

$\exists x \in S: x > 3$

## Source of Name

This entry was named for Augustus De Morgan.