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

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## Theorem

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

### Formulation 1

- $\neg \paren {\forall x: \map P x} \dashv \vdash \exists x: \neg \map P x$

### Formulation 2

- $\vdash \neg \paren {\forall x: \map P x} \iff \paren{ \exists x: \neg \map P x }$

In text, this can be summarised as:

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

## 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.

## Sources

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.2$: Some Remarks on the Use of the Connectives*and*,*or*,*implies*

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- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: $149$ - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 3$: Quantifiers: Relations between quantifiers $4$