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

Jump to navigation
Jump to search

## 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*

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: formulation 1/2If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 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 $4$