# De Morgan's Laws (Predicate Logic)/Assertion of Existence

## Contents

## Theorem

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

Then:

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

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

## Proof

## Source of Name

This entry was named for Augustus De Morgan.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.4$: The Syllogism: $150$ - 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 $3$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.1$: What is a Set?