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

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## Contents

## Theorem

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

Then:

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

*If everything***is**, there exists nothing that**is not**.

## Proof

## Source of Name

This entry was named for Augustus De Morgan.

## Sources

- 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 $1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.1$: The need for logic