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

## Contents

## Theorem

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

Then:

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

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

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg \forall x: \map P x$ | Premise | (None) | ||

2 | 2 | $\neg \exists x: \neg \map P x$ | Assumption | (None) | ||

3 | 3 | $\neg \map P {\mathbf a}$ | Assumption | (None) | for an arbitrary $\mathbf a$ | |

4 | 3 | $\exists x: \neg \map P x$ | Existential Generalisation | 3 | ||

5 | 2, 3 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 2, 4 | ||

6 | 2 | $\map P {\mathbf a}$ | Reductio ad Absurdum | 3 – 5 | Assumption 3 has been discharged | |

7 | 1, 2 | $\forall x: \map P x$ | Universal Generalisation | 6 | as $\mathbf a$ was arbitrary | |

8 | 2 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 1, 7 | ||

9 | 1 | $\exists x: \neg \map P x$ | Reductio ad Absurdum | 2 – 8 | Assumption 2 has been discharged |

$\Box$

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\exists x: \neg \map P x$ | Premise | (None) | ||

2 | 2 | $\forall x: \map P x$ | Assumption | (None) | ||

3 | 1 | $\neg \map P {\mathbf a}$ | Existential Instantiation | 1 | ||

4 | 2 | $\map P {\mathbf a}$ | Universal Instantiation | 2 | ||

5 | 1, 2 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 3, 4 | ||

6 | 1 | $\neg \forall x: \map P x$ | Proof by Contradiction: $\neg \mathcal I$ | 2 – 5 | Assumption 2 has been discharged |

$\blacksquare$

## Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Reductio ad Absurdum.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.

## 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

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.4$: The Syllogism: $149$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.2$: Some Remarks on the Use of the Connectives*and*,*or*,*implies* - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3.1 \ \text{(i)}$: Statements and conditions; quantifiers - 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 $4$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.1$