# De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 1

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## Theorem

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

Then:

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

## Proof

By the tableau method of natural deduction:

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

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

2 | 2 | $\neg \paren {\exists x: \neg \map P x}$ | 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: $\neg \EE$ | 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: $\neg \EE$ | 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: $\neg \EE$ | 3, 4 | ||

6 | 1 | $\neg \paren {\forall x: \map P x}$ | Proof by Contradiction: $\neg \II$ | 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.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3.1 \ \text{(i)}$: Statements and conditions; quantifiers

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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 $4$