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

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## Theorem

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

Then:

- $\forall x: \map P x \dashv \vdash \neg \paren {\exists x: \neg \map P x}$
*If everything***is**, there exists nothing that**is not**.

## Proof

By the tableau method of natural deduction:

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

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

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

3 | 2 | $\neg \map P {\mathbf a}$ | Existential Instantiation | 2 | for some arbitrary $\mathbf a$ | |

4 | 1 | $\map P {\mathbf a}$ | Universal Instantiation | 1 | ||

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

6 | 1 | $\neg \paren {\exists x: \neg \map P x}$ | Proof by Contradiction: $\neg \II$ | 2 – 5 | Assumption 2 has been discharged |

$\Box$

By the tableau method of natural deduction:

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

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

2 | 2 | $\neg \paren {\forall x: \map P x}$ | Assumption | (None) | ||

3 | 2 | $\exists x: \neg \map P x$ | Sequent Introduction | 2 | Denial of Universality | |

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

5 | 1 | $\forall x: \map P x$ | Reductio ad Absurdum | 2 – 4 | 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.

## Source of Name

This entry was named for Augustus De Morgan.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and 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 $1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic