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

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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:

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

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