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

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Theorem

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

Then:

$\neg \forall x: \neg \map P x \dashv \vdash \exists x: \map P x$
If not everything is not, there exists something that is.


Proof

By the tableau method of natural deduction:

$\neg \forall x: \neg \map P x \vdash \exists x: \map P x$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \forall x: \neg \map P x$ Premise (None)
2 2 $\neg \exists x: \map P x$ Assumption (None)
3 2 $\forall x: \neg \map P x$ Sequent Introduction 2 Denial of Existence
4 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 1, 3
5 1 $\exists x: \map P x$ Reductio ad Absurdum 2 – 4 Assumption 2 has been discharged

$\Box$


By the tableau method of natural deduction:

$\exists x: \map P x \vdash \neg \forall x: \neg \map P x$
Line Pool Formula Rule Depends upon Notes
1 1 $\exists x: \map P x$ Premise (None)
2 2 $\forall x: \neg \map P x$ Assumption (None)
3 1 $\map P {\mathbf a}$ Existential Instantiation 1
4 2 $\neg \map P {\mathbf a}$ Universal Instantiation 2
5 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 3, 4
6 1 $\neg \forall x: \neg \map P x$ 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.


Source of Name

This entry was named for Augustus De Morgan.


Sources