De Morgan's Laws (Predicate Logic)
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Theorem
These are extensions of De Morgan's laws of propositional logic.
They are used to connect the universal quantifier $\forall$ with the existential quantifier $\exists$.
They can be stated as:
| \(\displaystyle \) | \(\displaystyle \forall x: P \left({x}\right)\) | \(\dashv \vdash\) | \(\displaystyle \neg \exists x: \neg P \left({x}\right)\) | \(\displaystyle \) | If everything is, there's nothing that isn't. | ||
| \(\displaystyle \) | \(\displaystyle \forall x: \neg P \left({x}\right)\) | \(\dashv \vdash\) | \(\displaystyle \neg \exists x: P \left({x}\right)\) | \(\displaystyle \) | If everything isn't, there's nothing that is. | ||
| \(\displaystyle \) | \(\displaystyle \neg \forall x: P \left({x}\right)\) | \(\dashv \vdash\) | \(\displaystyle \exists x: \neg P \left({x}\right)\) | \(\displaystyle \) | If not everything is, there's something that isn't. | ||
| \(\displaystyle \) | \(\displaystyle \neg \forall x: \neg P \left({x}\right)\) | \(\dashv \vdash\) | \(\displaystyle \exists x: P \left({x}\right)\) | \(\displaystyle \) | If not everything isn't, there's something that is. |
Proof
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Source of Name
This entry was named for Augustus De Morgan.
