Intuitive Reason that Quantifier Order Matters

Take $P(x, y)$ to mean $y$ is a parent of $x$.

Then $\forall x \exists y P(x, y)$ means everybody has a parent, while $\exists y \forall x P(x, y)$ means there is someone who is the parent of every son and daughter.


Let $P(O,C)$ mean car $C$ is owned by owner $O$.

Then $\forall C \ \exists O \ P(O,C)$ means every car has an owner.

However, $\exists O \ \forall C \ P(O,C)$ means that some owner owns all cars.

Clearly these mean different things.


Take $P(x,y)$ to be "$x$ is friends with $y$".

One statement says 'everyone has a friend', the other says 'someone is friends with everyone'.

Tags:

Predicate Logic

Quantifiers

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