Order of quantifiers

Summary

In a statement with several quantifiers, the order of different quantifiers changes the meaning. For a predicate $P(x, y)$, the statement $\exists y \, \forall x \, P(x, y)$ implies $\forall x \, \exists y \, P(x, y)$, but the converse is false in general. Two quantifiers of the same type commute: $\forall x \, \forall y \, P(x, y)$ is equivalent to $\forall y \, \forall x \, P(x, y)$, and the same holds for $\exists$.

Remarks

On the integers, $\forall x \, \exists y \, (y > x)$ is true, but $\exists y \, \forall x \, (y > x)$ is false. In $\forall x \, \exists y$, the choice of $y$ can depend on $x$. The definitions of continuity and of uniform continuity differ only in the order of quantifiers.

Explanations

L1 human

"Every person has a mother" is $\forall x \, \exists y$: each person has their own mother. "Some person is the mother of every person" is $\exists y \, \forall x$: one mother for all. The first sentence is true, and the second is false.