Alternation depth of quantifiers

Example. $\forall a,\ \exists b,\ \forall c,\ \exists d,\ ad-bc=1$.

Meaning. For every $a$, there is $b$ such that, for every $c$, there is $d$ with $ad-bc=1$.

Permutation. If we tried $\exists d,\ \forall a,\ \exists b,\ \forall c,\ \ldots$, the single $d$ would have to work for all $a$ and $c$ simultaneously, which is much stronger.

Conclusion. Alternation depth controls expressiveness; the order of quantifiers cannot generally be changed.