Fallacy of Every and All

Fallacy
A statement containing universal quantifiers and existential quantifiers has a different meaning if the order of the quantifiers is reversed.

To not recognize such a shift in meaning is to commit the Fallacy of Every and All.

Counterexample
This is a Proof by Counterexample.

Let the Universe of Discourse be the Natural Numbers.


 * $\forall x \exists y : y > x$: for every $x$ there is some $y$ such that $y$ is greater than $x$.

By Peano's Second Axiom, this is true.


 * $\exists y \forall x: y > x$: there is some $y$ such that for every $x$, $y$ is greater than $x$.

By Peano's Second Axiom, this is false.