Fallacy of Every and All

Fallacy
A statement containing both 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
Proof by Counterexample:

Let $x$ and $y$ be 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.