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.

$\forall x \, \exists y : x = y$: for every $x$ there is some $y$ such that $x$ equals $y$.

Since $x = x$, this is true.

$\exists y \, \forall x: x = y$: there is some $y$ such that for every $x$, $x$ equals $y$.

Since no natural number equals both $1$ and $2$ at the same time, this is false.

$\blacksquare$

Sources

• 1995: Merrilee H. Salmon: Introduction to Logic and Critical Thinking: $\S 11.5$, Appendix $B$: Every and All