Fallacy of Every and All
Jump to navigation
Jump to search
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
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