Fallacy of Every and All
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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.
$\blacksquare$
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Sources
- Merrilee H. Salmon: Introduction to Logic and Critical Thinking (1995) $\S 11.5$, Appendix $B$: Every and All
