From what I understand,
in second order propositional logic, ∀¬x and ¬∀x are equivalent statements. Apparently these are not equal. ¬∀x ≡ ∃¬x
However, rendered into the English language, consider the following case study:
Statement: All human beings have appendixes. Response 1: That's not completely true. Response 2: That's completely not true.
Response 1, the implication seems to be that the responder knows about appendectomy, and that some people have had their appendix removed. In
Response 2, the implication seems stronger than that, and while the motivation for such an expression is likely more for emphasis than pure logic, the implication seems be that
No human beings have appendixes, which is obviously not true.
So with all due haste, the question: Which one of these is the correct word choice and why? Perhaps there are logicians in the audience that can enlighten us with specific reasoning.