1. ## First-order prefixes

Explain the difference between the first-order prefixes:
(a) $\displaystyle \exists x\forall y$ and $\displaystyle \forall x\exists y$;
(b) $\displaystyle \exists x\forall y\exists z$ and $\displaystyle \forall x\exists y\forall z$;
(c) $\displaystyle \forall x\exists y\forall z\exists w$ and $\displaystyle \exists x\forall y\exists z\forall w$.
First, I notice that each prefix can be obtained by negation from the other one in the pair. Also, I can show a formula such that different prefixes produce different truth values. For example, consider (c):
$\displaystyle \forall x\exists y\forall z\exists w(x+z=y+w)$ is true, but $\displaystyle \exists x\forall y\exists z\forall w(x+z=y+w)$ is false. I ask whether I missed something.

2. ## Re: First-order prefixes

Originally Posted by andrei
I ask whether I missed something.
Your answer is correct. Another thing you may add is whether $\displaystyle \exists x\forall y\,A(x,y)$ implies $\displaystyle \forall x\exists y\,A(x,y)$ and vice versa, and similarly for (b) and (c).