The answer to this question which has been provided by my book has me kind of confused and my book is quite ambiguous about what I am about to ask: Is it obligatory that all quantors must be bound to a different formula when the entire formula is in prenex normal form?
To clarify myself I will give you guys an example:
translate the following to prenex normal form:
$[\forall x \neg K(x) \vee \forall x \neg C(x) \vee \forall x \neg D(x)] $
Two answers are possible, namely:
- $\forall x [\neg K(x) \vee \neg C(x) \vee \neg D(x)]$
OR
1.$ [\forall x \neg K(x) \vee \forall x \neg C(x)[y/x] \vee \forall x \neg D(x)[z/x]] $
2.$ \forall x,y,z [\neg K(x) \vee \neg C(y) \vee \neg D(z)]$
So my guess would be that both of these considered prenex normal form. The provided answer to this question, however, only considers the second option. But going by the laws of distribution, option 1 should also be correct.
I would be grateful if anyone could confirm/reject my hypothesis!