A question about logical equivalence and bound variables within Predicate Logic

Question

Question 1: Are these two translated statements logically equivalent?

Statement A: ¬∃x(Nx ∧ Axc) (It is not the case, that for some x, x is a novelist and x admires Carol) Statement B: ∀x¬(Nx → Axc) (For all x, if x isn't novelist, then x does not admire Carol)

Translated from 'No novelists admire Carol'.

Question 2: If quantifiers (∀x, ¬∀x, ∃x, ¬∃x) bind variables, how would I know if the following statement is true or not? 'Everything in that store is either overpriced or poorly made', or symbolically, '∀x(Sx → (Ox ∨ Px))'.

If the 'x' in 'Sx', for example, is bound and not free, and I am not allowed to ask [with an intention of falsifying that statement] something like: 'Do they sell pistachios in here?', I have no way of finding out whether the pistachios are indeed overpriced and poorly made, and if the initial statement itself is true or not.

The fact that variables that occur after a quantifier are bound, and not free, is taken from Velleman's book

Thanks

Answer 1

Qn 1. No. Here's the correct equivalence:

¬∃x(Nx ∧ Axc)

is equivalent to

∀x¬(Nx ∧ Axc)

for pushing a negation pass a quantifier flips the quantifier to its dual. And that's equivalent to

∀x(Nx → ¬Axc)

by the classical equivalence of ¬(A ∧ B) with (¬A v ¬B) with (A → ¬ B).

Qn. 2 You find out whether ∀x(Sx → (Ox ∨ Px)) by finding out whether everything in that store is either overpriced or poorly made. That might be a practical problem, but it isn't a logical one!