Sorry for maybe a silly question but i need to understand how ¬(¬∀x ¬A(x)) equals ∀x ¬A(x)
In my mind, the negation before the parenthesis will be applied to both ¬∀x and ¬A(x). So it would look like this:
¬(¬∀x ¬A(x)) = ¬¬∀x ¬¬A(x)
A double negation would become positive and would then let ¬(¬∀x ¬A(x)) equal to ∀x A(x).
So, why would ¬(¬∀x ¬A(x)) equal ∀x ¬A(x)? Why wouldnt ¬(¬∀x ¬A(x)) equal ∀x A(x)?
The semantics of negation are very simple:
$\lnot \alpha$ is True iff $\alpha$ is false.
In your example, $\lnot (\lnot \forall x \lnot A(x))$ is True iff $\lnot \forall x \lnot A(x)$ is False iff $\forall x \lnot A(x)$ is True iff for all $x$, $\lnot A(x)$ is True iff for all $x$, $A(x)$ is false.
Perhaps the issue would become simpler if we fully parenthesize your expression: $$ \lnot \stackrel 1( \lnot \stackrel2( \forall x \stackrel3( \lnot \stackrel4( A\stackrel5( x \stackrel 5) \stackrel 4) \stackrel 3) \stackrel 2) \stackrel 1). $$ The superscripts count the nesting depth.
