Predicate Logics - double negation HELP me understand
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.