How do you properly discern between these quantification sentences?

Question

The first sentence being something like $$\text{For all $x$, $P(x)$}$$ meaning that $P$ for any $x$. And the second $$\text{(For all $x$ such that $P(x)$})\dots Q(x)$$ supposed to mean that $Q$ only for the $x$ for $P$. How do you formalize this? How do you write this in terms of the symbols appearing in FO logic? Is this the above even a "right" way to write those two notions? And is the latter statement really saying $$ \forall x \in \{s|P(s)\} \dots $$ Thanks in advance.

Answer 1

It depends what comes after the ellipsis. By itself "For all x $P(x)$" can be written as $\forall x, P(x)$ and this means that for each $x$, $P(x)$ is true. However if you put the words "implies $Q(x)$" after the ellipsis in the first phrase then you have $\forall x, P(x)\implies Q(x)$ which can be written as "For all x such that $P(x), Q(x)$". This latter expression means $\forall x,\neg{(Q(x)\wedge (\neg(P(x))}$, which is clearly different from $\forall x, P(x)$