are the predicates $\forall x P(x) \land Q(x)$ and $\forall x(P(x) \land Q(x))$, the same?

Question

are the predicates $\forall x P(x) \land Q(x)$ and $\forall x(P(x) \land Q(x))$, the same?

Answer 1

Unless you have some in-house parsing rule, no, they aren’t the same.

The natural parsing of the first is that it is $$\Bigl( \forall xP(x)\Bigr)\wedge Q(x).$$ The $\forall x$ only bounds the $P(x)$, so this formula has $x$ as a free variable (it appears unbounded in the second part of the conjunction).

On the other hand, the second formula is $$\forall x\Bigl( P(x)\wedge Q(x()\Bigr)$$ which has no free variables (is a sentence): $x$ is bound by $\forall x$ in all its occurrences.

Since the second formula is a sentence (no free variables) but the first formula has a free variable, they are not the same.

(Again, unless you have a parsing rule that gives $\wedge$ precedence so that when parsing the first formula you would first associate $P(x)\wedge Q(x)$, and then apply the quantifier.)