$P (x)$ and $Q (x)$ are predicates and $a$ is an element of the domain of the variable $x$

Question

$P (x)$ and $Q (x)$ are predicates and $a$ is an element of the domain of the variable $x$.

Assuming that $[∃x (Q (x) → P (x))] ∧ Q (a)$ is true, can we conclude that $P (a)$ is true?

Answer 1

Not necessarily. With the expression $$\exists x: Q(x)\rightarrow P(x)$$ you are indicating that there exists some x for which $P(x)\rightarrow Q(x)$ is true.

Then, knowing $Q(a)$ is true isn't enough to imply $P(a)$, since $a$ could be "one of those $x$s" that satisfy $P(x)\rightarrow Q(x)$, but it could also not be part of the set of $x$s that satisfy that. There just isn't enough information to discern which situation we are in.