Basic question on Implication

Question

Could anyone conceive of any predicates and Universe ( in mathematics, in the world, etc ) where we should use $\exists x ( P(x) \to Q(x) )$, and not necessarily $\forall x ( P(x) \to Q(x) )$ ?
I was thinking of some sittuation where the property of P implies property of Q for at least some individual on the domain, but not for all individuals on the domain. i think its non-existant sittuation ?
I tried to think , and the closest i got was to think that since we know that for some $n$, we have that $n$ is prime, plus $ 2^n-1$ is prime. So, i thought of using $\exists n$ ( $n$ is prime $\to$ $2^n -1$ is prime ) .
Would this be correct to use ?
Then i thought i could use $\exists n$ ( $n$ is prime $\land$ $2^n - 1 $ is prime ) instead, and those are not semantic equvialent, so now i'm kinda lost.

Answer 1

Consider $P(x) \equiv Cube(x)$, $Q(x) \equiv Small(x)$ in a world with a small cube $a$ and a big cube $b$.

Then $\exists x:P(x) \rightarrow Q(x)$ is true (due to $a$ being small), while $\forall x: P(x) \rightarrow Q(x)$ is false (because $b$ is a cube, but is not small).

Answer 2

$$P(x)\equiv(x\le 1),\qquad Q(x)\equiv(x=1)$$

$$\exists x(x\le1\implies x=1)$$ is true, but

$$\forall x(x\le1\implies x=1)$$ is false.

Answer 3

It is worth remarking that a claim of the form $\exists x ( P(x) \to Q(x) )$ is typically unlikely to be interestingly informative and worth saying.

Why so?

Well, $\exists x ( P(x) \to Q(x) )$ is true so long as $P(a) \to Q(a)$ is true for some case where $a$ newly dubs an element of the domain. But that material conditional will be true so long as its antecedent is false.

Hence, so long as there is something $a$ in the domain which doesn't satisfy $P$, we have $\exists x ( P(x) \to Q(x) )$.

So, if you know already that $P$ is not universally true of everything in the domain (so there is something in the domain who can dub $a$ where $P(a)$ is false), and this is probably the typical case, then $\exists x ( P(x) \to Q(x) )$ gives you no new information.