In my logic class, we explored whether various quantified statements implied others. We agreed that the statement "Everyone loves everyone" implies that "Everyone loves someone":
$Px$ = $x$ is a person
$Lxy$ = $x$ loves $y$
$$1. (∀x)(Px \implies (∀y)(Py \implies Lxy))$$
$$2.(∀x)(Px \implies (∃y)(Py ∧ Lxy))$$
We didn't go into it in class however it seems to me that this form only holds try when the inner and outer predicates are the same.
If I change the inner predicate to something else:
$Kx$ = $x$ is a kitten
$$3.(∀x)(Px \implies (∀y)(Ky \implies Lxy))$$
No longer implies that:
$$4.(∀x)(Px \implies (∃y)(Ky ∧ Lxy))$$
The contradictory case is that there are people, but in fact no kittens exist. In this case it is true that for every person, for everything else, its kitten-hood implies the person's love for it (antecedent $Kx$ is always false), but it is not the case for every person (anyone actually) that there exists a kitten that they love.
It worked, however, when the predicates were the same, because if there were people to do the loving then there were people available to be loved even if it was just a solitary narcissist.
Is my understanding correct? It seems counter-intuitive to reject 3 -> 4 when the form is so similar to 1 -> 2. Is there more to it than this?