Difference between $(\forall x.\rm{Cat}(x))\rightarrow(\forall y.\rm{Loves}(y,y))$ and $\forall x.\forall y.(\rm{Cat}(x)\rightarrow\rm{Loves}(y, y))$?

Question

I'm struggling to wrap my head around the difference between these two statements:

1. $(\forall x. \rm{Cat}(x)) \rightarrow (\forall y. \rm{Loves}(y, y)) $

2. $\forall x. \forall y. (\rm{Cat}(x) \rightarrow \rm{Loves}(y, y))$

Translated to plain English:
1. If everything is a cat, then everything loves itself.
2. If any cats exist, then everything loves itself.

Am I on the right track?

Answer 1

In order to clarify the difference between the propositions, consider their negations:

1. $ \neg ((\forall x.\operatorname{Cat}{(x)}) \rightarrow \forall y.\operatorname{Loves}(y,y)) ~\equiv~ (\forall x.\operatorname{Cat}{(x)}) \wedge \exists y.\neg \operatorname{Loves}(y,y) $
2. $ \neg (\forall x.\forall y. \operatorname{Cat}{(x)} \rightarrow \operatorname{Loves}{(y,y)}) ~\equiv~ \exists x.\exists y. \operatorname{Cat}{(x)} \wedge \neg \operatorname{Loves}{(y,y)} $

It follows that:

1. the first proposition is false if everything is a cat and some cat does not love itself; and
2. the second proposition is false if there is a cat and something does not love itself.

For reference, we also know that:

1. the first proposition is true if something is not a cat or everything loves itself; and
2. the second proposition is true if, whenever you consider any pair of things in the domain of discourse, either the first is not a cat or the second loves itself. These might or might not be the same things. For example, self-hating dogs and narcissistic cats satisfy this proposition.