# What is the difference between $\forall x [ (\exists y Pxy) \longrightarrow \phi ]$ and $\forall x \exists y [Pxy \longrightarrow \phi]$?

Suppose $$Pxy$$ is a $$2$$-ary predicate and $$\phi$$ is just some fixed arbitrary predicate form.

1. $$\forall x [ (\exists y Pxy) \longrightarrow \phi ]$$

2. $$\forall x \exists y [Pxy \longrightarrow \phi]$$

Is the only difference between 1. and 2. in the case where $$\exists y$$ is not true? i.e. We may be in an interpretation $$\mathcal{I}$$ where the universe/domain of discourse may be empty, so does that then make 1. true and 2. false?

So it seems to me that if we restrict ourselves to interpretations $$\mathcal{I}$$ whose domain of discourse is not empty then 1. and 2. are logically equivalent$$^1$$.

Remarks: $$^1$$I'm thinking of the interpretations in the context of first-order logic, but what about second-order logic or other higher orders of logic? (Though I have not studied anything beyond first order logic, but maybe someone can say something about this? But this probably is another question in its own right)

• See Prenex Noraml Form: 1 is equiv to $∀x∀y[Pxy⟶ϕ]$ May 15 at 13:27

Consider an interpretation with domain $$\mathbb N$$ and let $$\phi := (0=1)$$ (or any other formulas which is False).
Then interpret $$Pxy$$ with $$x < y$$.
In this interpretation, 1. will be $$∀x[∃y(x < y) ⟶ (0=1)]$$, which is False, while 2. is $$∀x∃y[(x < y) ⟶ (0=1)]$$, which is True.