Vacuous Domain Mixing For all and There Exists. May be this is a stupid question but I was thinking we know that suppose $D = \varnothing$ then $\forall\, x \in D \,P(x)$ is true vacuously and $\exists y\in D\, P(y)$ is false. What is you mix the two like $\forall x \in D \,\,\exists \,y \in D \,Q(x, y)$. Then is $Q$ true or false or neither?

$\bigl(\forall x \in D\bigr) \bigl(\exists y \in D\bigr) Q(x,y)$ has the form $\bigl(\forall x \in D\bigr) P(x)$, where $P(x)$ expands to $\bigl(\exists y \in D\bigr) Q(x,y)$. So it is vacuously true.

If you switch the order of the quantors, it becomes vacuously false, since you get a proposition of the form $\bigl(\exists y \in D\bigr) R(y)$ (with $R(y) = \bigl(\forall x \in D\bigr) Q(x,y)$).

When you have an empty domain $D$, only the outermost quantor matters, any outermost $\bigl(\forall x \in D\bigr)$ makes it vacuously true, $\bigl(\exists y \in D\bigr)$ vacuously false.