Hi Pete, I agree with your view on quantifiers and that one can view \\(\forall x \in X: \phi(x)\\) as \\(\bigwedge_{x \in X} \phi(x)\\) (similarly for "exists"). In a posetal category meets are products and the empty product (a limit of an empty diagram) is the terminal object. Specifically in the boolean poset \\(false \leq true\\), \\(false\\) is the initial object (reflecting the fact *ex falso quodlibet*, from falsehood it all follows) and \\(true\\) is the terminal one. But we'd need more apparatus to turn analogy into theorem.

**Added later**: in classical FOL you can say \\(\forall x \in \emptyset: \phi(x) \overset{(1)}{\iff} \neg \neg \forall x \in \emptyset: \phi(x) \overset{(2)}\iff \neg \exists x \in \emptyset: \neg \phi(x)\\), but \\(\exists x \in \emptyset: \neg \phi(x)\\) is false (no candidate \\(x\\)), so \\(\forall x \in \emptyset: \phi(x)\\) holds.

(1) by double negation, (2) by "cuantificational de Morgan".