If (condition) is true for all $x \in X$, does there exist an $x$ such that it has such a condition for all $x \in X$?

Question

If $\forall x \in X$, (condition) is true, is it then also true to say $\exists x$ such that (condition holds) $\forall x \in X$?

Answer 1

I don't understand the downvotes, this is actually an interesting question. It boils down to whether you allow vacuous quantifiers (many systems don't), that is quantifiers that do not bind any variable.

Let's say that your condition is $P$ and I'm going to change from the extensive formulation of $x \in X$ to $x$ has property $F$.

So $\forall x \in X, \text{(condition is true)}$ becomes $\forall x (Fx \to Px)$ which is a closed formula, call it $\phi$. What you're asking is whether $\exists x\phi$ is true... it depends, if you allow empty domains then no. If you allow for vacuous quantifiers (without empty domains) then it makes sense to let $\exists x\phi$ be logically equivalent to $\phi$, and the same with the universal quantifier.

Know though that these formulae do not correspond to any meaningful english sentence, it's just a pure logical shibboleth.