Is $\forall x$ meaningful when there's no (specified or implied) domain for $x$?

Question

Warning: XY problem

My recent question here revolved (among other things) on whether the intersection of all elements in an empty set of sets is a matter of definition or convention.

When working in a Topological space (X,T), I suggested that for $S \subseteq T$, the intersection is defined by (definition A): $$ \bigcap S := \bigcap_{A\in S} (A) =\{x\in X \ \vert\ \forall A \in S: x \in A\}. $$
In which case the intersection of the empty set follows from the definition.

Another user disagreed and said (admittedly, I can't quite follow) that (Edit: "Definition" B):

The definition would say that $\forall x : x \in \cap \emptyset $, while of course there is no set that contains everything; the notion that we restrict to elements of X, so that X is"everything", is a convention.

Question: is $\forall x$ always abuse-of-notation, and simply shorthand for $\forall x \in P$ for some implicit P? or is $\forall x$ actually meaningful without any (even implied) restriction to some set over which we quantify?

Because it looks to me like this statement first denies the existence of "the set that contains everything", then immediately proceeds to quantify over elements of that set. But, I'm a student and unsure of my reasoning.

Answer 1

My OP is rooted in a misunderstanding, I now realize, and thus somewhat beside the point. What the comment I referenced was hinting at is that:

1. In axiomatic set theory intersection of a collection of sets is only defined for a non-empty collection.
2. ...because otherwise expressions like $\{x | True \}$, the set of all sets, which lead to a Paradox, creep in.
3. The book I was working with addressed this directly by adopting the convention that all sets under discussion are restricted to be subsets of $X$. That amounts to using Definition A rather than the standard definition from ZFC [1] [2].
4. For clarity, you have to explicitly state the use of this convention, because it is different from the standard definition in set theory.
5. According to [3], when used in a logic context, $\forall x:P(x)$ always implies that x is "Limited to the Domain Of Discourse"

[1] Paul Halmos, Naive set theory, Springer-Verlag 1974, pp15.

[2] Wikipedia:Axiom_of_union#Relation_to_Intersection

[3] Wikipedia:Quantifier_(logic)

Answer 2

The definition you've given is entirely valid (in the context at hand). To determine the contents of $\cap S$. You take each $x \in X$ and ask if, for every $A \in T$ is $x \in A$? The answer will always be no because one such $A$ is the empty set (by requirements of the definition of $T$), and thus in the end, the set of $x$'s one obtains must be empty.