Consider the two first order symbolic statements:
$$\forall x \in S, \big( \phi(x) \rightarrow \phi'(x) \big)$$
and
$$\forall x , \big( \phi(x) \rightarrow \phi'(x) \big)$$
I interpret the first statement as saying, "For all objects in set $S$, one of three scenarios must be true for an arbitrary $x^* \in S$:
- if a specific $x^* \in S$ satisfies the $\phi$ property, then $x^*$ satisfies the $\phi'$ property.
- if a specific $x^* \in S$ does not satisfy the $\phi$ property, then $x^*$ satisfies the $\phi'$ property.
- if a specific $x^* \in S$ does not satisfy the $\phi$ property, then $x^*$ does not satisfy the $\phi'$ property.
I believe this is correct.
Now, more and more I am beginning to see symbolic statements of the second form, where there is no set that the $\forall x$ specifically refers to. How should I interpret this? I am becoming more acquainted with set theory (Zermelo–Fraenkel set theory, specifically) and so am reluctant to say that this particular $\forall x$ is referencing the universe.
At the moment, the only remedy I have implemented to address this confusion is by noticing that the set for which the $\forall x$ references is often times embedded within one of the $\phi$ statements. So, what I really do is "remove" that embedded property from $\phi$ and take it out to where the $\forall x$ symbol is located. In doing this, I feel like I am effectively changing the $\phi$ statement to some new statement $\psi$ (where...in a very brutally informal way "$\psi=\phi - \text{set for which x references}"$...i.e. $\psi$ retains all property descriptions of $\phi$ besides the set for which $x$ references.
Is this the correct approach...or do universal quantifiers not necessarily need to reference a specific set? If so, how exactly should I interpret the second statement? Any clarity would be greatly appreciated! Cheers~