When constructing models for a first order modal logic, we have to decide whether we want to have a fixed domain (the possibilist semantics) or a world-relative domain (the actualist semantics).
In the fixed-domain model, the quantifiers range over all possible objects. Some will introduce an existence predicate, in order to restrict statements to the "actual" world.
The SEP page on modal logic summarizes the situation like this:
Furthermore, those quantifier expressions of natural language whose domain is world (or time) dependent can be expressed using the fixed-domain quantifier ∃x and a predicate letter E with the reading ‘actually exists’. For example, instead of translating ‘Some Man exists who Signed the Declaration of Independence’ by
∃x(Mx & Sx),
the defender of fixed domains may write:
∃x(Ex & Mx & Sx),
thus ensuring the translation is counted false at the present time.
I'm having trouble grasping how "∃x(Ex & Mx & Sx)" fixes an interpretation to the actual world any more than "∃x(Mx & Sx)" does.
As I understand it, these expressions are always evaluated at a world. So "Ex" will only be true in worlds where Ex is satisfied. (For instance, assume that G is an interpretation function, and assume that "Ex" is only satisfied in worlds where G(x) \in G(E).)
But wouldn't the same be true for "Mx"? Why should "Ex" be satisfied in one world any more than "Mx"?
What am I not grasping here?
My reading of the Modal Logic entry of SEP is that it does not claim that "$\exists x (Ex \wedge Mx \wedge Sx)$" fixes an interpretation to the actual world.
Introduction of the predicate letter $E$ merely gives an adopter of constant-domain quantification the ability to write a sentence asserting the existence of a man who signed the Declaration of Independence that may be false in one world and true in another, accessible world.
Interpreting $Ex$ as "$x$ actually exists in this world" makes the sentence false in this world and true in the world of 1777. (Provided the obvious interpretations are given to $M$ and $S$ too.)
In 1775, however, no man had signed the Declaration of Independence. The sentence "$\exists x (Ex \wedge Mx \wedge Sx)$" is false in that world too, though the interpretation is not fixed to the actual world.