One of the ways in which Hamkins expresses the multiverse standpoint is as the assertion that there is no "absolute background concept of sets or even ordinals." He spells out examples of ill-founded and nonlinear metatheories of axioms in service of that statement. So it seems like the multiverse standpoint involves eschewing a moment of quantification over "the" multiverse, on pain of simply collapsing into a theory of a universe of universes (a second-order universe, then, as much as a multiverse otherwise described).
Now, although plural quantification is supposed to be "ontologically innocent" in such a way as to appeal to "non-realists" about sets in that PQ doesn't require understanding plural cardinality in terms of sets (one can quantify over multiple things without representing a singular set containing those things), yet its innocence seems as if it can be taken for a sort of "political neutrality": so that one can be a realist-pluralist who surmises that logical pluralism is reflected in the plurality of mathematical worlds by the by. In other words, plural quantification in multiverse theory would be such as:
- Plurally quantifying over possible universes themselves.
- Plurally quantifying over propositions about possible universes.
Reasoning: if one quantifies over multiple things in normal set theory, one forms a singular set of those things, and worse, the singleton axiom allows such constructions so generally as to conflict with the representation of an unrestricted set (which would not be contained in a singleton, not even a self-singleton). Diversified quantifier theory in this event grows mainly along the "cardinality quantifier" branch, which amounts to a diversification of singular quantification. Accordingly, the levels of cardinality (and hence possible quantification) describable/characterizable on a given such theory will affect one's sense of quantifying over things in general, per the theory; a theory according to which a given kind of cardinality doesn't exist can't quantify over objects that would have to be quantified over by that kind of cardinality, so such a theory lacks those objects to boot.
By contrast, plural quantification would not require specifying which infinite and nonstandard cardinals do or do not exist; even ∃xx as a representation does not mean singular quantification using the singular set {1} or {{0}} or {0, {0}} (or whatever). This can be seen in that a different way of indicating plural quantification could have been used than two iterates of x. Alternatively, if there is a singular quantifier that plays into plural quantification even so, and if it is a quantifier over dualities as such, we could "at worst" say that the dual singular quantifier can be used effectively even with just a "thin" theory of singular quantification in effect. So to say, if we have a concept of duality that is neutral with respect to standard or nonstandard natural numbers (if it adverts to "metatheoretic" natural numbers) as such, then we have a small, and fundamental, moment of singular quantification in play, but this does not undermine the remainder of the appeal to plural quantifiers.
Option (2) listed above might be even more "ontologically neutral" than (1), depending on how one differentiates possible worlds from sets of (modalized) propositions. At any rate, again, it seems as we if plurally quantify over possible axioms, we don't have to quantify over specific axioms so as to inadvertently "presuppose" a theory that we're actually trying to diversify off from (or apart from, in stronger cases).
If differences in a background theory of quantifiers allow us to express the concept of proper classes (as a "closest theoretical equivalent to universal sets"), does a parallel alternation in quantifier theory allow us to express the concept of set-theoretic multiverses (as an alternative "closest theoretical equivalent to a universal set")?
Further considerations/"evidence":
- Quantification defined modulo ur-elements/Quine atoms? Though one can arbitrarily imagine a universe with an ur-element that otherwise played the same role as the empty set in the common universe, or do something similar even with self-singletons, it still seems as if one can generically characterize the quantifier signature of such terms as possibly diverging from their WF(X) signature (i.e. even Quine atoms can be fit into a "relatively well-founded" world, the world WF(At) (for At = "(Quine) Atoms"). Importantly for present purposes, then, ill-founded cardinality (and ill-ordered cardinality, for that matter) might be open to generic representation in terms of plural quantification.
- Singleton-minus is a hypothetical set "so large that it can't fit inside a singleton." In other words, the singleton axiom, and any use of another axiom to justify singletons indiscriminately, is eventually violated in a universe with singleton-minus. Thus far this is a rather thin description, though, and one would like some positive, substantial account of such an object. Could plural quantification do the trick? We would be claiming, then, that something about singleton-minus's special elements "mandates" plural quantification over those elements, where for some reason the background logic permitted singular quantification over elements of preceding types. I don't know how intelligible that hypothesis is (it's meant in the vein of talk about how logical properties of other kinds figure in large-cardinal introductions, e.g. as with extendible or weakly/strongly compact cardinals), though.