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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:

  1. Plurally quantifying over possible universes themselves.
  2. 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":

  1. 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.
  2. 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.
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Thanks for asking this question and forcing my mind to expand. I'm going to take a stab at the answer and phrase it in general philosophical language since some of the specificity of the mechanics of quantification in the set-theoretic domain are beyond me. If I miss the mark, the blame is mine obviously.

If one accepts a set-theoretic multiverse with a realist interpretation, the as I see it, there would be indeed three constraints in discussion:

1 - The multiverse represents the primary and fundamental domain of discourse, and all presumptions of ontolological and existential quantification that vary are taken to delineate a distinct sub-domain of discourse, the universe.
2 - The universe of discourse in question therefore is nothing more than in nominalist thinking an adequate theory of set-theory thus shifting set-theoretic reason as a process to an mathematical object, which then makes the primary domain of discourse a metamathematical process.
3 - To accommodate modal realism as a basis of any set-theoretic discussion entails making all set-theoretic models qua objects real objects, and by extension requires the multiverse qua object to also be real, at least what I recall from scanning the paper (waiting on some additional resources to help me decipher some of the technicals of set-theory). To argue that because all of the parts are real, the whole must also be real sets off my sniff detector, because that seems to be the fallacy of composition, but that's just an intuition.

The end results might be a proposition such as:

There exists a multiverse which is the universal set of set-theoretic universes which can be predicated as real in the ontological sense.

Not that I'm quite smart enough to understand the question fully, but it just seems to me that you're looking for some clarification. You've suggested that what's missing from this proposition is the nature of the quantification itself used to reason over the metauniversal domain of discourse for which you suggest two variables, the universe and the proposition related to it affirmation. Perhaps you're looking to construct some sort of formal system to accompany this sort of discourse such that:

∃Q,∃P,∃S,∃!M:s∈M,real(S)->real(M),well-formed(P,S,Q)

That is to say, that in some formal multiuniverse theory, the ontology consists of quantifications to determine propositions, set-theories, and the multiverse, to constrain statements about the ontological commitment of the universe, and to determine what constitutes a legitimate statement in the theory by use of Q which represents a variable bound over a domain of discourse devoted to quantification itself. Intuitively, this would have to be the case, right? You cannot have a plurality of S without a plurality of Q since the axiomatic basis of any given S might diverge slightly, and such a divergence might entail a difference in the rules of quantification. I'm not a set-theorist, but it seems to me that moving from ZF to ZFC provides a necessary consequence in providing a new quantification rule regarding indexed sets. Thus formal modeling of S1 and S2 requires Q itself to contain statements BOTH containing ∃ and variables bound to P∈S AND P∈M. As such there must be quantification logic unique to M to cover the latter case and embrace the diversity of quantification in each S.

[D]oes a parallel alternation in quantifier theory allow us to express the concept of set-theoretic multiverses?

So the answer would seem to be yes by virtue of how formal systems are constructed and used in practice, and that the construction of such a model-theoretic system, while plainly more complicated because it requires a syntax and semantics for every S in M in addition to a syntax and semantics for M itself, isn't in principle divergent from proof and models as traditionally conceived. In fact, poking around, I found an article quantifier variance covering the gist of a thesis put forth by the philosopher Eli Hirsch which I would presume would answer your questions in detail.

But then, I'm not quite sure I'm not confused about the nature of your question to begin with because of my own lack of familiarity with the methodology of set-theory that you invoke as examples.

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    This is an exciting and fantastic question, btw.
    – J D
    Jan 25, 2023 at 21:07
  • I think that by bringing the issue of quantifier variance, directly named, to my attention, you have indeed provided me a window to an answer to my question. My OP reference to "ideological commitments" comes from the closing section of the SEP article on the overall topic, and much of what they say there appears identical or parallel to what you say here. Jan 25, 2023 at 22:32
  • I had a chance to read the introduction to the Hirsch's book, and it seems his thesis is essentially that no natural language and natural language ontological expression is a privileged frame ostensibly because there seems to be a deeper ontological process that is independent of language per se. That is, if one is a realist, then no schema of ontology in English or French or any other language is superior to the others in the basic metaphysical skill of recognizing what is and isn't real. He does address on xvi anti-realist sentiments, such as quantifier logic would determine reality...
    – J D
    Jan 26, 2023 at 8:49
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    Plurally quantified intuitionistic logic is definitely a strong idea. It would smooth out a reference to "all real numbers" without having a specific "set of all real numbers," for example. The gray area where the Continuum is made to be a proper class could be traversed without class-scale quantification over absolute infinity (I imagine that for Kantian reasons intuitionists would prefer not quantifying over absolute infinity!). Jan 26, 2023 at 9:20
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    Interesting thought. Just like limits and neighborhoods are introduced to make rigorous the intuitions of continuity, Various formal quantification logics would make rigorous the intuitions of existence. I wish I were an academic. I feel like there's a paper in here somewhere to be produced. :D
    – J D
    Jan 26, 2023 at 9:28

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