Wednesday, January 24, 2024

What plurals are there?

Plural quantification is meant to be a logical way of avoiding some technical and/or conceptual difficulties with sets and second-order quantification. Instead of quantifying over one thing, one quantifies over pluralities. Thus, a theist might say: For all xs, God thinks of the xs in their interrelationship.

What plurals are there? Intuitively, for any finite list of objects, there is a plurality of precisely those objects. After all, we can easily have a sentence about any finite plurality of things we have names for: Alice, Bob and Carl like each other. But what furthe pluralities are there?

An expansive proposal is plural comprehension: the axiom schema that says that for any formula F with free variables that include y, for any values of the free variables other than y, there are xs such that y is one of the xs iff F. Unlike the comprehension schema in naive set theory, there does not seem to be any direct Russell-type paradox for plural comprehension, because the xs are not in general an object, but multiple objects.

But plural comprehension on its own does not seem to quite settle what plurals there are. Suppose we have a plurality of nonempty disjoint sets. We can for instance ask: Is there a plurality of objects that includes exactly one object from each of these sets? If (a) there is a set of these disjoint sets, and (b) the Axiom of Choice holds for sets, then the answer is affirmative by plural comprehension. But of course whether the Axiom of Choice holds for sets is itself not philosophically settled, and further not every plurality of sets is such that there is a set of the sets in the plurality.

Observations of this sort show that plural quantification is not as metaphysically innocent as it may seem. You might have hoped that there is no further metaphysical commitment in allowing for plural quantification than in singular quantification. But we can now have substantive questions about what pluralities there are even after we have fixed what singular objects there are, even if we assume plural comprehension. For instance, suppose we think that the objects are the physical objects of the world plus the elements of a model of ZF set theory with ur-elements and with the negation of the Axiom of Choice. We can know what all the objects are, and it still not be decided what pluralities there are. For in the case of a set of disjoint nonempty sets that lacks a choice set, as far as I can tell, there still might be a "choice plurality" (a plurality that has exactly one object from each of the disjoint sets) or there might not be one. (And if you say, well, the Axiom of Choice is obviously true, I may try to come back with a similar issue regarding Choice for proper classes.)

Or I might make a similar point about the Continuum Hypothesis (CH). The following story seems quite coherent. Every uncountable subset of the real numbers is in a bijection with the set of reals (i.e., CH is true), but there is an uncountable plurality of real numbers not in bijection with the plurality of reals. (It's easy to define bijections of pluralities in terms of pluralities of pairs.) But it's also coherent that CH is true, but there is no such uncountable plurality of reals--i.e., that CH is true for sets but its analogue for pluralities is false.

We might try to get out of this by insisting that, necessarily, the right set theory has to have a stronger version of the Schema of Separation that allows for formulas free plural variables and for the plural-membership relation. But that's conceding that the theory of pluralities is metaphysically non-innocent, because now what pluralities there are will constrain what objects there are!

So the question of what restrictions we put on plurals is a really substantive question.

Next note that following point. There seem to be two particularly simple and non-arbitrary answers to the Special Composition Question which asks which pluralities compose a whole: nihilism (there are no non-trivial cases of composition) and universalism (every plurality composes a whole). But once we have realized that it is a substantive question what pluralities there are, it seems that what objects there are and affirming universalism, even with mereological essential thrown in, doesn't settle the question of what wholes there are. There is substantial metaphysics to be done to figure out what pluralities there are!

I say the above with a caution: there are various technicalities I am glossing over, and I wouldn't be surprised if some of them turned out to be really important.

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