Plural quantification and the continuum hypothesis

Some people, including myself, are concerned that plural quantification may be quantification over sets in logical clothing rather than a purely logical tool or a free lunch. Here is a somewhat involved argument in this direction. The argument has analogues for mereological universalism and second-order quantification (and is indeed a variant of known arguments in the last context).

The Continuum Hypothesis (CH) in set theory says that there is no set whose cardinality is greater than that of the integers and less than that of the real numbers. In fact, due to the work of Goedel and Cohen, we know that CH is independent of the axioms of Zermelo-Fraenkel-Choice (ZFC) set theory assuming ZFC is consistent, and indeed ZFC is consistent with a broad variety of answers to the question of how many cardinalities there are between the integers and the reals (any finite number is a possible answer, but there can even be infinitely many). While many of the other axioms of set theory sound like they might be just a matter of the logic of collections, neither CH nor its denial seems like that. Indeed, these observations may push one to think that there many different universes of sets, some with CH and others with an alternative to CH, rather than a single privileged concept of “true sets”.

Today I want to show that plural quantification, together with some modal assumptions, allow one to state a version of CH. I think this pushes one to think analogous things about plural quantification as about sets: plural quantification is not just a matter of logic (vague as this statement might be) and there may even be a plurality of plural quantifications.

This is well-known given a pairing function. But I won’t assume a pairing function, and instead I will do a bunch of hard work.

The same approach will give us a version of CH in Monadic Second Order logic and in a mereology with arbitrary fusions.

Let’s go!

Say that a possible world w is admissible provided that:

1. w is a multiverse of universes

2. for any two universes u and v and item a in u, there is a unique item b in v with the same mass as a

3. the items in each universe are well-ordered by mass

4. for each item in each universe there is an item in the same universe with bigger mass

5. for each item c in a universe u if a is not of the least mass in u, a has an immediate predecessor with respect to mass in u.

The point of (3)–(5) is to ensure that each universe has a least-mass item and that there are only countably many items. If we assumed that masses are real numbers, we would just need (3) and (4).

Say that pluralities of items xx in a universe u and yy in a universe v of an admissible world correspond provided that for all natural numbers i, if a is in u and b is in v and a and b have equal mass, then a is among xx if and only if b is among yy. Two individual items correspond provided that they have equal mass.

Say that an admissible possible world w is big provided that:

6. at w: there is a plurality xx of items such that (a) for any universe u and any plurality yy of items in u, there is a universe v such that the subplurality of items from xx that are in v corresponds to yy and (b) there are no distinct universes u and v each with an item in common with xx such that the subpluralities of xx consisting of items in u and v, respectively, correspond to each other.

The bigness condition ensures that we have at least continuum-many universes.

Say that the head of a universe u is the item u in the universe that has least mass. Say that two items are neighbors provided that they are in the same universe. We can identify universes with their heads.

Say that a plurality hh of heads of universes is countable provided that:

• There is a plurality xx of items such that each of the hh has exactly one neighbor among the xx and no two items of xx correspond.

The plurality xx defines a mapping of each head in hh to one of its neighbors, and the above condition ensures each distinct pair of heads is mapped to non-corresponding neighbors, and that ensures there are countably many hh.

Say that a plurality hh of heads of universes is continuum-sized provided that:

• There is a plurality xx of items such that each head z among the hh has a neighbor among the xx, and for any universe u and any plurality yy of the items of u, there is a unique head z among the hh such that the plurality of its neighbors corresponds to z.

The plurality xx basically defines a bijection between hh and the subpluralities of any fixed universe.

Given pluralities gg and hh of heads of universes, say pluralities xx and yy of items define a mapping from gg to hh provided that:

• There are pluralities xx and yy of items such that for each item a from gg, if uu is the plurality of a’s neighbors among the xx, then there is a unique item b among the hh such that the plurality vv of b’s neighbors among the yy corresponds to uu.

If a and b are as above, we say that b is the value of a under the mapping defined by xx and yy. Here’s how this works: xx defines a map of heads in gg to pluralities of their respective neighbors and yy defines a map of some of the heads in hh to pluralities of their respective neighbors, and then the correspondence relation can be used to match up heads in gg with heads in hh.

We now say that the mapping defined by xx and yy is injective provided that distinct items in gg never have the same value under the mapping. (This is only going to be possible if gg is continuum-sized.)

If there are xx and yy that define an injective mapping from gg to hh, then we say that |gg| ≤ |hh|. If we have |gg| ≤ |hh| but not |hh| ≤ |gg|, we say that |gg| < |hh|.

The rest is easy. The Continuum Hypothesis for the heads in big admissible w says that there aren’t pluralities of heads gg and hh such that gg is not countable, hh is continuum-sized, and |gg| < |hh|.

We can also get analogues of the finite alternatives to the Continuum Hypothesis. For instance, an analogue to 2^ℵ₀ = ℵ₃ says that there are pluralities bb, cc and dd of heads such that bb is not countable, dd is continuum sized and |bb| < |cc| < |dd|, but there are not pluralities aa, bb, cc and dd with aa not countable, dd continuum-sized and |aa| < |bb| < |cc| < |dd.

Mereology, plural quantification and free lunches

It is sometimes claimed that arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a new way of talking without any deep philosophical (or at least metaphysical) commitments.

I think this is false.

Consider this Axiom of Choice schema for mereology:

1. If for every x and y such that ϕ(x) and ϕ(y), either x = y or x and y don’t overlap, and if every x such that ϕ(x) has a part y such that ψ(y), then there is a z such that for every x such that ϕ(x), there is common part y of x and z such that ψ(y).

Or this Axiom of Choice schema for pluralities:

2. If for all xx and yy such that ϕ(xx) and ϕ(yy) either xx and yy are the same or have nothing in common, then there are zz that have exactly one thing in common with every xx such that ϕ(xx).

If arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a handy way of talking, then whether (1) or (2) is correct is just a verbal question.

But (1) and (2) respectively imply mereological and plural Banach-Tarski paradoxes:

3. If z is a solid ball made of points, then it has five pairwise non-overlapping parts, of which the first two can be rigidly moved to be pairwise non-overlapping and compose another ball of the same size as z, and the last three can likewise be so moved.

4. If the xx are the points of a solid ball, then there are aa, bb, cc, dd and ee which have nothing pairwise in common and such that together they make up xx, and there are rigid motions that allow one to move aa and bb into pluralities that have nothing in common but make up a solid ball of the same size as xx and to move cc, dd and ee into pluralities that have nothing in common and make up another solid ball of the same size.

Conversely, assuming ZF set theory is consistent, there is no way to prove (3) and (4) if we do not have some extension to the standard axioms of mereology or plurals like the Axiom of Choice. The reason is that we can model pluralities and mereological objects with sets of points in three-dimensional space, and either (3) or (4) in that setting will imply the Banach-Tarski paradox for sets, while the Banach-Tarski paradox for sets is known not to be provable from ZF set theory without Choice.

But whether (3) or (4) is true is not a purely verbal question.

One reason it’s not a purely verbal question is intuitive. Banach-Tarski is too paradoxical for it or its negation to be a purely verbal thing.

Another is a reason that I gave in a previous post with a similar argument. Whether the Banach-Tarski paradox holds for sets is not a purely verbal question. But assuming that the Axiom of Separation can take formulas involving mereological terminology or plural quantification, each of (3) and (4) implies the Banach-Tarski paradox for sets.

Restricted composition and laws of nature

Ted Sider famously argues for the universality of composition on the grounds that:

1. If composition is not universal, then one can find a continuous series of cases from a case of no composition to a case of composition.

2. Given such a continuous series, there won’t be any abrupt cut-off in composition.

3. But composition is never vague, so there would have to be an abrupt cut-off.

Consider this argument that every velocity is an escape velocity:

4. If it’s not the case that every velocity is an escape velocity from a spherically symmetric body of some fixed size and mass, then one can find a continuous series of cases from a case of insufficiency to escape to a case of sufficiency to escape.

5. Given such a continuous series, there won’t be any abrupt cut-off in escape velocity.

6. But escape velocity is never vague, so there would have to be an abrupt cut-off.

It’s obvious that we should deny (5). There is an abrupt cut-off in escape velocity, and there is a precise formula for what it is: (2GM/r)^1/2 where G is the gravitational constant, M is the mass of the spherical body, and r is its radius. As the velocity of a projectile gets closer and closer to the (2GM/r)^1/2, the projectile goes further and further before turning back. When the velocity reaches (2GM/r)^1/2, the projectile goes out forever. There is no paradox here.

Why think that composition is different from escape velocity? Why not think that just as the laws of nature precisely specify when the projectile can escape gravity, they also precisely specify when a bunch of objects compose a whole?

My suspicion is that the reason for thinking the two are different is thinking that composition is something like a “logical” or maybe “metaphysical” matter, while escape is a “causal” matter. Now, universalists like David Lewis do tend to think that the whole is a free lunch, nothing but the “sum of the parts”, in which case it makes sense to think that composition is not something for the laws of nature to specify. But if we are not universalists, then it seems to me that it is very natural to think of composition in a causal way: when a proper plurality of xs are arranged a certain way, they cause the existence of a new entity y that stands in a composed-by relation to the xs, just as when a projectile has a certain velocity, that causes the projectile to escape to infinity.

Some may be bothered by the fact that laws of nature are often taken to be contingent, and so there would be a world with the same parts as ours but different wholes. That would bother one if one thinks that wholes are a free lunch. But if we take wholes seriously, it should no more bother us than a world where particles behave the same way up to time t₁, and then behave differently after t₁ because the laws are different.

Humeans have good reason to reject the above view, though. If the laws of composition are to match our intuitions about composition, they are likely to be extremely complex, and perhaps too complex to be part of the best system defining the laws on a Humean account of laws. But if we are not Humeans about laws, and think the simplicity of laws is merely an epistemic virtue, the explanatory power of laws of composition might make it reasonable to accept very complex such laws.

That said, we all have reject the simple causal version of the above view, where a proper plurality composing a whole causes the whole’s existence. For instance, I am composed by a plurality of parts that includes my hair, but my hair is not a cause of my existence: I would have just as much existed had I never developed hair. So a more complex version of the causal view is needed: initial parts (maybe the DNA in the zygote that I started as) causally contribute to the existence of the whole, but the causal relation runs in a different direction with respect to later parts, like teeth: perhaps I and my teeth together cause the teeth to be parts of me.

(I don’t endorse the more complex causal view either. I prefer, but still do not endorse, an Aristotelian alternative: when y is in a certain condition, it causes the existence of all of the parts. This is much neater because the causation always runs in the same direction.)

Causing via a part

Assume this plausible principle:

1. If a part x of z causes w, then z causes w.

Add this controversial thesis:

2. For any x and y, there is a z that x and y are parts of.

Thesis (2) is a consequence of mereological universalism, for instance.

Finally, add this pretty plausible principle:

3. All the parts of a physical entity are physical.

Here is an interesting consequence of (1)–(3):

4. If there is any non-physical entity, any entity that has a cause has a cause that is not a physical entity.

For if w is an entity that has a cause x, and y is any non-physical entity, by (2) there is a z that x and y are both parts of. By (3), z is not physical. And by (1), z causes w.

In particular, given (1)–(3) and the obvious fact that some physical thing has a cause, we have an argument from causal closure (the thesis that no physical entity has a non-physical cause) to full-strength physicalism (the thesis that all entities are physical). Whatever we think of causal closure and physicalism, however, it does not seem that causal closure should entail full-strength physicalism.

Here is another curious line of thought. Strengthen (2) to another consequence of mereological universalism:

5. The cosmos exists, i.e., there is an entity c such that every entity is a part of c.

Then (1) and (5) yield the following holistic thesis:

6. Every item that has a cause is caused by the cosmos.

That sounds quite implausible.

We could take the above lines of thought to refute (1). But (1) sounds pretty plausible. A different move is to take the above lines of thought to refute (2) and (5), and thereby mereological universalism.

All in all, I suspect that (1) fits best with a view on which composition is quite limited.

Distant collaboration

Suppose mereological universalism is true, and that I make a pizza and an alien a long time ago in galaxy far, far away (or even in another universe) makes a sandwich. Then I and the alien have engaged in an amazing collaboration spanning time and space, and maybe even across universes, and produced a fusion of a pizza and a sandwich. Surely I cannot so very easily collaborate in the production of things with beings so far off!

Do all positive truths have truthmakers?

Consider this thesis:

1. Every positive true proposition has a truthmaker.

This seems plausible. But I think it is only reasonable to accept (1) if one accepts:

2. Any plurality of objects has a mereological sum or fusion which essentially has the members of the plurality as parts.

To see this, consider some plurality, the xs of existing things. Then, surely:

3. The proposition, E!xx, that the xs exist is positive.

But what object is suited to be the truthmaker of E!xx? The truthmaker of E!xx will have to be some object o with the property that, necessarily, if o exists, so do all the xs. Our best candidate for that object is some object that has all the xs as essential parts. But we also don’t want to include irrelevancies in the truthmaker, so we shouldn’t include in o anything that overlaps none of the xs. In other words, o will very plausibly be the mereological sum of the xs.

Since I don’t believe in fusions, I have to deny (1). But at least I may be able to accept:

4. Every positive true proposition has a plural truthmaker,

where a plural truthmaker of p is a plurality of objects that collectively make p true. Note that pluralities need not in general be objects themselves, so we do not have the same problem as above.

Fusions and organisms

Suppose you believe the following:

1. For any physical objects, the xs, there is a physical object y with the following properties:
   1. each of the xs is a part of y;
   2. it is an essential property of y that it have the parts it does; and
   3. necessarily, if all the actual proper parts of y exist, then y exists as well.

For instance, on the standard version of mereological universalism, it seems we could just take y to be the fusion of the xs. And on some versions of monism, we could take y to be the cosmos.

But it seems (1) is false if organisms are physical objects and if particles survive ingestion. For suppose that there is exactly one x, Alice, who is a squirrel, and at t₁ we find a y that satisfies (1). And now suppose that at t₂ there comes into existence a nut whose simple parts are not already parts of y, and at t₃ this nut has been eaten and fully digested by Alice. Suppose no parts of y have ceased to exist between t₁ and t₃. Then y exists at t₃ by (c), and has Alice as a part of itself (by (a) and (b)), and the simple particles of the nut are parts of y by transitivity as they are parts of Alice. Hence y has gained parts, contrary to (b), a contradiction.

(Note that the argument can be run modally against a four-dimensionalist version of (1).)

The mereological universalist’s best bet may be to deny that fusions satisfy (c). Normally, we think that the only way for a fusion to perish is for one of its proper parts to perish. But there may be another way for a fusion to perish, namely by certain kinds of changes in the mereological structure of the fusion’s proper parts, and specifically by one of the fusion’s proper parts gaining a part that wasn’t already in the fusion.

Here is another problem for (1), though. Suppose that Alice the squirrel is the only physical object in the universe. Now consider a y satisfying (1)(a)–(b). Then y is distinct from Alice because y has different modal properties from Alice: Alice can survive annihilation of one of her claws while y cannot by (b). But this violates the Weak Supplementation mereological axiom, since all of y’s parts overlap Alice. So we cannot combine fusions as normally conceived of (since the normal conception of them includes classical mereology) with organisms.

A way out of both problems is to say that there are two different senses of parthood at issue: fusion-parthood and organic-parthood, and there is no transitivity across them. This is a serious ideological complication.

Artifacts and non-naturalism

One of the reasons to be suspicious of artifacts is that it seems magical to think we have the power to create a new object just by thinking about things a certain way while manipulating stuff. If Bob gets some clay and exercise his fingers by randomly kneading it, he doesn’t make a sculpture or any other new object out of it. But if his identical twin Carl intends to shape the clay into a sculpture, and in doing so moves his fingers in exactly the same way that Bob did, and produces exactly the same shape, then—assuming artifacts exist—he creates a new object, a sculpture. It seems magical that our thoughts should affect what object exists in the world, even when the thoughts make no difference to our manipulation of the world.

When I discussed arguments with this in my Mid-Sized Objects graduate seminar, I found, however, that there was a lot of friendliness towards the view that, yes, we are capable of this magic, though some demurred at the word “magic”. And in particular, a student pointed out that we are in the image of a God who can create.

This has made me think that a non-naturalist can think that our thoughts have effects that are not screened by the movements of our bodies. Thus, it could well be that Carl’s thoughts causes the world to be different. For instance, on a hylomorphic view, Carl could have the power to create a scu;tural form for a piece of clay by his thoughts. Or on a variant of Markosian’s brute composition view, Carl could have the power simply to cause a new object composed of the clay.

In fact, this suggests an interesting new argument against physicalism, where physicalism is understood as the claim that all causal powers reduce to those of physics. Intuitively, the correct ontology includes more things than van Inwagen’s ontology of particles and organisms and but not all the things from the mereological universalist’s bloated ontology. In particular, intuitively, the correct ontology does include Carl’s new sculpture, but Bob hasn’t produced anything new, and hence the correct ontology seems to require a non-natural “magical” power over composition facts to be found in Carl’s (and presumably, albeit in this context unexercised, Bob’s) mind. And if our ontology is to include, as common-sense would suggest, galaxies, planets, mountains and rocks, we need powers in things to produce such objects—i.e., to ensure that their particulate parts do compose something—and these powers are not to be found in physics.

Markosian’s apparently preferred version of the brute composition view can almost accommodate this. On that version, the composition facts supervene on the arrangement of particles: there are infinitely many necessary truths that specify which arrangements of particles compose. But these necessary truths would include lots of arbitrary parameters (e.g., encoding the difference between some stones that are just lying there and a hillock). We don’t want necessary truths with arbitrary parameters. It is much better if any such arbitrary parameters are relocated to the laws of nature or, better, the causal powers of things.

Artifacts, Aristotelianism and naturalism

One of the main reasons I don’t believe in (complex) artifacts is that the existence of an artifact would have to depend on our intentions. Whether some stones make up a sculpture depends on whether they were piled with the intention of making a sculpture or just tossed in a heap to provide raw materials. And it is incredible that just because one thinks about something in a particular way while executing a series of physical actions, a material object comes into being, and if one doesn’t think in this way, but executes the same series of physical actions, there are just raw materials in a heap rather than a thing. This just seems like magic.

It has, however, just occurred to me that I may have been thinking too much like a naturalist. We human beings already have a broad array of amazing non-natural powers. By promising, I create an obligation for myself, and by requesting, I create a reason for you. By reproducing, two humans produce a new thinking being. Why couldn’t human beings (and perhaps other tool-using animals) also be gifted with the basic power to create a form for a bunch of physical objects, a power which they exercise by executing some physical movements with particular intentions, much as I change my own normative status by using my vocal chords with particular intentions?

That our intentions should affect what material objects there are is also a bit less magical when one has an Aristotelian ontology. For on an Aristotelian ontology, “material objects” are not purely material: they have immaterial form. Yes, all this is a bit magical. But on Aristotelian ontology, all beings are a little magical, and we are especially so, being minded.

That said, I still find it hard to believe that we can create artifacts.

But all this suggests an interesting argument against naturalism:

1. We can bring complex artifacts into existence.

2. Mereological universalism is false.

3. If naturalism is true, we can bring complex artifacts into existence if and only if mereological universalism is true.

4. So, naturalism is not true.

But I am still not sure (1) is true.

Mereology and constituent ontology

I’ve just realized that one can motivate belief in bare particulars as follows:

1. Constituent ontology of attribution: A thing has a quality if and only if that quality is a part of it.

2. Universalism: Every plurality has a fusion.

3. Weak supplementation: If x is a proper part of y, then y has a part that does not overlap x.

4. Anti-bundleism: A substance (or at least a non-divine substance) is not the fusion of its qualities.

For, let S be a substance. If S has no qualities, it’s a bare particular, and the argument is done.

So, suppose S has qualities. By universalism, let Q be the fusion of the qualities that are parts of S. This is a part of S by uncontroversial mereology. By anti-bundleism, Q is a proper part of S. By weak supplementation, S has a part P that does not overlap Q. That part has no qualities as a part of it, since if it had any quality as a part of it, it would overlap Q. Hence, P is a bare particular. (And if we want a beefier bare particular, just form the fusion of all such Ps.)

It follows that every substance has a bare particular as a part.

[Bibliographic notes: Sider thinks that something like this argument means that the debate between constituent metaphysicians overlap bare particulars is merely verbal. Not all bare particularists find themselves motivated in this way (e.g., Smith denies 1).]

To me, universalism is the most clearly false claim. And someone who accepts constituent ontology of attribution can’t accept universalism: by universalism, there is fusion of Mt. Everest and my wedding ring, and given constituent ontology, the montaineity that is a part of Everest and the goldenness of my ring will both be qualities of EverestRing, so that EverestRing will be a golden mountain, which is absurd.

But universalism is not, I think, crucial to the argument. We use universalism only once in the argument, to generate the fusion of the qualities of S. But it seems plausible that even if universalism in general is false, there can be a substance S such that there is a fusion Q of its qualities. For instance, imagine a substance that has only one quality, or a substance that has a quality Q₁ such that all its other qualities are parts of Q₁. Applying the rest of the argument to that substance shows that it has a bare particular as a part of it. And if some substances have bare particular parts, plausibly so do all substances (or at least all non-divine substances, say).

If this is right, then we have an argument that:

5. You shouldn’t accept all of: constituent ontology, weak supplementation, anti-bundleism and anti-bare-particularism.

I am an anti-bundleist and an anti-bare-particularist, but constituent ontology seems to have some plausibility to me. So I want to deny weak supplementation. And indeed I think it is plausible to say that the case of a substance that has only one quality is a pretty good counterexample to weak supplementation: that one quality lacks even a weak supplement.

A method for blocking deflation of ontological debates

Consider Hirsch-type deflationary views on which many differences in ontology are simply verbal differences. A standard case is nihilism and universalism about composition: the nihilist says that multiple things can never compose a whole and the universalist says that every plurality must compose a whole. The deflationist sees the two views as notational variants. The universalist’s sentences describe the same facts as the nihilist’s. We can maybe even translate with little if any loss between the two idioms, replacing the nihilist’s quantifiers with quantifiers restricted to simples on the universalist’s side, and replacing the universalist’s quantifiers with plural quantification, or quantification over sets, or some other device acceptable to the nihilist.

Note, first, that in this particular case there is a bit of a problem. The universalist might allow for composed objects that have no simple parts—“gunk”. The claim that possibly there is gunk is one that cannot be translated into any statement in the nihilist’s language that has a hope of being true. The nihilist’s usual way of translating a universalist’s statement is to use plural quantification. So the statement that possibly there is gunk is going to get translated into something like the statement that possibly there is a plurality of things none of which is a simple. But that’s obviously false given nihilism, since the nihilist’s quantifiers can only quantify over simples, and so the statement basically says that there are simples none of which is a simple. Thus, we have a genuine, non-verbal disagreement.

So the only way we can take a nihilist-universalist disagreement to be merely verbal is if both theorists deny the possibility of gunk. I think they should deny the possibility of gunk.

Here is a second case, where disagreement on composition cannot be deflated. Consider a brutal composition view like Markosian’s. On this view, there will be possible worlds with the same simple objects standing in the same non-mereological relations but differing as to composition facts. For instance, in one world there might be three rocks that make up a whole and in the other world the very same three rocks do not make up a whole, even though they are arranged in exactly the same way. Any nihilist or universalist description of the two worlds will be unable to distinguish such worlds, but on a brutal composition view, there can be such pairs. Here we have a real disagreement, one that cannot be taken to be merely verbal. The brutal composition theorist has more possibilities than the nihilist and universalist. And the brutal composition theorist’s statement that the two worlds differ in composition facts but not in non-mereological facts either has no translation into either nihilist or universalist language or translates into something that is clearly false on the given theory.

The brutal compositionalist has an additional “degree of freedom”, as the scientist would say, on her theory than the nihilist or universalist does. The case here is similar to those dualists who believe in the possibility of zombies. While the disagreement between a dualist who thinks the mental supervenes on the physical and the pure physicalist could seem to be merely verbal to some (though I think it’s a mistake to see it that way), the disagreement between a dualist who thinks that the mental does not supervene on the physical and the pure physicalist is certainly not merely verbal.

In general, thus, the modal ramifications of theories can block deflationary moves. One theory may allow for a possibility that simply rules out the other theory (e.g., gunk ruling out nihilism), or one theory may posit contingent facts that do not supervene on reality as describable in the other theory (the brutal composition or zombie cases).

This leaves the possibility that there will be some ontological debates that are merely verbal. Perhaps the debate between the nihilist and the anti-gunk universalist is merely verbal. But that some pairs of ontological theories disagree merely verbally is not a very interesting deflationary thesis.

Moreover, I think that once we see that there are nearby debates that are clearly not merely verbal, the plausibility of the deflationary move in the cases that looks more verbal goes down. Once we realize that among the views under discussion there is a brutal composition view on which there is a possible world just like ours but where nihilism contingently holds and a possible world just like ours but where universalism contingently holds, it becomes pretty clear that holding nihilism to hold necessarily will also differ from holding universalism to hold necessarily. (That said, there may be particular variants on universalism that just are notational variants on nihilism. Say, ones where the quantifiers are stipulated in terms of plural quantification over simples.)

Quantifier and Predicate Variance

Suppose we decide to speak with a quantifier family (a quantifier family includes ∃ and ∀, but may also include things like "many" and "most" and "at least three", and maybe even two-place quantifiers, all ranging over the same domain) that makes arbitrary pluralities of things have a fusion, i.e., a mereologically universalist quantifier family. According to the Quantifier Variance thesis, this decision to extend quantifiers is a linguistic decision that needs merely pragmatic justification, a decision that introduces a quantifier perhaps different from the one most ordinarily used.

This sounds like a decision solely about quantifiers. Not so. For now we need to say something about the meaning of predicates applied to variables bound by these quantifiers. For instance, we need to be able to meaningfully say using the new "there is" whether there is something whose mass is 455 tons, i.e., whether ∃x(Mass(x, 455T)). (Using van Inwagen quantifiers which range over simples and organisms, unless there are alien organisms much larger than blue whales, there isn't anything of that mass.) We can give a semantics for the universalist quantifiers in terms of plural quantification, but we need to account for how much a plurality masses. Intuitively, the mass of a plurality is the sum total of the masses of the simples in the plurality (some technical problems: what if there is gunk? do we count the mass-equivalent of the energy of the bonds between the simples?), and so ∃x(Mass(x, 455T)) provided that there are ys which plurally mass 455T. I suppose extending mass in this way once one has extended the quantifiers is pretty obvious.

But not all predicates extend in an obvious way. For instance, consider the predicate that says that something is spatially extended. Does that predicate apply to the fusion of the number seven with the Empire State Building? Here we have a decision to make, roughly about what it is to be plurally extended: Do we say that for the ys to be plurally extended, each one of them must be extended, or is it enough that one of them is extended? The former decision will fit better with the intuition that extended objects are material. The latter with the intuition that occupying space is sufficient for extension. And either way, we will have some technicalities (can't a plurality of unextended points make up something extended?). Or take the causal relation. Did the people who built the Empire State Building cause the fusion of the number seven with the Empire State Building? It's hard to say. And aesthetic properties will be particularly hairy (is the fusion of Beethoven's Ninth with Michelangelo's David beautiful? is it a work of art?).

Thinking about such examples makes it clear that the linguistic decision to speak with a new quantifier family needs to come along with a correlate decision about the semantics of predicates extended to work with this new quantifier family (a decision that could, sometimes, be simply to leave a predicate underdefined or vague). This means that it is a bit misleading to talk of "Quantifier Variance". The relevant thesis is "Quantifier and Predicate Variance". (And we may also need to have "name variance", unless we consider names to be a kind of quantifier.)

Note that none of this is an issue if one creates a new quantifier merely by domain restriction. It's domain expansion that generates the problems.

Extension and mereological universalism

Plausibly, a fusion of extended objects is extended. Also, plausibly, an extended object has a size. Now suppose, as is surely possible, that there are two universes that aren't spatiotemporally connected, and an extended object A in one and another extended object B in another. Then the fusion of A and B would be an extended object that has no size, since there is no meaningful distance between a part of A and a part of B. Hence, given our assumptions about extended objects, mereological universalism--the thesis that necessarily all pluralities have a fusion--is false.

From unrestricted composition to unrestricted caninity

According to unrestricted composition (UC) for any plurality of things there is a whole that is exactly composed of them. Sider offers a continuity argument for UC. Here's a vivid formulation. Let the Ps be the particles in the even-numbered books on one of my bookshelves. If UC is false, then in the actual world the Ps will be a paradigm case of something that doesn't compose a whole. But there is a world where the Ps compose a dog. And between these two worlds there is a continuous sequence of worlds where the Ps gradually migrate from their every-second-book positioning to their canine positioning. It is absurd to think that suddenly somewhere in this continuous sequence the particles come to compose something. So, Sider concludes, they compose something all along, even in the actual world.

But to a hylomorphist, the argument as I've put it simply fails. There is no world where the Ps compose a dog, since a dog—or any other complex entity—is not composed of matter, but of matter and form. The argument can, however, be reformulated. Say that the Ps materially compose an F provided that the Ps are material and together with some form compose an F. Then the argument gets off the ground. In the actual world, the Ps do not materially compose anything while in the final world they materially compose something. Where along the line do they come to materially compose something?

Now, however, the story is underdescribed. For we have failed to say in which worlds in the sequence there is a substantial form of the dog informing the Ps. Facts about substantial forms should not be assumed to supervene on facts about the arrangement of the particles. There could be zombie dogs that are nothing but heaps of particles looking like a dog. In other words, it's a contingent matter whether a certain kind of arrangement of particles materially composes something—if there is a form informing them, then they compose and if not, not.

Of course, there is a question of explanation: Why is there no form informing the Ps in the actual world but there is one in the the non-zombie dog worlds? But the answers aren't particularly troublesome. Maybe the laws of nature explain that. Maybe God just decides when to create forms and make them inform particles.

However, there is a final move that Sider can make. Instead of asking in which worlds the Ps (materially) compose something, he could ask which arrangements of particles are such that something could be materially composed of the particles in that arrangement. Of course the dog-like arrangement is like that. And the even-numbered-book arrangement is not. So where is the transition in the continuous deformation of the even-numbered-book arrangement into the dog-like arrangement?

This is an interesting question for the hylomorphist. It is closely to the question of what forms there could be (cf. the discussion here and in the Murphy book referenced in the comments there). The hylomorphist could take an unrestricted view. There is a sufficiently wide variety of possible forms and defects that any possible arrangement of matter is compatible with being informed by some form—perhaps defectively. There could be a possible world where something looking just like our even-numbered-book arrangement is a highly defective (it doesn't grow or reproduce) plant.

Nonetheless, there is a remaining problem. While the even-numbered-book arrangement may be apt for materially composing a defective plant, it's surely inapt for materially composing a dog. So there will seem to be a discontinuous transition between those arrangements that can and those that cannot materially compose a dog. One answer here is that "dog" is vague. This doesn't fit with traditional Aristotelian views, though, on which all dogs have an exactly similar form, and so one could meaningfully ask about the range of arrangements that could be informed by a form that's exactly like that. But perhaps the Aristotelian can yield some ground here. Another answer would be unrestricted canine composition: any material arrangement could materially compose a dog, albeit a highly defective one. I am somewhat drawn to this strange view. Yet is it that strange? I think I can imagine a dog continuously deforming into the even-numbered-book arrangement but where rather than dying the dog comes to be more and more defective. I am dualist enough that I can even imagine the dog being conscious throughout the process.

The vagueness argument against restricted compositionality

Lewis and Sider have argued that if restricted compositionality is true—some but not all pluralities of two or more objects compose a whole—then there will be cases where it's vague how many objects there are. For instance, imagine two universes, A and B, each with the same finite set of n particles with the same intrinsic properties. But in A, the particles are neatly arranged into galaxies, trees, tables, buildings, etc. And in B there is just a blooming buzzing confusion. If restricted compositionality holds, then, assuming there are no immaterial objects, universe B has exactly n or at most n+1 objects—it's just too messy to have any cases of composition, except perhaps for the universe as a whole (that's why it might be n+1 rather than n). But A is much like our universe, and so we would expect lots of cases of composition, and hence the number of objects will be a lot more than n+1, say n+m for some large m. However, we can now imagine a continuous sequence of universes ranging from A to B, differing continuously in how the particles are arranged. As we move that continuous sequence, the number of objects will have to change from no more than n+m to n+1. But it is incredible that the object count should sharply change due to a very tiny shift in particle positions. Instead, the object count will at times be vague. But how many objects there are is a matter of which sentences using universal quantification, conjunction, negation and identity are true. But quantification, conjunction, negation and identity are not vague. So we have vagueness where we cannot have vagueness.

There may be some technical problems with the argument as I formulated it, given the assumption of no immaterial objects. Maybe we can't do without immaterial entities like God or numbers. One could reformulate the argument to restrict the counting to material entities, but "material" might actually be a vague term. Perhaps the best thing to do is to assume that these universes have no immaterial contingent entities, and then just count contingent entities. Contingency shouldn't be a vague matter, after all. The Aristotelian may balk at this. For it may well be that a necessary condition for a bunch of material entities to compose a whole that they have a form, and forms are immaterial but contingent. Maybe, though, "form" is not vague, and so we can just count the contingent non-forms.

But talking of forms suggests a more serious difficulty. If there are Aristotelian forms, then how many material objects there are may well not supervene on how material objects are spatiotemporally arranged and what intrinsic properties they have. For objects to come to compose a whole, there must come into existence a form. There is nothing absurd about there being sharp laws of nature specifying under which precise conditions a form comes into existence. There is no need for the laws of nature to be continuous (and the possibility of fundamental discreteness is empirically still open). Or perhaps God decides on a case-by-case basis whether to create a form. Then there is no vagueness as to how many material objects there are: the number of material objects equals the number of forms of material objects that in fact inform some matter (the souls of the resurrected are forms of material objects but temporarily fail to inform any matter). Of course in transitional cases we won't have much confidence whether some objects compose a whole, but that's just because we are unable to see forms except through their normal effects.