1. The special composition question and some physics-based answers

What are the necessary and sufficient conditions, under which a set^{Footnote 1} of material objects S composes something? In other words: what is the criterion—that is, a condition that is both sufficient and necessary—ψ such that:

$$\psi (S){\rm\ {iff}}\ {\rm{the}}\ {\rm{objects}}\ {\rm{in}}\ {\rm{set}}\ S\ {\rm{compose}}\ ({{Comp}})\ {\rm{an}}\ {\rm{object}}\ x:\,\exists x(Comp(S,x))?$$

This is a version of the so-called Special Composition Question (SCQ). SCQ was famously introduced in the metaphysics literature by van Inwagen (Reference van Inwagen1987),^{Footnote 2} and it has been driving the debate on composition ever since. Answers to SCQ can be broadly divided in two camps: extreme or moderate answers.^{Footnote 3} According to extreme answers, ψ is irrelevant for composition: either composition always occurs—that is, a set S of entities composes a further entity under any ψ whatsoever (mereological universalism)—or composition never occurs—that is, a set of entities S composes a further entity under no ψ whatsoever (mereological nihilism).^{Footnote 4} Moderate answers single out a nonempty and nontrivial criterion ψ—one that neither fails for every S, nor holds for every S—for composition to occur. Despite their initial attractiveness, satisfactory moderate answers are hard to come by (see van Inwagen Reference van Inwagen1990). Recently, different physics-based answers to the SCQ have been put forward in the literature. This is a very welcome development. Metaphysical considerations should be sensitive to insights from empirical science. The main focus of this paper will be on the so-called Bound State Answer (BSA), suggested by McKenzie (Reference McKenzie2011) and recently advocated by McKenzie and Muller (Reference Kerry and Muller2017) and Waechter and Ladyman (Reference Waechter and Ladyman2019). However, before we enter into some of its details, it is worth introducing another physics-based answer, namely, the entanglement answer. The reason for that will be clear in due course.^{Footnote 5} Roughly, according to the latter, a set S of material objects composes a further entity iff the members of S are entangled—thus, ψ is “being entangled.”^{Footnote 6} The entanglement answer, which is considered by Calosi and Tarozzi (Reference Calosi and Tarozzi2014), is discussed and discarded by McKenzie and Muller (Reference Kerry and Muller2017) on the grounds that it is extensionally equivalent to mereological universalism:

As I noted above, McKenzie and Muller go on to defend another physics-based answer. According to such an answer, ψ amounts to “being in a (common) bound state.” While the formulations of the BSA due to McKenzie and Muller (Reference Kerry and Muller2017) and Waechter and Ladyman (Reference Waechter and Ladyman2019) differ in details the spirit is very much the same. I will mostly follow McKenzie and Muller (Reference Kerry and Muller2017) for a simple reason. I find the reasons they give in favor of the BSA controversial. Waechter and Ladyman (Reference Waechter and Ladyman2019) appeal to some of those same reasons and suggest others as well. I am prepared to concede that those other reasons do provide support for the BSA.^{Footnote 7} Given that I will be mostly—but not exclusively—concerned with the reasons in favor of the BSA, rather than with details of formulation, I will stick mostly to McKenzie and Muller (Reference Kerry and Muller2017). That being said, the discussion will give me the chance to deal with Waechter and Ladyman (Reference Waechter and Ladyman2019) as well.

The rest of the paper is structured as follows. I will first discuss the BSA and the reasons in its favor (§2). I will then contest those reasons (§3) and compare the BSA with a further restricted answer due to van Inwagen, namely, Fastening (§4). This is important, insofar as van Inwagen objects to Fastening. It is a substantive question whether the BSA is vulnerable to the same objection. Taken together, §3 and §4 provide a critical assessment of the BSA. In light of the above, I go on to suggest a different general overlook on physics-based answers to SCQ, which helps reevaluate them (§5). A brief conclusion follows (§6).

2. The BSA and its virtues

The core of the BSA can be summed up as follows: a set S of material objects forms a composite object iff those material objects interact and are in a common bound state; that is, they are in the potential well that results from their mutual interaction (McKenzie and Muller Reference Kerry and Muller2017, 234). A bound state is a state where the constituent objects have a potential energy that is greater in absolute value than their kinetic energy.^{Footnote 8} Non-bound states are sometimes referred to as scattering states. Restricting our attention to potentials that go to zero at infinity, the criterion for distinguishing between bound and scattering states can be roughly phrased as follows:

(1) $$\eqalign{\matrix{ {{\bf Bound \ State} \Rightarrow} {E_s} \lt 0} \hfill \cr{{\bf Scattering\ State} \Rightarrow {E_s} \ge 0} \hfill \cr } $$

where ${E_s}$ is the (expectation-value of the) energy of the physical system s—e.g., a particle (Griffiths Reference Griffiths1995, 51–52), or a composite system.

Without entering nitpicking technicalities, a little more precision will be useful. I follow McKenzie and Muller (Reference Kerry and Muller2017) almost verbatim. Let S be a nonempty set of material objects, let $Comp(S,x)$ stand for: the objects in S compose object x, and, finally, let $x \sqsubseteq y$ stand for: Object x is part of y. Then:

${\bf{BS}}{{\bf{A}}_1}$ If S contains a single object, then:

(2) $$Comp\,(S,x)\;\;{\rm{iff}}\;\;S = \left\{ x \right\}$$

${\bf{BS}}{{\bf{A}}_2}$ If S contains at least two distinct objects, then:

(3) $$Comp\,(S,x)\;\;{\rm{iff}}\;\;\forall y \in S\;(y \sqsubseteq x) \wedge {E_x} \lt 0$$

Informally, condition (3) says that (i) every $y \in S$ is part of x, and (ii) the total energy of the system x is less than 0; that is, the members of S are in common bound state.

Direct Part x is a direct part of y— $x\,{ \sqsubseteq _d}\,y$ —iff there is a set S such that the objects in S compose y and contain x:

(4) $$x{ \sqsubseteq _d}\ y\;\;{\rm{iff}}\;\;\exists S\,(Comp\,(S,y) \wedge x \in S)$$

Part x is part of y iff there is a finite sequence of direct parts that begins with x and ends with y:

(5) $$x \sqsubseteq y\;\;{\rm{iff}}\;\;\exists {i_1}, \ldots ,{i_n}:x{ \sqsubseteq _d}{i_1} \ldots { \sqsubseteq _d} \ldots {i_n}{ \sqsubseteq _d}y$$

The parts that arise for $i \ge 1$ are called Indirect Parts. I will use $x{ \sqsubseteq _i}y$ for such a case. This exhausts the core of the BSA. ${\rm{BS}}{{\rm{A}}_1}$ and ${\rm{BS}}{{\rm{A}}_2}$ provide the answer to SCQ, McKenzie and Muller contend, whereas Direct Part and Part allow us to recover parthood and other mereological notions (e.g. proper part, overlap, and so on)—via the usual mereological definitions.

Before going over the reasons in favor of the BSA, it is worth noting that McKenzie and Muller do not seem to take composition (Comp) as a primitive, as they explicitly define it in in terms of $ \sqsubseteq $ . They go on to define $ \sqsubseteq $ in terms of Comp. This might be problematic. As of now, I just want to point out that they have three different notions of parthood, that is, $ \sqsubseteq $ , ${ \sqsubseteq _d}$ , and ${ \sqsubseteq _i}$ , one of which figures twice: once in the definition of Comp and once as defined in Part. I will return to all this in §3.

Let us then move on to discuss the reasons in favor of the BSA, or its “virtues” as I shall call them. McKenzie and Muller (Reference Kerry and Muller2017) lists four of them, two of which are discussed by Waechter and Ladyman (Reference Waechter and Ladyman2019) as well. I shall label them (i) Moderation, Conservativeness, and Extensional Adequacy; (ii) Precision; (iii) Simplicity; and (iv) Parsimony.^{Footnote 9}

Moderation, Conservativeness, and Extensional Adequacy. The first virtue of BSA is its moderation. BSA is a moderate answer to SCQ. Some, but not all, material objects are in a bound state. Thus some sets of material objects, but not every non-empty set, compose something, contra nihilisim and universalism, respectively. Given that

this counts as a reason in favor of the BSA. In effect, the proposal is in line with (some) common sense judgments about composite objects. A hydrogen atom is sanctioned as a bona fide composite object, whereas a trout-turkey—that is, the “mereological fusion” of the undetached front half of a trout and the undetached back half of a turkey—is not.^{Footnote 10} Thus, the first reason seems to be one of moderation and conservativeness: the BSA is conservative, insofar as it aligns with common sense moderate judgments about composition, judgments that are “honed through immersion in physical science” (McKenzie and Muller Reference Kerry and Muller2017, 235). Relatedly, Waechter and Ladyman (Reference Waechter and Ladyman2019) claim that the BSA provides a moderate answer to the SCQ that is extensionally adequate. In particular, Waechter and Ladyman (Reference Waechter and Ladyman2019) claim that

The Swadesh list is a list of words that have cognates in virtually all linguistic communities.^{Footnote 11} Many such words refer to ordinary composite objects. I take it that this “Swadesh list” argument is relevantly similar to the conservativeness argument above, so that it is warranted to discuss them together.

Precision. Both McKenzie and Muller (Reference Kerry and Muller2017) and Waechter and Ladyman (Reference Waechter and Ladyman2019) note that there is an influential argument in the literature to the point that every moderate answer to SCQ entails metaphysical indeterminacy or vagueness, rather than less worrisome forms of indeterminacy, such as epistemic or semantical indeterminacy.^{Footnote 12} By contrast, the BSA offers a sharp criterion for composition. In effect, a set of objects compose iff those objects are in a common bound state. This ultimately boils down to (1), which offers, upon inspection, a precise, non-vague criterion.

Simplicity. The BSA is simple, insofar as it offers the very same criterion of composition for different kinds of material objects. No matter whether fundamental particles, molecules, mid-size dry goods, or planets are at stake, the BSA will always tell the same story: they compose something iff they are in a common bound state. Compare this with some other moderate answers, e.g., the answer van Inwagen calls Series. Here is van Inwagen:

According to Series, different relations will account for composition of different kinds of objects. Fundamental particles, molecules, mid-size dry goods, and planets will compose atoms, cells, cathedrals, and planetary systems by instantiating very different relations. The simplicity that is lost in a disjunctive answer like Series is retained in the BSA. Other things being equal, disjunctive moderate answers are less simple than nondisjunctive ones. Other things being equal, we should prefer the latter.

Parsimony. It is widely agreed that parthood has some formal features; e.g., it is widely agreed that it is a partial order. Usually, these formal features are assumed axiomatically.^{Footnote 14} But, given the BSA, they need not be. In effect, Reflexivity and Transitivity (at least) can be proven. Reflexivity follows from Part and ${\rm{BS}}{{\rm{A}}_1}$ .^{Footnote 15} Transitivity follows from Part.^{Footnote 16} Insofar as these features of the parthood relation need not be further axiomatic assumptions, the BSA is parsimonious.

3. Measure for measure

There is no denying that the virtues of the BSA are attractive. But how virtuous is the BSA, really? In the following sections I will attempt to evaluate the “measure” of the aforementioned virtues. Unfortunately, at a closer scrutiny, the BSA will turn out to be less virtuous than it first appears. But then again, who isn’t?^{Footnote 17}

3.2. Measuring precision

Once again, it is indisputable that the BSA gives a precise, non-vague moderate answer to the SCQ. The question is whether precision per se is really a virtue that is so hard to obtain. I contend that precision in itself is quite easy. And is something really a virtue, if it is so easy? What I am really challenging here is not the measure of precision, as much as its worthiness. Any moderate answer to the SCQ takes the following form:

(6) $$\forall S\,(\exists x\,(x \in S) \wedge \psi \,(S) \leftrightarrow \exists z\,(Comp\,(S,z))$$

where $\neg \forall S\,(\psi \,(S)) \wedge \neg \forall S\,(\neg \psi \,(S))$ holds.^{Footnote 20} Equation (6) claims—roughly—that criterion ψ provides necessary and sufficient conditions for members of S to compose z. Pick any non-vague ψ whatsoever, plug it into (6), and the result will be a precise, nonvague moderate answer to the SCQ. Suppose ψ is “having negative charge.” The corresponding moderate answer to the SCQ will be as precise as the BSA. Unfortunately, it will entail that only things with negative charge undergo composition. Or suppose that ψ is the ancestral relation of topological connection. Insofar as topological connection is not vague, neither is the resulting moderate answer. Let us restrict our attention to binary fusions, that is, fusions of two (atomic) parts. Then, the following is—according both to McKenzie and Muller’s and to Waechter and Ladyman’s own standards—a precise moderate answer to the SCQ:

(7) $$x \circ y \leftrightarrow \exists w\,\boldsymbol(Comp\ \boldsymbol(\left\{ {x,y} \right\},w\boldsymbol)$$

where $ \circ $ is mereological overlap, defined as usual:

(8) $$x \circ y\;\;{\rm{iff}}\;\;\exists z\,(z \sqsubseteq x \wedge z \sqsubseteq y)$$

Insofar as $ \sqsubseteq $ is not vague, $ \circ $ is not vague. Once again, (7) is precise, yet it is not plausible, for it states that two things compose another iff they overlap. All these examples teach the same lesson. Precision per se is not difficult to get. The real difficulty, and thus the real value, lies in specifying a precise ψ that also meets further desiderata for composition we might care about. In particular, if we want our restricted answer to the SCQ to align with common sense judgments, the task is to put forward a precise ψ that at the same time sanctions commonsensical judgments about composition. To put it differently: it is the combination of conservativeness and precision that should be considered a virtue (for moderate answers). And I already made my case about the “real conservativeness” of the BSA.

3.4. Measuring parsimony

The BSA is allegedly parsimonious, insofar as some formal features of the relation of parthood can be proved, rather than assumed axiomatically. McKenzie and Muller focus on Reflexivity and Transitivity. The argument in the previous section has some bearing on Reflexivity. It need not be assumed axiomatically, only insofar as one assumes ${\rm{BS}}{{\rm{A}}_1}$ and distinguishes two cases of composition, self-composition and composition of two (or more) objects. This detracts from Simplicity. Or so I argued. In effect, one can make a case that ${\rm{BS}}{{\rm{A}}_1}$ is assumed only to guarantee Reflexivity.^{Footnote 25} Assuming ${\rm{BS}}{{\rm{A}}_1}$ has the same costs that assuming Reflexivity directly has. There is actually no parsimony here. Let us move then to Transitivity.

In what follows it will be crucial to qualify parthood as mereological whenever needed—the reason will be obvious in a moment. Thus, at the risk of sounding repetitive, I will indeed explicitly add that qualification when necessary. The relation of mereological parthood axiomatized in (classical) mereology is usually taken as a primitive, and the notion of mereological fusion is defined in terms of mereological parthood as follows:

(9) $$Fus\,(x,S{) = _{df}}\forall y\,(y \in S) \to (y \sqsubseteq x) \wedge \forall z(z \sqsubseteq x \to \exists w\,(w \in S \wedge z \circ w))$$

Informally, x is a fusion of (the members of a) set S iff every member of S is a mereological part of x, and every mereological part of x overlaps at least a member of S. But the two notions are interdefinable. Starting with a notion of Fusion, in mereology we define mereological parthood as follows (see, e.g., van Inwagen Reference van Inwagen1987, 25):

(10) $$x \sqsubseteq y\;\;{\rm{iff}}\;\;\exists S\,(Fus\,(y,S \cup \left\{ x \right\}))$$

That is to say that x is a mereological part of y iff there is a set S such that y is the fusion of the members of S and x. Taking Fus and Comp as mutual converses, x is a mereologiocal part of y iff there is a set S such that the members of $S \cup \left\{ x \right\}$ compose y. And, naturally, $x \in S \cup \left\{ x \right\}$ . Hence, what (10) claims is that x is a mereological part of y iff there is a set ${S^*}$ such that the objects in ${S^*}$ compose y and contain x. This is verbatim the notion of Direct Parthood. In other words, the argument above shows that the usual definition of mereological parthood given in terms of fusion in mereology is what McKenzie and Muller call Direct Parthood ${ \sqsubseteq _d}$ , not what they define in Part $ \sqsubseteq $ . So, the question of whether we can prove that mereological parthood is transitive boils down to the question of whether ${ \sqsubseteq _d}$ is transitive. This is problematic because, as McKenzie and Muller themselves point out, it turns out that ${ \sqsubseteq _d}$ is not transitive after all. Waechter and Ladyman (Reference Waechter and Ladyman2019) are explicit about this. Let me flesh out in some detail an example McKenzie and Muller briefly mention. Three quarks ${q_1},{q_2},{q_3}$ compose a proton p, for they all lie in a common potential well—hence they are in a common bound state:

(11) $$Comp\,\big(\left\{ {{q_1},{q_2},{q_3}} \right\},p\big)$$

From (11) we get that, for every ${q_i}$ , ${q_i}\,{ \sqsubseteq _d}\,p$ . The proton p and an electron e compose a hydrogen atom H, insofar as they are both, once again, in a common potential well:

(12) $$Comp(\left\{ {p,e} \right\},H)$$

From (12) we derive that $p\,{ \sqsubseteq _d}\,H$ . Transitivity will dictate that, for each ${q_i}$ , ${q_i}\,{ \sqsubseteq _d}\,H$ . That is, applying (4)—or (10)—there exists a set S such that $Comp(S,y)$ , and ${q_i} \in S$ , for each ${q_i}$ . Clearly, $S = \left\{ {{q_1},{q_2},{q_3},e} \right\}$ . Unfortunately, H does not compose S according to the BSA, for its members are not in common potential well; that is, they are not in a common bound state. Hence, transitivity of ${ \sqsubseteq _d}$ fails.^{Footnote 26} In effect, the relation McKenzie and Muller define in Part is basically the transitive closure of ${ \sqsubseteq _d}$ . It is no wonder that it is transitive. The transitive closure of any relation is transitive.

This last argument can be generalized. The general point is that it is dubious that we should ascribe the merits of the alleged derivability of the formal profile of some parthood relation to the physical details behind the BSA. To argue for this claim, let me introduce a construction due to Fine (Reference Fine2010, 567–68). Suppose we start from a very general—and flexible—composition operation ∑. ∑ is flexible, insofar as it can take any number of arguments, $0,1, \ldots ,n$ , for any n. Then we can define the notions of component and parthood*—the counterparts of McKenzie and Muller’s Direct Part and Part—as follows:

Component. x is a component of y iff y is the result of applying ∑ to x and (possibly) some other object.

Parthood*. x is part of y iff there is a sequence of objects ${x_1}, \ldots ,{x_n}$ , $n \gt 0$ , for which $x = {x_1}$ , $y = {x_n}$ , and ${x_i}$ is a component of ${x_{i + 1}}$ for $i = 1, \ldots ,n - 1$ .

Then, independently of any physical details about ∑, parthood* is reflexive and transitive.^{Footnote 27} This shows that details about “being in a bound state” are irrelevant when it comes to prove some formal features of some particular part-like relation—like parthood*. They simply follow from taking an operation of general composition as primitive (and basic), rather than a relation such as parthood. This is important especially in the case of McKenzie and Muller for, as should by now be clear, they take Parthood as both a primitive and a defined notion. The point here is that if one takes it as it is defined in Part—and one should take it as a defined notion if one wants to derive some formal features, rather than assuming them axiomatically—one ends up with Component or Direct Parthood. In effect, as Fine himself remarks, the notion of mereological parthood is equivalent to that of component, rather than parthood*.^{Footnote 28} This is in line with the argument I offered. And, as Fine points out,

[I]f part is understood as component, it would be a substantive question whether the relation is transitive. (Fine Reference Fine2010, 569)

Note that McKenzie and Muller are interested in offering an answer to the SCQ, as it is understood in metaphysics. They are explicit:

[W]e are claiming (…) that the Bound State Proposal identifies the sort of composition that is relevant to the Special Composition Question discussed in metaphysics. (McKenzie and Muller Reference Kerry and Muller2017, 240; italics mine)

The SCQ, as it is understood in metaphysics, is cashed out in terms of component, not of parthood*, using Fine’s terminology. Or, in McKenzie and Muller’s terminology, it is phrased in terms of Direct Parthood. It is a substantive question whether that relation is transitive. And it turns out that it is not. To conclude: the BSA can recover the Reflexivity and Transitivity of some relation in the vicinity of mereological parthood. And this is not because of some details about bound states, or some other physical details about composition, but rather because of some general formal features.

But, in the end, what’s in a name? That which we call a rose by any other name would smell as sweet.^{Footnote 29} And saying that a leg is a tail does not make it a tail.^{Footnote 30} Calling parthood a relation in the vicinity of mereological parthood does not make that relation mereological parthood, as it is understood in the SCQ. And as far as that relation goes, we cannot prove that it is transitive. In fact, given the BSA, we can prove that it is not.

4. The BSA and Fastening

While going through some possible moderate answers to the SCQ, van Inwagen discusses what he calls Fastening:

On the face of it, the BSA seems fairly analogous to Fastening. In effect, it looks like a way of using physics to make Fastening more precise, to improve van Inwagen’s original formulation, as he himself would put it. This is because scattering states are exactly those states, in which particular forces are responsible for “separating” components.^{Footnote 31}

The analogy is worth pointing out for different reasons. First because, to my knowledge, it has not been pointed out.^{Footnote 32} Second, if borne out, it can be used to reply to an objection due to Markosian (Reference Markosian1998). Markosian writes:

In light of this general objection, Markosian contends, the expression “the x-s are fastened together” should be taken as a primitive. And this is a great theoretical cost. But, if the analogy holds, perhaps we can simply reply that we do know what it means for the x-s to be fastened together: it means that they are in a common bound state. And that latter notion can be defined as in (1).

Finally, the analogy is worth pointing out because van Inwagen objects to Fastening on the grounds that it delivers unwanted results. If you and I shake hands and we suddenly become paralyzed we become fastened. Yet, according to van Inwagen:

Following Markosian (Reference Markosian1998), let me call this the “Paralyzed Handshakers” objection. In light of the analogy above, consider the following argument. Suppose a speck of galactic matter is caught in the gravitational well of the Earth. Does this add to the furniture of the universe or merely diminish its capacity to be rearranged? Is this the complete explanation of the generation of the new thing that the speck of matter and the Earth compose?

There seems to be a worry here. Let me phrase it this way. Either the BSA is relevantly similar to Fastening, or it is not. If it is not, an account of the relevant difference is owed. If it is, then one has to address the Paralyzed Handshaker objection—or my galactic speck of matter variant. Either the objection was compelling in the first case, or it was not. If it was not, then we should have gone with Fastening all along. The BSA is just a way to make Fastening precise. However, one would need to motivate why the objection was not compelling. I know of no such discussion. If the objection was compelling against Fastening, but it is not compelling against the BSA, then, once again, one needs to explain why, especially under the assumption at work here, namely, that the BSA and Fastening are relevantly similar. Let me lay my cards on the table. I do not think that these difficulties are insuperable. As a matter of fact, I will suggest some strategies myself in the next section. Yet, it seems fair to say that more work needs to be done. This concludes my critical assessment of the BSA.

5. Mereological pluralism

In light of the above, one may wonder whether I think the BSA should simply be rejected. I am actually more sympathetic than one might infer from the arguments in the previous sections. Here I want to suggest different ways to look at the BSA—and other physics-based answers—that sidestep many of the worries raised above. I should be explicit upfront and confess that I will not provide an argument to the point that these are the only ways, the best ways, or the correct ways of looking at the BSA. I will limit myself to putting some suggestions on the table.

It is perhaps instructive to start by going back to the entanglement answer to the SCQ. As I briefly pointed out in §2, the entanglement answer and the BSA are not extensionally equivalent. The easiest way to appreciate it is to note that two entangled particles need not be in a bound state. Now, one can push the point that quantum mechanics need to treat (multiparticle)^{Footnote 33} entangled systems as composite systems to account for experimentally detectable correlations. In other words, one might treat entanglement as sufficient for composition.^{Footnote 34} This would provide an alleged physics-based counterexample to the BSA. It also seems that now we have two physics-based answers that deliver different results.

I suggest that this is significant and can be taken at face value. Perhaps the moral to be drawn from the discussion above is that physics provides us with different ways to build wholes out of some components, so to speak. Let me expand on this. As far as I can see, there are—at least—two different ways to develop the suggestion.

The first option is that we distinguish simple mereological composition from ψ-composition and we define the latter in terms of the former:

ψ-composition: A set S of material objects ψ composes a further object x iff the members of S mereologically compose x and $\psi (S)$ .

Mereological composition is necessary but not sufficient for ψ-composition. Different ψ will deliver different kinds of composition, different kinds of wholes, and different kinds of parts. Suppose ψ is “being in a common bound state.”^{Footnote 35} Then we will get a notion of Bound-composition, a notion of Bound-whole, and a notion of Bound-part. Or, suppose that ψ is “being in an entangled state.” Then we will get a notion of Entanglement-composition, a notion of entangled-whole, and a notion of entangled- part.

This suggestion will not help answer the SCQ, for the SCQ is crucially understood in terms of mereological composition. Yet, it will help with some worries raised in §3. For one, depending on ψ, alignment with common sense judgments should arguably not be considered a desideratum in the first place. Some such conditions are clearly beyond the scope of common sense judgments. It will also help answer some worries about the formal profile of parthood relations. One can wholeheartedly accept that mereological parthood is indeed transitive, whereas the defined notion of ψ-parthood is not. But this is far from problematic. Whether ψ-parthood is transitive will depend on the exact ψ.^{Footnote 36} Finally, it should be noted that this can help to assuage—if not undermine—the Paralyzed Handshakers objection, or my speck of galactic matter counterpart. The worry was that the BSA seemed incapable of explaining the “addition” to the furniture of nature, rather than the diminished capacity of rearranging some of its already existing items. In the case at hand, one would say that the BSA does provide an explanation of the fact that an already existing mereological sum becomes a new kind of whole; namely, a bound-whole.

Now, the option I just discussed is a somewhat conservative option. There is a second, more radical one, that can be put forward. According to such an option, the arguments above suggest genuine mereological pluralism. Mereological pluralism is the view that

Or, as McDaniel puts it:

The thought here is that there are different parthood relations that are not definable in terms of mereological parthood.^{Footnote 37} This is where mereological pluralism departs from ψ-composition above, for the latter holds that any ψ-parthood relation is definable in terms of mereological parthood. As Fine points out, from the fact that pluralists hold that different notions of parthood cannot be defined in terms of mereological parthood, it does not follow that they cannot be defined at all. In effect, a way to do so is exactly the one I presented in ${\rm{\S}}$ 3.4. One can start with different operations of composition ∑ that cannot be interdefined and then go on to define different notions of Component and Parthood*. In other words: one can endorse both compositional pluralism and mereological pluralism.

As far as I can see, this proposal will have the same consequences as the previous one vis-à-vis the SCQ, the alignment to common sense, and, mutatis mutandis,^{Footnote 38} the response to the Paralyzed Handshakers objection. The question about the formal profile of different parthood relations is, however, more interesting, and it is worth spending a few words on it. On the one hand, one may hold the view that the different notions of parthood might not share the very same formal profile. In particular, some of them might be transitive; some would not be. Bound-parthood can be then identified with McKenzie and Muller’s Direct Parthood, and failure of transitivity would not be problematic after all. On the other hand, one might simply insist that different notions of parthood might well have different formal profiles, and yet, the partial ordering axioms are constitutive of any relation that aspires to be a parthood relation. In this case, one should then insist that the way, in which a bound-part is a part is not really given by Direct Parthood, but rather by its transitive closure; namely, the relation that McKenzie and Muller define in Part. This is reminiscent of the discussion by Fine (Reference Fine2010) that focuses on an alleged notion of set-theoretic parthood:

It should be noted that the discussion above would not help McKenzie and Muller’s case for Parsimony. Their claim was that they could prove transitivity of the notion of parthood that is at stake in the original SCQ. That is mereological parthood. And the suggestion at hand is exactly that bound-parthood is not mereological parthood.

As I pointed out already, I concede that the discussion above does not provide a fully fledged argument in favor of any of these two ways of looking at the BSA and other physics-based answers to questions of composition in general. I am afraid this deserves an independent scrutiny. It should be enough for now to have laid these possibilities on the table.

6. Conclusion

To take stock: The BSA represents a very welcome development in the debate on the physics and metaphysics of composition. On the one hand, I believe our metaphysics should be informed by empirical science. On the other hand, as I argued, at a closer look the BSA is less conservative and less simple than expected (or desired), its precision is arguably overrated, and its ability to get some formal features of mereological parthood relation for free—that is, the formal features of the specific parthood relation that is employed in cashing out the original SCQ—is dubious. However, I also suggested different ways to look at the proposal—and at other relevant ones—which promise to have significant and fruitful ramifications. In particular, I suggested a less revisionary and a more revisionary understanding of the physics-based answer(s) in question. There is a sense, in which both these understandings are card-carrying pluralist proposals.^{Footnote 39} It should in conclusion be noted that an overall pluralistic attitude toward composition in physics has been suggested, if not advocated, in a number of places; for instance, Healey (Reference Healey2013) and Ceravolo and French (Forthcoming). An echo of such a general pluralistic attitude might be heard in this passage of Ladyman and Ross:

I do not share Ladyman and Ross’s extreme skepticism toward the “abstract composition relation”—if this is meant to be mereological composition. Perhaps the disparity of the wholes that we are presented with in physics calls for some pluralism after all. Arguments in its favor will have to wait. We cannot do the whole work at once, only some parts.