Defending the Junk Argument *

Defending the Junk Argument* . . 

 Introduction

Unrestricted composition is one of the most controversial doctrines of classical mereology. An objection has been put forward to the view that mereological composition is necessarily unrestricted by appealing to the possibility of so-called ‘junky’ worlds. A world is junky if everything in that world is a proper part of something (i.e. ∀x∃y(x < y) is true).¹ e junk argument, due to Bohn [, , ], proceeds as follows:

(i) If a world is junky, then the unrestricted fusion axiom for mereology is false at that world.

(ii) Junky worlds are metaphysically possible.

(iii) erefore, the unrestricted fusion axiom for mereology is either metaphysically contingent or necessarily false.

Much attention has been devoted to the defence of premise (ii). I do not wish to challenge the case for (ii).² However, more recently premise (i) has come under ﬁre. For example, Contessa [] and Spencer [] have each provided independent reasons that ‘universalists’ should dispatch the argument by rejecting premise (i). In this paper, I undertake a defence of premise (i) against a variety of objections. In §, I put forward a new objection to premise (i): Bohn’s defence of it presupposes far too much. In particular, I show that Bohn’s suppositions entail mereological extensionalism; that is, these assumptions rule out the view that objects with the same proper parts may be distinct. Two of the most prominent non-extensional mereologies are introduced: the unsupplemented view, and the mutual parts view. A non-extensionalist, then, might have strong reasons for accepting universalism but rejecting the argument.

*is paper is a  . Please do not cite without permission. Comments are welcome. ¹is is the converse of the notion of a gunky world in which everything in that world has something as a proper part. ²But see Watson [] for discussion. Bohn’s main defence involves three criteria for metaphysical possibility: conceivability, advocacy, and consistency. But if we can conceive of junky worlds, and several prominent philosophers have taken the idea seriously, and there are no logical contradictions lurking, then we are hard pressed to deny the mere possibility of the world being junky. ([], ) For the record, I do not think these three criteria are suﬃcient for metaphysical possibility, but the matter is too complicated for discussion here.  | . . 

In §§–, I show that one can defend premise (i) from a much weaker set of assumptions. Hence variants of the junk argument can be shown to apply to non-extensional mereological systems as well. I address unsupplemented mereologies in § which explicitly reject one of Bohn’s assumptions. In order to avoid begging the question against such views, I show that a variant of the junk argument survives even without appeal to any kind of supplementation. § argues that the mutual parts theorist who accepts unrestricted composition can avoid Bohn’s argument. However, the kind of junk compatible the the mutual parts view is only counterfeit — it satisfies the definition in letter, but not in spirit. A revision in the definition of junk yields the desired result. § takes on Contessa’s objection to premise (i). Contessa contends that those who accept unrestricted composition should only accept the existence of binary sums rather than infinitary fusions. I argue this conception of unrestricted composition is problematic: it is in conflict with the existence of gunk satisfying an intuitive remainder principle. Ithenconsiderapossibleresponse based on the mereology of Bostock [], and suggest that it’s not really a version of unrestricted fusion and so changes the terms of the debate. It may well be the most plausible version of a junky mereology, but it hardly constitutes a counterinstance to premise (i). In§, Iconsiderarecentresponsetopremise(i)bySpencer[]. Spencer’s view is that there is no absolutely unrestricted universal quantifier. So, any statement of the unrestricted fusion axiom will simply not rule out the existence of junky worlds. I argue that in order for this response to provide a counterexample to premise (i), we need some further argument. First, since junky worlds are defined using the universal quantifier, what Spencer’s argument allows appears to be worlds that are inexpressibly junky. But the metaphysical possibility of inexpressibly junky worlds needs an independent defence, and is not given (even implicitly) by the acceptance of premise (ii). Second, we may be simply unable to quantify over all the parts of something, or unable to quantify over all the simples. But neither of these options will yield junky worlds. ird, even if we have a principled reason to think that our quantifier is indefinitely extensible upwards along parthood chains, it’s not clear that the resulting expression of the unrestricted fusion axiom is genuinely unrestricted in the required sense. ese defences of premise (i) show that it is particularly resilient and can be based on extremely minimal assumptions. e upshot, then, is that anyone who wants to reject the junk argument must do so by rejecting premise (ii) instead.  | . . 

 Analyticity & Extensionalism

In his argument for premise (i), Bohn relies on the following assumptions, where < is proper parthood, ≤ improper parthood, and ◦ mereological overlap:

Transitivity (x < y ∧ y < z) → x < z

Weak Supplementation x < y → ∃z(z ≤ y ∧ ¬z ◦ x))

Weak Supplementation says that if something has a proper part, it must have another part disjoint from the ﬁrst. In combination with transitivity, this entails the asymmetry and irreﬂexivity of <.

Irreﬂexivity x ̸< x

Asymmetry x < y → y ̸< x

Bohn’s argument for premise (i) is as follows:

[A]ssume universalism is true in all possible worlds and that some possible world w is junky. en universalism is true in w. Consider the plurality of everything there is in w, call it aa. By universal instantiation, ∃y(aa compose y); by existential instantiation, aa compose U. Bydeﬁnition[ofjunk],itistruein w that ∀x∃y(x < y). By universal instantiation, ∃y(U < y). But U is composed of everything in w, and hence, by deﬁnition, everything in w is a part of it, including itself. But then nothing can have U as a proper part because if so, by [weak] supplementation, there would be something that shares no part with U,andhenceisnotapartof U, which contradicts that everything is a part of U. So, if some possible world is junky, then universalism is not necessarily true. Q.E.D. ([, p. , fn ])

Now, the principles use in the argument are admittedly fairly plausible; however, they are far from uncontroversial.³ In fact, as we will see below, there is a growing number of mereologists who accept unrestricted composition but reject at least one. If that weren’t reason enough to generalise the junk argument, it turns out that these ‘minimal’ principles are enough, when combined with unrestricted composition, to yield full classical extensional mereology. is includes perhaps the

³Indeed, Bohn regards these principles as analytic of parthood. I assume at least some mereological principles without existential import are analytically true, and in virtue of that are necessarily true too. […] Minimal Extensional Mereology is a minimum of mereological necessary truths. is system includes (among other things) the asymmetry and transitivity of proper parthood, as well as a weak supplementation principle. ([], ) I do not think weak supplementation is analytic, but I appreciate more needs to be said. See [XXXXX].  | . . 

most controversial principle of all — extensionality which says that composite objects with the same proper parts are identical.⁴ e argument for this claim is due to Varzi []. Informally, imagine you had a counterexample to extensionality; a statue s and its clay c, suppose, are distinct objects with the same proper parts. By asymmetry, s ̸< c, and c ̸< s.⁵ By unrestricted fusion, there must be a sum s + c (which is distinct from either s or c, since neither is part of the other). Now, s < s + c; however, there is no part of s + c that is disjoint from s,asanypartof c is also part of s by supposition. is violates weak supplementation. Hence, there can be no such counterexample to extensionality. ere are two ways to avoid Varzi’s argument; as a result, there are two main approaches to mereology without extensionality: the unsupplemented view, and the mutual parts view. Both of these approaches are compatible with unrestricted composition; indeed, many proponents of these views themselves accept it. Since each explicitly rejects one of Bohn’s assumptions, the junk argument begs the question against them. But it does so needlessly, or so I shall argue.

 Weak Supplementation & Junk

e ﬁrst main non-extensionalist option is the unsupplemented approach which drops the weak supplementation axiom above. e unsupplemented approach has access to a well-developed mereology that includes unrestricted fusion. Here is a candidate list of axioms, where ≤, ◦, and F are deﬁned as above.

Asymmetry x < y → y ̸< x

Transitivity (x < y ∧ y < z) → x < z

Unrestricted Composition ∀xx∃y F(y, xx).

Fusions appearing in the unrestricted composition axiom are deﬁned as follows.⁶

Fusion F(t, xx) iﬀ xx ≤ t ∧ ∀y(xx ≤ y → t ≤ y)

Here, xx ≤ t means that each x that is among xx is part of t. On this view, proper parthood is still a strict partial order. e major change here is that weak supplementation is explicitly rejected. Some unsupplemented mereologists simply drop the

⁴See Varzi [] to get a ﬂavour of the debate. ⁵For, suppose c < s. Asymmetry implies s ̸< c. Hence, s and c are not a counterexample to extensionality. Mutatis mutandis for the supposition that s < c. ⁶is deﬁnition of fusion is weaker than the one used in Varzi’s argument. For current purposes, it seems prudent to use the weakest plausible form of the fusion axiom.  | . . 

axiom and leave it at that.⁷ Others (primarily Gilmore []) have suggested a replacement axiom to capture the intuition behind weak supplementation without risk of extensionality.⁸ But Bohn’s defence of premise (i) explicitly appeals to weak supplementation. is begs the question against unsupplemented anti-extensionalists, many of whom do accept unrestricted composition. ere is, however, a straightforward argument from transitivity and asymmetry (in place of weak supplementation) that unrestricted composition entails the negation of junk. Let xx range over absolutely everything in the model. Since composition is unrestricted, there is some u such that F(u, xx). Assuch,wehave xx ≤ u ∧ ∀y(xx ≤ y → u ≤ y) (by the definition of fusion). Now, to satisfy the definition of junk suppose uˆ ̸= u and u ≤ uˆ. Hence, u < uˆ. By asymmetry, uˆ ̸< u, and thus uˆ ≰ u. But that contradicts the supposition that xx contained absolutely everything in the model. So, there is no such uˆ, and hence the model is not junky. Since ≤ is a partial order and F(t, xx) is only true whenever t is the least upper bound of xx, this argument corresponds exactly to the well-known algebraic fact that complete lattices are always bounded. None of this depends on any supplementation whatsoever, and so not on weak supplementation. As a result, a version of the junk argument applies to anti-extensionalists of this stripe too. Now, there is some debate over the correct notion of fusion for anti-extensionalists. In fact, some have held that the least upper bound definition above is inadequate.⁹ An anti-extensionalist of this sort would have reason again to reject the junk argument, due to the inadequate and unintended formulation of unrestricted fusion. However, there is a further generalisation of the junk argument which applies to any adequate notion of fusion whatsoever. Surely, any adequate notion of mereological fusion must be such that it is an upper bound of the things it fuses (even if not a least upper bound). at is, if F(t, xx) then xx ≤ t. One cannot think t is the fusion of the cats if some cat fails to be part of t. But then, given unrestricted fusion (of any sort), we know that every subset of the domain has an upper bound. is implies that the entire domain has an upper bound. But then (in the presence of the partial order axioms at least) this implies that the model has a maximal element. Maximal elements are by definition precisely those elements x for which there is no y such that x < y. Any model that has a maximal element

⁷Proponents include: [, , , ]. Also see Caplan et. al. [] who express sympathy for the view. See these proponents for the case for the conceivability. It is also clear the approach is consistent (the axioms are satisﬁed in any complete lattice; for quasi-supplementation, consider complete join semi-lattices with at least two distinct minimal elements). So by Bohn’s own criteria, the approach would appear to be metaphysically possible. It should at least be regarded as an available position in logical space. ⁸e relevant axiom is: Quasi-Supplementation x < y → ∃w∃z(z ≤ y ∧ w ≤ y ∧ ¬z ◦ w)) However this addition is strictly optional; there is a version of the junk argument that goes through without it. ⁹See for example the arguments in Varzi [] and [XXXXX].  | . . 

cannot be a junky world. As a result, junky worlds are incompatible with unrestricted fusion of any reasonable type.

 Asymmetry & Junk

e second option for anti-extensionalists is the mutual parts view which drops the asymmetry axiom for proper parthood (or equivalently for our purposes, the antisymmetry axiom for improper parthood: (x ≤ y ∧ y ≤ x) → x = y.) is allows for two distinct objects to be parts of each other. As it turns out, this is suﬃcient for avoiding extensionality principles of all sorts.is approach has a growing number of advocates, who make a strong case mutual parts approach.¹⁰ e following axioms give a precise characterisation of the mutual parts approach, where ≤ and ◦ are deﬁned in the usual way.

Transitivity (x < y ∧ y < z) → x < z

Remainder y ≰ x → ∃z∀w(w ≤ z ↔ (w ≤ y ∧ ¬w ◦ x))

Unrestricted Composition ∀xx∃y F(y, xx).

By rejecting asymmetry, the mutual parts view has access to an even stronger supplementation principle — the remainder principle, which says that if y fails to be part of x, then there is an object composed of all and only the non-x-overlapping parts of y — x’s complement in y.¹¹ is principle is again optional; however, we will return to this principle below (§).¹² But even in the presence of unrestricted fusion, this approach does not fall prey to the junk argument. ere are models of this mereology that also satisfy Bohn’s junk principle, stipulating that everything has a proper part.

Junk ∀x∃y(x < y)

One such model is given by considering an a and b that are mutual parts such that everything else in the model is part of both. Such a model is consistent with unrestricted fusion, since a and b both count as fusions of the other. As is evident, a and b are distinct, so in fact there are two universal objects. Now, is this a junky world? Yes, since a < b, a has everything as a proper part,

¹⁰Proponents of dropping antisymmetry include [, , , , , , ]. ey also have access to a well-developed mereology (which includes unrestricted fusion) with a class of well-defined models. us, the approach is consistent. It appears the approach satisfies Bohn’s own criteria for possibility. At least very least, then, asymmetry should be considered worthy of consideration. ¹¹Simons [, p. ] is responsible for the name. Varzi [] and Hovda [] calls it Complementation. ¹²e definition of fusion here is the same as above. However, the weakness of this definition is made up for by the corresponding strength of the remainder principle. (See[], §. for more details).  | . . 

but is itself a proper part of something (namely, of b). Likewise, since b < a, b is has everything as a part but is itself a proper part of something (namely, a). So, on the current definition of junk, the junk argument fails against the mutual parts view as premise (i) is simply false. However, the kind of junk displayed above is not the sort of junk one has in mind when levelling the argument against opponents of unrestricted composition. In fact, all the above model shows is that the definition of junk is inadequate in this context. ankfully, we can replace this definition with a different one that does rule out models of the above sort.

Junk* ∀x∃y(x < y ∧ y ̸< x)

So, a junky* world is a world in which everything is a proper part of something that is not also ∗ one of its own proper parts. In other words, junk guarantees that everything is a non-mutual proper part. is sort of junk is incompatible with the above mereology. And moreover, it shows that the junk argument does not rely on any assumption of asymmetry, either.

 Binary Sums & Junk

I have been ignoring a complication. I have been using ‘unrestricted composition’ in the usual way to mean that whenever you have some things, xx, there is a fusion of them. Contessa [] distinguishes between strong and weak versions of the thesis.

Strong Unrestricted Composition For any plurality xx, there is an object that is their fusion.¹³

Weak Unrestricted Composition For any pair of objects x1 and x2, there is an object that is their fusion.

He contends that those who accept unrestricted composition should accept only the weak version and reject the strong. So, for Contessa mereological universalists ought to accept only the existence of binary sums, and reject inﬁnitary fusions. Let us set aside the fact that the junk argument was clearly originally levelled against the strong version of the thesis; moreover, ignore the fact that no one who calls herself a universalist accepts the weak thesis but not the strong. A natural reply springs to mind: the weak thesis isn’t really a form of mereological universalism at all, is it? After all, such a view rejects that certain kinds

¹³It should be noted that Bohn [] states the thesis as follows: “at is, assume that necessarily, any collection of things composes something” (p. ). Likewise, Contessa states the strong thesis as follows: “for any collection of objects, there is always an object that is their mereological sum” (p. ). But, given the ambiguity involved in the notion of collection, we should be clear that it is only non-empty collections or pluralities that have fusions; virtually no universalist thinks that the empty set has a fusion.  | . .  of fusions exist — it accepts that composition holds only in some cases (the ﬁnitary ones) and rejects that it holds under all cases (which would include the inﬁnitary ones).¹⁴ Contessa considers this thought and writes,

However, according to weak mereological universalism, that should not be under- stood as a restriction on composition, for, according to weak mereological universalism, mereological composition would be restricted only if there were pairs of objects that, under some circumstances, did not have a mereological sum and, since junky worlds do not feature any such pair of objects, they do not constitute a counterexample to weak mereological universalism. ([], p. )

I am not sure precisely what Contessa has in mind, but the passage seems to suggest the idea that according to the weak thesis the composition relation is a essentially binary. at is, the composition relation by its very nature takes two objects as inputs and outputs some further object. On that understanding of composition, weak universalists think composition is unrestricted. Infini- tary cases of compositions are no counterexample to the weak thesis because infinitary collections of objects aren’t even ‘composition-apt’. Composition doesn’t fail to obtain, it fails to apply. I wish to argue argue the weak conception of unrestricted composition faces a problem not faced by its strong counterpart: in particular, it is in tension with a standard principle of mereology, the remainder principle above. But before making this argument, it should be noted that (contrary to Bohn []) the weak thesis is compatible with the existence of gunky worlds — worlds in which everything has a proper part.¹⁵ e weak universalist might accept the possible existence of objects with infinitely many proper parts, so long as these objects may be constructed pair-wise. To see why, consider a non- junky but gunky model M; in particular, suppose M has no infinite ascending <-chains, but has

¹⁴Bohn [] considers a similar view, calling it a ‘restricted view of composition’ in his original paper, and provides an argument against it (see footnote  below). ¹⁵Bohn claims: at means that if the world is finite, it contains a universal object U, while if the world is infinite with mereological atoms, it is junky. e problem with this principle is that it is incompatible with the world being gunky. A gunky world is such that everything in it has a proper part. If everything in the world has a proper part, then every fusion in the world is infinitely divisible, which means that every fusion in the world is of infinite cardinality, which again means that no fusion in the world is of finite cardinality. Since our principle implies that necessarily all fusions are of finite cardinality, it must therefore be incompatible with the possibility of gunk. ([, p. ], emphasis mine) Unfortunately, this is not a sound argument. As we will see below, it is possible that a gunky world have only finitary fusions. e problem is with the emphasised portion of the passage. Just because the model is infinite (and indeed, every object infinitely divisible), it does not follow that every fusion must be infinite. After all, the fusion of two objects is a binary operation, even if those two objects have infinitely-many proper parts.  | . . 

inﬁnite descending <-chains.¹⁶ It follows that for every non-empty subset A of M, there is some ﬁnite subset F of A such that the least upper bound of F is the least upper bound of A.¹⁷ Since we can take all the members of F and fuse them pair-wise, it follows that every non-empty subset A has a fusion. is means that the weak thesis is compatible with the existence of gunky worlds. Here’s another way to put the point. Start with a universal object u. Now divide u into two

parts u1 and u2. Choose, say, u2 and divide it into two parts u3 and u4, and so on. Is there any reason for thinking this process must come to an end? e weak thesis is compatible with an answer of ‘no’. ′ Now, drop the requirement that the world be non-junky; so, wehaveanewmodel M which is both junky and gunky (i.e. what Bohn [] calls ‘hunky’). We no longer have the result that every ′ non-empty subset A of M has a sum. And here is where we run into trouble with the remainder principle. Recall:

Remainder y ≰ x → ∃z∀w(w ≤ z ↔ (w ≤ y ∧ ¬w ◦ x))

′ Take some gunky object a in our model M , and consider some b < a. Since a ≰ b, the remainder principle requires that there be some z composed of all and only the non-b-overlapping parts of a. But that set of non-b-overlappers might be infinite! And (unlike above) there’s no guarantee here that there’s any corresponding finite set F which has the same sum. As a result, the weak thesis is in conflict with the remainder principle. ere are a few avenues of reply: one might (a) reject the remainder principle, (b) reject the ′ existence of models like M (i.e. reject the possibility of hunky worlds), or (c) accept both, and allow some cases of infinitary composition. Against, (a), I would urge that the complementation principle is an important part of classical mereology; it’s what makes mereology a Boolean algebra after all. e resulting mereology would be highly non-classical in ways unrelated to the existence of junk.¹⁸ Against (b), while not incoherent, it would be odd if in defending the possibility of junk one was forced into denying its compossibility with gunk. Why should it be the case that any world with proper parthood chains that continue infinitely upward cannot have any proper parthood chains that continue infinitely downward? Absent some principled metaphysical reason, it is a dialectically weak position to find oneself defending, at any rate.¹⁹

¹⁶It is obvious that weak unrestricted composition is compatible with non-junky worlds (after all, all finite models with be non-junky); the weak thesis doesn’t force junk any more than it forces a universal object. But it is much less obvious to see that it is compatible with gunk. ¹⁷More specifically, the weak thesis and partial order axioms guarantee that M is a (semi-)lattice.∨ ∨ So, if M satisfies the ascending chain condition, then for every A ⊆ M there is some finite F ⊆ A such that A = F. For a proof of this fact, see Davey and Priestly [, p. ], eorem .(i). ¹⁸For more discussion, see Hovda [], especially footnote . ¹⁹See also the arguments in Bohn [, §].  | . . 

So I think the best option is to go for (c). In fact, this is the choice of the only mereologist (that I am aware of, at least) who accepts the weak thesis but not the strong thesis. Bostock [, ch.  §] devised his mereology for the foundations of arithmetic in the rational numbers; as such his mereology is quite robust. It allows for gunk and junk, and also satisﬁes the remainder principle. e main departure from classical mereology that Bostock proposes is a variant fusion axiom, which (in our notation) is:

Moderately Unrestricted Composition ∀xx(∃y(xx ≤ y) → ∃zF(z, xx))

Here again the notion of fusion in play is least upper bounds as above.²⁰ What this principle says is that if xx have an upper bound, then they must have a least upper bound. ese structures are standardly called bounded complete lattices, which need not be complete. In other words, if some inﬁnite collection is bounded from above, then it must have a fusion. is is compatible with the existence of junk, though, since not all inﬁnite collections are required to be bounded from above. Unfortunately, this version of principle makes it plain that the view is not really a version of mereological universalism at all. e fusion axiom is plainly restricted — only collections that are bounded have fusions. And so, I view this response as changing the terms of the debate. It may well be the most plausible version of a junky mereology, but it hardly constitutes a counterinstance to premise (i).

 Unrestricted Quantiﬁcation & Junk

Spencer [] has recently argued against premise (i) by asserting that its defense relies on presup- posing an absolutely unrestricted universal quantiﬁer.²¹ at is, the incompatibility of junk and unrestricted composition requires the supposition that there are some things xx — call them ‘all things’ — such that everything is amongst them.

[T]his argument clearly relies on the premise that there are some things that all things are amongst. If there are no such things, then the argument is unsound. In fact, unrestricted composition and [the denial of ‘all things’] together entail junk. ([, p. ]) ′ ²⁰Bostock [, p. ] actually uses a variant notion of fusion: F (t, xx) iff ∀y(y ◦ t ↔ y ◦ xx). In the presence of the remainder principle, though, the two notions are equivalent. ²¹is will come as no surprise to Bohn, who is quite explicit about this, writing: “Universalism [is defined as] ∀xx∃y(xx compose y), where the quantifiers are unrestricted and ‘xx’ is a plural variable taking one or more things as its value” ([, p. ]).  | . . 

Whilethisapproachisthebestattemptatavoidingpremise(i), itislessthanclearthatthedenialof unrestricted quantification will deliver the compossibility of unrestricted composition and junky worlds in the required sense. e first argument is simple: junky worlds are themselves defined using the universal quantifier. So, if our universal quantifier fails to quantify over everything, those objects not in its scope will fail to have a fusion. However, it will not follow that the world is junky, since it will not follow that ∀x∃y(x < y) is true. at’s because those objects that fall outside the scope of the unrestricted fusion axiom will also fall outside the scope of the quantifier in the statement of junk. What might be maintained are worlds that are inexpressibly junky. But the metaphysical possibility of inexpressibly junky worlds has not been argued for. It needs an independent defence — a defence that goes well beyond the defence of premise (ii). One could respond that there is a difference between absolutely unrestricted singular quantification and absolutely unrestricted plural quantification. e former is used in the definition of junk, while the latter is used in the fusion axiom. Spencer might hold that, while there is no way of plurally quantifying over all things, there is a way of singularly doing so.²² If so, then the denial of ‘all things’ could be expressed as follows: ¬∃xx∀y(y is among xx).²³ However, it’s not clear to me how stable this view is. Consider the plurality of things — zz — that satisfy the predicate ∃x(z = x). Given plural comprehension, there must be such a plurality. And if ∃ ranges over absolutely everything, then zz are ‘all things’. Of course, one might reject plural comprehension, but that comes with its own set of costs. Here is a second reason why we the failure of absolutely general quantification might not yield the existence of junky worlds. Suppose there is no plurality of all things. It could still very well be the case that there is some yy such that the fusion of yy is the universal object. Why? Because the fusion of yy can have far more parts than just the yy! Suppose we can quantify over every macro-level object but we cannot quantify over all their parts. It may well be that the world gets ‘too big’ as we move to more and more finely grained levels of reality. We may be simply unable to quantify over all things because we cannot quantify over all the parts of things.²⁴ Another possibility: it may be that there are too many mereological simples to quantify over. In that case, we couldn’t quantify over all things because we couldn’t quantify over all the smallest things. In either of the two scenarios above, we can mount an argument from Zorn’s Lemma.

Zorn’s Lemma Let M be any non-empty partially ordered set. If every chain has an upper bound,

²²Indeed, Spencer considers the possibility of this view. (See §, pp. –.) ²³is is not the way Spencer expresses it however. He expresses the denial of ‘all things’ thus: ¬∃xx∀yy(yy are among xx). ²⁴Spencer considers this option and suggests it would be an ‘interesting metaphysical result’.  | . . 

then M has a maximal element.

Suppose M here is a model of mereology and the partial order on M is just the parthood relation. A chain is defined to be a totally ordered subset of M; so, a subset such that for every x and y, either x ≤ y or y ≤ x. If we can quantify over objects at some macro-level specification, by unrestricted fusion (of any sort), we will have it that every parthood chain has an upper bound. By Zorn’s Lemma, we know that M has at least one maximal element. And again, a maximal element entails the failure of junk. ²⁵ Likewise, if there are too many simples to form a plurality, those simples may not be parts of anything (due to the corresponding weakness of the unrestricted composition axiom). Still, each individual simple will be an upper bound of that (trivial) chain containing itself. Every other chain will have an upper bound by unrestricted fusion. Hence, by Zorn’s Lemma, we will have a maximal element of the domain — in this case, perhaps lots! But the existence of such maximal elements mean that this cannot be a model of a junky world. is is all perfectly compatible with the failure of the universal quantifier to be absolutely general. What is required, then, is a reason to think that our quantifier fails to quantify over absolutely everything going upwards along parthood chains, rather than the failure occurring as we go downwards. It’s not as if Spencer has nothing to say on this front; but it goes beyond simply the mere supposition that our quantifiers fail to be absolutely unrestricted. e best form of this view, I think, would motivate the failure of unrestricted quantification via indefinite extensibility. at is, whenever we have ‘all the things’ we can construct some new thing which has all previous things as parts. So, at each level n in the construction, there is some further thing at level n + 1 which failed to be quantified over at level n. is typed approach is seen clearly by indexing our quantifiers. us, a kind of ‘junk’ becomes expressible:

Typed Junk ∀nx∃n+1y(x < y)

Likewise, the corresponding version of ‘unrestricted’ composition becomes:

Typed Composition ∀nxx∃n+1y F(y, xx)

On this view, junk is compatible with typed composition. However, one should immediately begin to wonder whether this typed version of composition is really unrestricted after all. How

²⁵For a concrete example, imagine the domain of all subsets of the [0, 1] interval of the real numbers. Imagine that pluralities can take at most countably many subsets of that interval. Let yy be the plurality containing two intervals: [0, 0.5] and [0.5, 1]. e fusion of yy is the whole unit, even though there is no plurality of all subsets of the unit.  | . .  are we supposed to admit that our quantifiers fail to be unrestricted, and yet accept that composition is unrestricted is they exhibit the very same phenomena? We might say that junk is still incompatible with absolutely unrestricted composition; an reword premise (i) accordingly. Of course, the reply will come: but there is no such thing as absolutely unrestricted composition — some variant of our typed composition is the best anyone can do! I am not sure how to adjudicate this debate over whether something counts as ‘unrestricted’ or not. e issues are too complex to delve into here. But I’ll just note that if the reply is correct, then it depends on the arguments for the metaphysical possibility of indefinite extensibility, and the corresponding sort of junk generated from it. And so it still seems to me to boil down to a case of requiring a robust defense of premise (ii). I have argued that the junk argument can be modified to apply to virtually any mereology — extensional or not — that accepts unrestricted composition of any sort. I’ve shown that the retreat the the weak version of mereological universalism is not a promising move. It either requires more widescale revisions to mereological structures, or is explicitly a form of restricted composition. Moreover, I’ve argued that rejecting absolutely general quantification does not (at least by itself) ensure the possibility junky worlds. One might allow for inexpressible gunk, this option goes well beyond anything that premise (ii) provides. Or if one expresses junk and composition via type restrictions, then it’s unclear that the view really does represent a form of unrestricted mereological composition. I conclude that premise (i) of the junk argument is undefeated. It remains for proponents of unrestricted composition to explain precisely why junk is impossible.²⁶ University of St Andrews St Andrews, Fife  ,  ac@st-andrews.ac.uk

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