As anyone who has flown out of a cloud knows, the boundaries of acloud are a lot less sharp up close than they can appear on theground. Even when it seems clearly true that there is one, sharplybounded, cloud up there, really there are thousands of water dropletsthat are neither determinately part of the cloud, nor determinatelyoutside it. Consider any object that consists of the core of thecloud, plus an arbitrary selection of these droplets. It will looklike a cloud, and circumstances permitting rain like a cloud, andgenerally has as good a claim to be a cloud as any other object inthat part of the sky. But we cannot say every such object is a cloud,else there would be millions of clouds where it seemed like there wasone. And what holds for clouds holds for anything whose boundarieslook less clear the closer you look at it. And that includes justabout every kind of object we normally think about, including humans.Although this seems to be a merely technical puzzle, even atriviality, a surprising range of proposed solutions has emerged, manyof them mutually inconsistent. It is not even settled whether asolution should come from metaphysics, or from philosophy of language,or from logic. Here we survey the options, and provide several linksto the many topics related to the Problem.

• 1. Introduction
• 2. Nihilism
• 3. Overpopulation
• 4. Brutalism
• 5. Relative Identity
• 6. Partial Identity
• 7. Vagueness
  • 7.1 Argument from Duplication
  • 7.2 Argument from Similarity
  • 7.3 Argument from Meaning
• 8. Composition is not Identity
• 9. Rethinking Parthood
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

In his (1980), Peter Unger introduced the “Problem of theMany”. A similar problem appeared simultaneously in P. T. Geach(1980), but Unger’s presentation has been the most influentialover recent years. The problem initially looks like a special kind ofpuzzle aboutvague predicates, but that may be misleading. Some of the standard solutions toSorites paradoxes do not obviously help here, so perhaps the Problem reveals somedeeper truths involving the metaphysics of material constitution, orthe logic of statements involving identity.

The puzzle arises as soon as there is an object without clearlydemarcated borders. Unger suggested that clouds are paradigms of thisphenomenon, and recent authors such as David Lewis (1993) and NeilMcKinnon (2002) have followed him here. Here is Lewis’spresentation of the puzzle:

Think of a cloud—just one cloud, and around it a clear blue sky.Seen from the ground, the cloud may seem to have a sharp boundary. Notso. The cloud is a swarm of water droplets. At the outskirts of thecloud, the density of the droplets falls off. Eventually they are sofew and far between that we may hesitate to say that the outlyingdroplets are still part of the cloud at all; perhaps we might bettersay only that they are near the cloud. But the transition is gradual.Many surfaces are equally good candidates to be the boundary of thecloud. Therefore many aggregates of droplets, some more inclusive andsome less inclusive (and some inclusive in different ways thanothers), are equally good candidates to be the cloud. Since they haveequal claim, how can we say that the cloud is one of these aggregatesrather than another? But if all of them count as clouds, then we havemany clouds rather than one. And if none of them count, each one beingruled out because of the competition from the others, then we have nocloud. How is it, then, that we have just one cloud? And yet we do.(Lewis 1993: 164)

The paradox arises because in the story as told the following eightclaims each seem to be true, but they are mutually inconsistent.

0. There are several distinct sets of water dropletss_k such that for each such set, it is not clearwhether the water droplets ins_k form acloud.
1. There is a cloud in the sky.
2. There is at most one cloud in the sky.
3. For each sets_k, there is an objecto_k that the water droplets ins_k compose.
4. If the water droplets ins_i composeo_i, and the objects ins_jcomposeo_j, and the setss_iands_k are not identical, then the objectso_i ando_j are notidentical.
5. Ifo_i is a cloud in the sky, ando_j is a cloud in the sky, ando_i is not identical witho_j,then there are at least two clouds in the sky.
6. If any of these setss_i is such that itsmembers compose a cloud, then for any other sets_j, if its members compose an objecto_j, theno_j is a cloud.
7. Any cloud is composed of a set of water droplets.

To see the inconsistency, note that by 1 and 7 there is a cloudcomposed of water droplets. Say this cloud is composed of the waterdroplets ins_i, and lets_j beany other set whose members might, for all we can tell, form a cloud.(Premise 0 guarantees the existence of such a set.) By 3, the waterdroplets ins_j compose an objecto_j. By 4,o_j is not identicalto our original cloud. By 6,o_j is a cloud, andsince it is transparently in the sky, it is a cloud in the sky. By 5,there are at least two clouds in the sky. But this is inconsistentwith 2. A solution to the paradox must provide a reason for rejectingone of the premises, or a reason to reject the reasoning that led usto the contradiction, or the means to live with the contradiction.Since none of the motivations for believing in the existence ofdialetheia apply here, let us ignore the last possibility. And since 0 followsdirectly from the way the story is told, let us ignore that option aswell. That leaves open eight possibilities.

(The classification of the solutions here isslightlydifferent from that in Chapter One of Hud Hudson’s “AMaterialist Metaphysics of the Human Person.” But it has a deepdebt to Hudson’s presentation of the range of solutions, whichshould be clear from the discussion that follows.)

2. Nihilism

Unger’s original solution was to reject 1. The concept of acloud involved, he thought, inconsistent presuppositions. Since thosepresuppositions were not satisfied, there are no clouds. This is arather radical move, since it applies not just to clouds, but to anykind of sortal for which a similar problem can be generated. And,Unger pointed out, this includes most sortals. As Lewis puts it,“Think of a rusty nail, and the gradual transition from steel… to rust merely resting on the nail. Or think of a cathode,and its departing electrons. Or think of anything that undergoesevaporation or erosion or abrasion. Or think of yourself, or anyorganism, with parts that gradually come loose in metabolism orexcretion or perspiration or shedding of dead skin” (Lewis 1993:165).

Despite Lewis’s presentation, the Problem of the Many is not aproblem about change. The salient feature of these examples is that,in practice, change is a slow process. Hence whenever a cathode, or ahuman, is changing, be it by shedding electrons, or shedding skin,there are some things that are not clearly part of the object, norclearly not part of it. Hence there are distinct sets that each have agood claim to being the set of parts of the cathode, or of the human,and that is what is important.

It would be profoundly counterintuitive if there were no clouds, or nocathodes, or no humans, and that is probably enough to reject theposition, if any of the alternatives are not also equallycounterintuitive. It also, as Unger noted, creates difficulties formany views about singular thought and talk. Intuitively, we can pickout one of the objects composed of water droplets by the phrase‘that cloud’. But if it is not a cloud, then possibly wecannot. For similar reasons, we may not be able to name any suchobject, if we use any kind of reference-fixing description involving‘cloud’ to pick it out from other objects composed ofwater droplets. If the Problem of the Many applies to humans as wellas clouds, then by similar reasoning we cannot name or demonstrate anyhuman, or, if you think there are no humans, any human-like object.Unger was happy to take these results to be philosophical discoveries,but they are so counterintuitive that most theorists hold that theyform a reductio of his theory. Bradley Rettler (2018) argues that thenihilist has even more problems than this. Nihilism solves somephilosophical problems, such as explaining which of 0–7 is false. But,he argues, for any problem it solves, there is a parallel problemwhich it does not solve, but rival solutions do solve. For instance,if you think of the problem here as a version of aSorites paradox, nihilism does not help with versions of the paradox which concernpredicates applied to simples.

It is interesting that some other theories of vagueness have adoptedpositions resembling Unger’s in some respects, but without theextreme conclusions. Matti Eklund (2002) and Roy Sorensen (2001) haveargued that all vague concepts involve inconsistent presuppositions.Sorensen spells this out by saying that there are some inconsistentpropositions that anyone who possesses a vague concept should believe.In the case of a vague predicateF that is vulnerable to aSorites paradox, the inconsistent propositions are that some thingsareFs, some things are notFs, any object thatclosely resembles (in a suitable respect) something that isFis itselfF, and that there are chains of ‘suitablyresembling’ objects between anF and a non-F.Here the inconsistent propositions are that a story like Lewis’sis possible, and in it 0 through 7 are true. Neither Eklund norSorensen conclude from this that nothing satisfies the predicates inquestion; rather they conclude that some propositions that we findcompelling merely in virtue of possessing the concepts from which theyare constituted are false. So while they don’t adoptUnger’s nihilist conclusions, two contemporary theorists agreewith him that vague concepts are in some sense incoherent.

3. Overpopulation

A simple solution to the puzzle is to reject premise 2. Each of therelevant fusions of water droplets looks and acts like a cloud, so itis a cloud. As with the first option this leads to some verycounterintuitive results. In any room with at least one person, thereare many millions of people. But this is not as bad as saying thatthere are no people. And perhaps we don’t even havetosay the striking claim. In many circumstances, we quantifyover a restricted domain. We can say, “There’s nobeer,” even when there is beer in some non-salient locales. Withrespect to some restricted quantifier domains, it is true that thereis exactly one person in a particular room. The surprising result isthat with respect to other quantifier domains, there are many millionsof people in that room. The defender of the overpopulation theory willhold that this shows how unusual it is to use unrestrictedquantifiers, not that there really is only one person in the room.

The overpopulation solution is not popular, but it is not withoutdefenders. J. Robert G. Williams (2006) endorses it, largely becauseof a tension between the supervaluationist solution (that will bediscussed in section 7) and what supervaluationism says about theSorites paradox. James Openshaw (2021) and Alexander Sandgren(forthcoming) argue that the overpopulation solution is true, and eachoffer a theory of how singular thought about the cloud is possiblegiven overpopulation. Sandgren also points out that there might bemultiple sources of overpopulation. Even given a particular set ofwater droplets, some metaphysical theories will say that there aremultiple objects those droplets compose, which differ in theirtemporal or modal properties.

Hudson (2001: 39–44) draws out a surprising consequence of theoverpopulation solution as applied to people. Assume that there arereally millions of people just where we’d normally say there wasexactly one. Call that person Charlie. When Charlie raises her arm,each of the millions must also raise their arms, for the millionsdiffer only in whether or not they contain some borderline skin cells,not in whether their arm is raised or lowered. Normally, if two peopleare such that whenever one acts a certain way, then so must the other,we would say that at most one of them is acting freely. So it lookslike at most one of the millions of people around Charlie are free.There are a few possible responses here, though whether a defender ofthe overpopulation view will view this consequence as being morecounter-intuitive than other claims to which she is already committed,and hence whether it needs a special response, is not clear. There aresome other striking, though not always philosophically relevant,features of this solution. To quote Hudson:

Among the most troublesome are worries about naming and singularreference … how can any of us ever hope to successfully referto himself without referring to his brothers as well? Or how might wehave a little private time to tell just one of our sons of ouraffection for him without sharing the moment with uncountably many ofhis brothers? Or how might we follow through on our vow to practicemonogamy? (Hudson 2001: 39)

4. Brutalism

As Unger originally states it, the puzzle relies on a contentiousprinciple of mereology. In particular, it assumes mereologicalUniversalism, the view that for any objects, there is an object thathas all of them as its fusion. (That is, it has each of those objectsas parts, and has no parts that do not overlap at least one of theoriginal objects.) Without this assumption, the Problem of the Manymay have an easy solution. The cloud in the sky isthe objectup there that is a fusion of water droplets. There are many other setsof water droplets, other than the set of water droplets that composethe cloud, but since the members of those sets do not compose anobject, they do not compose a cloud.

There are two kinds of theories that imply that only one of the setsof water droplets is such that there exists a fusion of its atoms.First, there areprincipled restrictions on composition,theories that say that thexs compose an objectyiff thexs areF, for some natural propertyF. Secondly, there arebrutal theories, which sayit’s just a brute fact that in some cases thexscompose an object, and in others they do not. It would be quite hardto imagine a principled theory solving the Problem of the Many, sinceit is hard to see what the principle could be. (For a more detailedargument for this, set against a somewhat different backdrop, seeMcKinnon 2002.) But a brutal theory could work. And such a theory hasbeen defended. Ned Markosian (1998) argues that not only doesbrutalism, the doctrine that there are brute facts about when thexs compose ay, solve the Problem of the Many, theaccount of composition it implies fits more naturally with ourintuitions about composition.

It seems objectionable, in some not easy-to-pin-down way, to rely onbrute facts in just this way. Here is how Terrence Horgan puts theobjection:

In particular, a good metaphysical theory or scientific theory shouldavoid positing a plethora of quite specific, disconnected,suigeneris, compositional facts. Such facts would be ontologicaldanglers; they would be metaphysically queer. Even though explanationpresumably must bottom out somewhere, it is just not credible—oreven intelligible—that it should bottom out with specificcompositional facts which themselves are utterly unexplainable andwhich do not conform to any systematic general principles. (Horgan1993: 694–5)

On the other hand, this kind of view does provide a particularlystraightforward solution to the Problem of the Many. As Markosiannotes, if we have independent reason to view favourably the idea thatfacts about when some things compose an object are brute facts, whichhe thinks is provided by our intuitions about cases of composition andnon-composition, the very simplicity of this solution to the Problemof the Many may count as an argument in favour of brutalism.

5. Relative Identity

Assume that the brutalist is wrong, and that for every set of waterdroplets, there is an object those water droplets compose. Since thatobject looks for all the world like a cloud, we will say it is acloud. The fourth solution accepts those claims, but denies that thereare many clouds. It is true that there are many fusions of atoms, butthese are all the samecloud. This view adopts a positionmost commonly associated with P. T. Geach (1980), that two things canbe the sameF but not the sameG, even though theyare bothGs. To see the motivation for that position, and adiscussion of its strengths and weaknesses, see the article onrelative identity.

Here is one objection that many have felt is telling against therelative identity view: Letw be a water droplet that isins₁ but nots₂. The relativeidentity solution says that the droplets ins₁ compose an objecto₁, andthe droplets ins₂ compose an objecto₂, and thougho₁ ando₂ are different fusions of water droplets, theyare the same cloud. Call this cloudc. Ifo₁ is the same cloud aso₂,then presumably they have the same properties. Buto₁ has the property of havingw as apart, whileo₂ does not. Defenders of the relativeidentity theory here deny the principle that if two objects are thesameF, they have the same properties. Many theorists findthis denial to amount to areductio of the view.

6. Partial Identity

Even ifo₁ ando₂ exist, andare clouds, and are not the same cloud, it does not immediately followthat there are two clouds. If we analyze “There are twoclouds” as “There is anx and ay suchthatx is a cloud andy is a cloud, andxis not the same cloud asy” then the conclusion willnaturally follow. But perhaps that is not the correct analysis of“There are two clouds.” Or, more cautiously, perhaps it isnot the correct analysis in all contexts. Following some suggestionsof D. M. Armstrong’s (Armstrong 1978, vol. 2: 37–8), DavidLewis suggests a solution along these lines. The objectso₁ ando₂ are not the samecloud, but they arealmost the same cloud. And in everydaycircumstances (AT) is a good-enough analysis of “There is onecloud”

(AT)

There is a cloud, and all clouds arealmost identicalwith it.

As Lewis puts it, we ‘count by almost-identity’ ratherthan by identity in everyday contexts. And when we do, we get thecorrect result that there is one cloud in the sky. Lewis notes thatthere are other contexts in which we count by some criteria other thanidentity.

If an infirm man wishes to know how many roads he must cross to reachhis destination, I will count by identity-along-his-path rather thanby identity. By crossing the Chester A. Arthur Parkway and Route 137at the brief stretch where they have merged, he can cross both bycrossing only one road. (Lewis 1976: 27)

There are two major objections to this theory. First, as Hudson notes,even if we normally count by almost-identity, we know how to count byidentity, and when we do it seems that there is one cloud in the sky,not many millions. A defender of Lewis’s position may say thatthe only reason this seems intuitive is that it is normally intuitiveto say that there is only one cloud in the sky. And that intuition isrespected! More contentiously, it may be argued that it is agood thing to predict that when we count by identity we getthe result that there are millions of clouds. After all, the only timewe’d do this is when we’re doing metaphysics, and we havenoted that in the metaphysics classroom, there is some intuitive forceto the argument that there are millions of clouds in the sky. It wouldbe a brave philosopher to endorse this as avirtue of thetheory, but it may offset some of the costs.

Secondly, something like the Problem of the Many can arise even whenthe possible objects are not almost identical. Lewis notes thisobjection, and provides an illustrative example to back it up. Asimilar kind of example can be found in W. V. O. Quine’sWord and Object (1960). Lewis’s example is of a housewith an attached garage. It is unclear whether the garage is part ofthe house or an external attachment to it. So it is unclear whetherthe phrase ‘Fred’s house’ denotes the basic house,call it the home, or the fusion of the home and the garage. What isclear is that there is exactly one house here. However, the home mightbe quite different from the fusion of the home and the garage. It willprobably be smaller and warmer, for example. So the home and thehome–garage fusion are not even almost identical. Quine’sexample is of something that looks, at first, to be a mountain withtwo peaks. On closer inspection we find that the peaks are not quiteas connected as first appeared, and perhaps they could be properlyconstrued as two separate mountains. What we could not say is thatthere arethree mountains here, the two peaks and theirfusion, but since neither peak is almost identical to the other, or tothe fusion, this is what Lewis’s solution implies.

But perhaps it is wrong to understand almost-identity in this way.Consider another example of Lewis’s, one that Dan López deSa (2014) argues is central to a Lewisian solution to the problem.

You draw two diagonals in a square; you ask me how many triangles; Isay there are four; you deride me for ignoring the four largetriangles and counting only the small ones. But the joke is on you.For I was within my rights as a speaker of ordinary language, and youcouldn’t see it because you insisted on counting by strictidentity. I meant that, for somew,x,y,z, (1)w,x,y, andz aretriangles; (2)w andx are distinct, and … and soarey andz (six clauses); (3) for any trianglet, eithert andw are not distinct, or …ort andz are not distinct (four clauses). And by‘distinct’ I meant non-overlap rather than non-identity,so what I said was true. (Lewis 1993, fn. 9)

One might think this is the general way to understand countingsentences in ordinary language. SoThere is exactly one Fgets interpreted asThere is an F, and no F is wholly distinctfrom it; andThere are exactly two Fs gets interpretedasThere are wholly distinct things x and y that are both F, andno F is wholly distinct from both of them, and so on. Lewiswrites as if this is an explication of the almost-identity proposal,but this is at best misleading. A house with solar panels partiallyoverlaps the city’s electrical grid, but it would be verystrange to call them almost-identical. It sounds like a similar, butdistinct, proposed solution.

However we understand the proposal, López de Sa notes that ithas a number of virtues. It seems to account for the puzzle involvingthe house, what he calls the Problem of the Two. If in generalcounting involves distinctness, then we have a good sense in whichthere is one cloud in the sky, and Fred owns one house.

There still remain two challenges for this view. First, one couldstill follow Hudson and argue that even if we ordinarily understandcounting sentences this way, we do still know how to count byidentity. And when we do, it seems that there is just one cloud, notmillions of them. Second, it isn’t that clear that we alwayscount by distinctness, in the way López de Sa suggests. If Isay there are three ways to get from my house to my office, Idon’t mean to say that these three are completely distinct.Indeed, they probably all start with going out my door, down mydriveway etc., and end by walking up the stairs into my office. So thegeneral claim about how to understand counting sentences seemsfalse.

C. S. Sutton (2015) argued that we can get around something like thesecond of these problems if we do two things. First, the ruleisn’t that we don’t normally quantify over things thatoverlap, just that we don’t normally quantify over things thatsubstantially overlap. We can look at a row of townhouses and say thatthere are seven houses there even if the walls of the house overlap.Second, the notion of overlap here is not sensitive to the quantity ofmaterial that the objects have in common, but to their functionalrole. If two objects play very different functional roles, she arguesthat we will naturally count them as two, even if they have a lot ofmaterial in common. This could account for a version of the getting towork example where the three different ways of getting to work onlydiffer on how to get through one small stretch in the middle. That is,if there are three (wholly distinct ways) to get from B to C, and theway to get from A to D is to go A-B-C-D, then there is a good sense inwhich there are three ways to get from A to D. Sutton’s theoryexplains how this could be true even if the B-C leg is a short part ofthe trip.

David Liebesman (2020) argued for a different way of implementing thiskind of theory. He argues that the kinds of constraints that Lewis,López de Sa and Sutton have suggested don’t getincorporated into our theory of counting, but into the properinterpretation of the nouns involved in counting sentences. It helpsto understand Liebesman’s idea with an example.

There is, we’ll presumably all agree, just one colour in YvesKlein’sBlue Monochrome. (It says so in the title.) ButBlue Monochrome has blue init, and it has ultramarine in it, and blue doesn’t equalultramarine. What’s gone on? Well, says Liebesman, whenever anoun occurs under a determiner, it needs an interpretation. Thatinterpretation will almost never be maximal. When we ask how manyanimals a person has in their house, we typically don’t mean tocount the insects. When we say every book is on the shelf, wedon’t mean every book in the universe. And typically, therelevant interpretation of the determiner phrase (like ‘everybook’, or ‘one cloud’) will exclude overlappingobjects. Typically but not, says Liebesman, always. We can say, forinstance, that every shade of a colour is a colour, and in thatsentence ‘colour’ includes both blue and ultramarine.

This offers a new solution to the Problem of the Many. He argues thatall of 0 to 7 are true, but they are true for differentinterpretations of phrases including the word ‘cloud’.When we interpret it in a way that rules out overlap, then 6 is false.When we interpret it in the maximal way, like the way we interpret‘colour’ inEvery shade of a colour is a colour,then 2 is false. But to get the inconsistency, we have to equivocate.It’s an easy equivocation to make, since each of the meanings isone that we frequently use.

7. Vagueness

Step 6 in the initial setup of the problem says that if any of theo_i is a cloud, then they all are. There are threeimportant arguments for this premise, two of them presented explicitlyby Unger, and the other by Geach. Two of the arguments seem to befaulty, and the third can be rejected if we adopt some familiar,though by no means universally endorsed, theories of vagueness.

7.1 Argument from Duplication

The first argument, due essentially to Geach, runs as follows.Geach’s presentation did not involve clouds, but the principlesare clearly stated in his version of the argument. (The argument showsthat if ano_k is a cloud for arbitraryk,we can easily generalize to the claim that for everyi,o_i is a cloud.)

D2 implies thatbeing a cloud is anintrinsic property. The idea is that by changing the world outside the cloud,we do not change whether or not it is a cloud. There is, however,little reason to believe this is true. And given that it leads to arather implausible conclusion, that there are millions of clouds wherewe think there is one, there is some reason to believe it is false. Wecan argue directly for the same conclusion. Assume many more waterdroplets coalesce around our original cloud. There is still one cloudin the sky, but it determinately includes more water droplets than theoriginal cloud. The fusion of those water droplets exists, and we mayassume that they did not change their intrinsic properties, but theyare now apart of a cloud, rather than a cloud. Even ifsomething looks like a cloud, smells like a cloud and rains like acloud, it need not be a cloud, it may only be a part of a cloud.

7.2 Argument from Similarity

Unger’s primary argument takes a quite different tack.

Since we only care about the conditionalif o_jis a cloud, so is o_k, it is clearly acceptable toassume thato_j is a cloud for the sake of theargument. And S3 is guaranteed to be true by the setup of the problem.The main issue then is whether S2 is true. As Hudson notes, thereappear to be some clear counterexamples to it. The fusion of a cloudwith one of the water droplets in my bathtub is clearly not a cloud,but by most standards it differs minutely from a cloud, since there isonly one droplet of water difference between them.

7.3 Argument from Meaning

The final argument is not set out as clearly, but it has perhaps themost persuasive force. Unger says that if exactly one of theo_i is a cloud, then there must be a‘selection principle’ that picks it over the others. Butit is not clear just what kind of selection principle that could be.The underlying argument seems to be something like this:

The idea behind M2 is that word meanings are not brute facts aboutreality. As Jerry Fodor put it, “if aboutness is real, it mustbe really something else” (Fodor 1987: 97). Something makes itthe case thato_j is the unique thing (around here)that satisfies our term ‘heap’. Maybe that could bebecauseo_j has some unique properties that make itsuitable to be in the denotation of ordinary terms. Or maybe it issomething about our linguistic practices. Or maybe it is somecombination of these things. But something must determine it, andwhatever it is, we can (in theory) say what that is, by giving somekind of principled explanation of whyo_j is theunique cloud.

It is at this point that theories ofvagueness can play a role in the debate. Two of the leading theories ofvagueness, epistemicism and supervaluationism, provide principledreasons to reject this argument. The epistemicist says that there aresemantic facts that are beyond our possible knowledge. Arguably we canonly know where a semantic boundary lies if that boundary was fixed byour use or by the fact that one particular property is a natural kind.But, say the epistemicists, there are many other boundaries that arenot like this, such as the boundary between the heaps and thenon-heaps. Here we have a similar kind of situation. It is vague justwhich of theo_i is a cloud. What that means isthat there is a fact about which of them is a cloud, but we cannotpossibly know it. The epistemicist is naturally read as rejecting thevery last step in the previous paragraph. Even if something (probablyour linguistic practices) makes it the case thato_j is the unique cloud, that need not be somethingwe can know and state.

The supervaluationist response is worth spending more time on here,both because it engages directly with the intuitions behind thisargument and because two of its leading proponents (Vann McGee andBrian McLaughlin, in their 2001) have responded directly to thisargument using the supervaluationist framework. Roughly (and for moredetail see the section on supervaluations in the entry onvagueness) supervaluationists say that whenever some terms are vague, there areways of making them more precise consistent with our intuitions on howthe terms behave. So, to use a classic case, ‘heap’ isvague, which to the supervaluationist means that there are some pilesof sand that are neither determinately heaps nor determinatelynon-heaps, and a sentence saying that that object is a heap is neitherdeterminately true nor determinately false. However, there are manyways toextend the meaning of ‘heap’ so itbecomes precise. Each of these ways of making it precise is called aprecisification. A precisification isadmissible iffevery sentence that is determinately true (false) in English is true(false) in the precisification. So ifa is determinately aheap,b is determinately not a heap andc is neitherdeterminately a heap nor determinately not a heap, then everyprecisification must make ‘a is a heap’ true and‘b is a heap’ false, but some make‘c is a heap’ true and others make it false. To afirst approximation, to be admissible a precisification must assignall the determinate heaps to the extension of ‘heap’ andassign none of the determinate non-heaps to its extension, but it isfree to assign or not assign things in the ‘penumbra’between these groups to the extension of ‘heap’. But thisis not quite right. Ifd is a little larger thanc,but still not determinately a heap, then the sentence “Ifc is a heap so isd” is intuitively true. Asit is often put, following Kit Fine (1975), a precisification mustrespect ‘penumbral connections’ between the borderlinecases. Ifd has a better case for being a heap thanc, then a precisification cannot makec a heap butnotd. These penumbral connections play a crucial role in thesupervaluationist solution to the Problem of the Many. Finally, asentence is determinately true iff it is true on all admissibleprecisifications, determinately false iff it is false on alladmissible precisifications.

In the original example, described by Lewis, the sentence “Thereis one cloud in the sky” is determinately true. None of thesentences “o₁ is a cloud”,“o₂ is a cloud” and so on aredeterminately true. So a precisification can make each of these eithertrue or false. But, if it is to preserve the fact that “There isone cloud in the sky” is determinately true, it must makeexactlyone of those sentences true. McGee and McLaughlinsuggest that this combination of constraints lets us preserve what isplausible about M2, without accepting that it is true. The term‘cloud’ is vague; there is no fact of the matter as towhether its extension includeso₁ oro₂ oro₃ or …. If therewere such a fact, there would have to be something that made it thecase that it includedo_j and noto_k, and as M3 correctly points out, no such factsexist. But this is consistent with saying that its extension doescontain exactly one of theo_i. The beauty of thesupervaluationist solution is that it lets us hold these seeminglycontradictory positions simultaneously. We also get to capture some ofthe plausibility of S2—it is consistent with thesupervaluationist position to say that anything similar to a cloud isnot determinately not a cloud.

Penumbral connections also let us explain some other puzzlingsituations. Imagine I point cloudwards and say, “That is acloud.” Intuitively, what I have said is true, even though‘cloud’ is vague, and so is my demonstrative‘that’. (To see this, note that there’s nodeterminate answer as to which of theo_i it picksout.) On different precisifications, ‘that’ picks outdifferento_i. But on every precisification itpicks out theo_i that is in the extension of‘cloud’, so “That is a cloud” comes out trueas desired. Similarly, if I named the cloud ‘Edgar’, thena similar trick lets it be true that “Edgar” is vague,while “Edgar is a cloud” is determinately true. So thesupervaluationist solution lets us preserve many of the intuitionsabout the original case, including the intuitions that seemed tounderwrite M2, without conceding that there are millions of clouds.But there are a few objections to this package.

• Objection: There are many telling objections tosupervaluationist theories of vagueness.

  • Reply: This may be true, but it would take us well beyond thescope of this entry to outline them all. See the entries onvagueness (the section on supervaluation) and theSorites Paradox for more detail.

• Objection: The supervaluationist solution makes someexistentially quantified sentences, like “There is a cloud inthe sky” determinately true even though no instance of them isdeterminately true.

  • Reply: As Lewis says, this is odd, but no odder than thingsthat we learn to live with in other contexts. Lewis compares this to“I owe you a horse, but there is no particular horse that I oweyou.”

• Objection: The penumbral connections here are notprecisely specified. What is the rule that says how much overlap isrequired before two objects cannot both be clouds?

  • Reply: It is true that the connections are not preciselyspecified. It would be quite hard to carefully analyze‘cloud’ to work out exactly what they are. But that wecannot say exactly what the rules are is no reason for saying no suchrules exist, any more than our inability to say exactly what knowledgeis provides a reason for saying that no one ever knows anything.Scepticism can’t be proventhat easily.

• Objection: The penumbral connections appealed to here areleft unexplained. At a crucial stage in the explanation, it seems tobe just assumed that the problem can be solved, and that it is adeterminate truth that there is one cloud in the sky.

  • Reply 1: We have to start somewhere in philosophy. This kindof reply can be spelled out in two ways. There is a‘Moorean’ move that says that the premise that there isone cloud in the sky is more plausible than the premises that wouldhave to be used in an argument against supervaluationism.Alternatively, it might be claimed that the main argument forsupervaluationism is an inference to the best explanation. In thatcase, the intuition that there is exactly one cloud in the sky, but itis indeterminate just which object it is, is something to beexplained, not something that has to beproven. This is thekind of position defended by Rosanna Keefe in her bookTheories ofVagueness. Although Keefe does not apply thisdirectlyto the Problem of the Many, the way to apply her position to theProblem seems clear enough.

  • Reply 2: The penumbral connections we find for most words aregenerated by the inferential role provided by the meaning of theterms. It is because the inference from “This pile of sand is aheap”, and “That pile of sand is slightly larger than thisone, and arranged roughly the same way,” to “That pile ofsand is a heap” is generally acceptable that precisificationswhich make the premises true and the conclusions false areinadmissible. (We have to restrict this inferential rule to the casewhere ‘this’ and ‘that’ are ordinarydemonstratives, and not used to pick out arbitrary fusions of grains,or else we get bizarre results for reasons that should be familiar bythis point in the story.) And it isn’t too hard to specify theinferential rule here. The inference from“o_j is a cloud” and“o_j ando_k massivelyoverlap” to “o_k is a cloud” isjust as acceptable as the above inference involving heaps. Indeed, itis part of the meaning of ‘cloud’ that this inference isacceptable. (Much of this reply is drawn from the discussion of‘maximal’ predicates in Sider 2001 and 2003, though sinceSider is no supervaluationist, he would not entirely endorse this wayof putting things.)

• Objection: The second reply cannot explain the existenceof penumbral connections between ‘cloud’ anddemonstratives like ‘that’ and names like‘Edgar’. It would only explain the existence of thosepenumbral connections if it was part of the meaning of names anddemonstratives that they fill some kind of inferential role. But thisis inconsistent with the widely held view that demonstratives andnames aredirectly referential. (For more details on debates about the meanings of names, see theentries onpropositional attitude reports andsingular propositions.)

  • Reply: One response to this would be to deny the view thatnames and demonstratives are directly referential. Another would be todeny that inferential roles provide theonly penumbralconstraints on precisifications. Weatherson 2003b sketches a theorythat does exactly this. The theory draws on David Lewis’sresponse to some quite different work on semantic indeterminacy. Asmany authors (Quine 1960, Putnam 1981, Kripke 1982) showed, thedispositions of speakers to use terms are not fine-grained enough tomake the language as precise as we ordinarily think it is. As far asour usage dispositions go, ‘rabbit’ could mean undetachedrabbit part, ‘vat’ could mean vat image, and‘plus’ could mean quus. (Quus is a function defined overpairs of numbers that yields the sum of the two numbers when they areboth small, and 5 when one is sufficiently large.) But intuitively ourlanguage is not that indeterminate: ‘plus’ determinatelydoes not mean quus.

    Lewis (1983, 1984) suggested the way out here is to posit a notion of‘naturalness’. Sometimes a termt denotes theconceptC₁ rather thanC₂ notbecause we are disposed to uset as if it meantC₁ rather thanC₂, but simplybecauseC₁ is a more natural concept.‘Plus’ means plus rather than quus simply because plus ismore natural than quus. Something like the same story applies to namesand demonstratives. Imagine I point in the direction of Tibbles thecat and say, “That is Edgar’s favourite cat.” Thereis a way of systematically (mis)interpreting all my utterances so‘that’ denotes the Tibbles-shaped region of space-timeexactly one metre behind Tibbles. (We have to reinterpret what‘cat’ means to make this work, but the discussions ofsemantic indeterminacy in Quine and Kripke make it clear how to dothis.) So there’s nothing in my usage dispositions that makes‘that’ mean Tibbles, rather than the region of space-timethat ‘follows’ him around. But because Tibbles is morenatural than that region of space-time, ‘that’ does pickout Tibbles. It is the very same naturalness that makes‘cat’ denote a property that Tibbles (and not the trailingregion of space-time) satisfies that makes ‘that’ denoteTibbles, a fact that will become important below.

    The same kind of story can be applied to the cloud. It is because thecloud is a more natural object than the region of space-time a mileabove the cloud that our demonstrative ‘that’ denotes thecloud and not the region. However, none of theo_iare more natural than any other, so there is still no fact of thematter as to whether ‘that’ picks outo_j oro_k. Lewis’s theorydoes not eliminate all semantic indeterminacy; when there are equallynatural candidates to be the denotation of a term, and each of them isconsistent with our dispositions to use the term, then the denotationof the term is simply indeterminate between those candidates.

    Weatherson’s theory is that the role of each precisification isto arbitrarily make one of theo_i more naturalthan the rest. Typically, it is thought that the denotation of a termaccording to a precisification is determined directly. It is a factabout a precisificationP that, according to it,‘cloud’ denotes propertyc₁. OnWeatherson’s theory this is not the case. What theprecisification does is provide a new, and somewhat arbitrary,standard of naturalness, and the content of the terms according to theprecisification is then determined by Lewis’s theory of content.The denotations of ‘cloud’ and ‘that’according to a precisificationP are those concepts andobjects that are the mostnatural-according-to-P of theconcepts and objects that we could be denoting by those terms, for allone can tell from the way the terms are used. The coordination betweenthe two terms, the fact that on every precisification‘that’ denotes an object in the extension of‘cloud’ is explained by the fact that the very same thing,naturalness-according-to-P, determines the denotation of‘cloud’ and of ‘that’.

• Objection (From Stephen Schiffer 1998): Thesupervaluationist account cannot handle speech reports involving vaguenames. Imagine that Alex points cloudwards and says, “That is acloud.” Later Sam points towards the same cloud and says,“Alex said that that is a cloud.” Intuitively, Sam’sutterance is determinately true. But according to thesupervaluationist, it is only determinately true if it is true onevery precisification. So it must be true that Alex said thato₁ is a cloud, that Alex said thato₂ is a cloud, and so on, since these are allprecisifications of “Alex said that that is a cloud.” ButAlex did not say all of those things, for if she did she would becommitted to saying that there are millions of clouds in the sky, andof course she is not, as the supervaluationists have been arguing.

  • Reply: There is a little logical slip here. LetP_i be a precisification of Sam’s word‘that’ that makes it denoteo_i. Allthe supervaluationist who holds that Sam’s utterance isdeterminately true is committed to is that for eachi,according toP_i, Alex said thato_i is a cloud. And this will be true if thedenotation of Alex’s word ‘that’ is alsoo_i according toP_i. So as longas there is a penumbral connection between Sam’s word‘that’ and Alex’s word ‘that’, thesupervaluationist avoids the objection. Such a connection may seemmysterious at first, but note that Weatherson’s theory predictsthat just such a penumbral connection obtains. So if that theory isacceptable, then Schiffer’s objection misfires.

• Objection (From Neil Mackinnon 2002 and Thomas Sattig2013): It is part of our notion of a mountain that facts aboutmountain-hood are not basic. If something is a mountain, there arefacts in virtue of which it is a mountain, and nearby things are not.Yet on each precisification of ‘mountain’, thiswon’t be true; it will be completely arbitrary which fusion ofrocks is a mountain.

  • Reply: It is true that on any precisification there will beno principled reason why this fusion of rocks is a mountain, andanother is not. And it is true that there should be such a principledreason; mountainhood facts are not basic. But that problem can beavoided by the theory that denies that “the supervaluationistrule [applies] to any statement whatever, never mind that thestatement makes no sense that way” (Lewis 1993, 173).Lewis’s idea, or at least the application of Lewis’s ideato this puzzle, is that we know how to understand the idea thatmountainhood facts are non-arbitrary: we understand it as a claim thatthere is some non-arbitrary explanation of which precisifications of‘mountain’ are and are not admissible. If we must applythe supervaluationist rule to every statement, including the statementthat it is not arbitrary which things are mountains, thisunderstanding is ruled out. Lewis’s response is to deny that therule must always be applied. As long as there is some sensible way tounderstand the claim, we don’t have to insist on applying thesupervaluationist machinery to it.

    That said, it does seem like this is likely to be somewhat of aproblem for everyone (even the theorist like Lewis who uses thesupervaluationist machinery only when it is helpful). Sattig himselfclaims to avoid the problem by making the mountain be a maximal fusionof candidates. But for any plausible mountain, it will be vague andsomewhat arbitrary what the boundary is between being amountain-candidate and not being one. The lower boundaries ofmountains are not, in practice, clearly marked. Similarly, there willbe some arbitrariness in the boundaries between the admissible andinadmissible precisifications of ‘mountain’. We may haveto live with some arbitrariness.

• Objection (From J. Robert G. Williams 2006): When we lookat an ordinary mountain, there definitely is (at least) one mountainin front of us. That seems clear. The vagueness solution respects thisfact. But in many contexts, people tend to systematically confuse“Definitely, there is anF” with “There isa definiteF.” Indeed, the standard explanation of whySorites arguments seem, mistakenly, to be attractive is that thisconfusion gets made. Yet on the vagueness account, there is nodefinite mountain; all the candidates are borderline cases. So byparity of reasoning, we should expect intuition to deny that there isdefinitely a mountain. And intuition does not deny that, it loudlyconfirms it. At the very least, this shows a tension between thestandard account of the Sorites paradox, and the vagueness solution tothe Problem of the Many.

  • Reply: This is definitely a problem for the views that manyphilosophers have put forward. As Williams stresses, it isn’t onits own a problem for the vagueness solution to the Problem of theMany, but it is a problem for the conjunction of that solution with awidely endorsed, and independently plausible, explanation of theSorites paradox. In his dissertation, Nicholas K. Jones (2010) arguesthat the right response is to give up the idea that speakers typicallyconfuse “Definitely, there is anF” with“There is a definiteF”, and instead use adifferent resolution of the Sorites.

Space prevents a further discussion of all possible objections to thesupervaluationist account, but interested readers are particularlyencouraged to look at Neil McKinnon’s objection to the account(see the Other Internet Resources section), which suggests thatdistinctive problems arise for the supervaluationist when there reallyare two or more clouds involved.

Even if the supervaluationist solution to the Problem of the Many hasresponses to all of the objections that have been levelled against it,some of those objections rely on theories that are contentious and/orunderdeveloped. So it is far from clear at this stage how well thesupervaluationist solution, or indeed any solution based on vagueness,to the Problem of the Many will do in future years.

8. Composition is not Identity

Some theorists have argued that the underlying cause of the problem isthat we have the wrong theory about the relation between parts andwholes. Peter van Inwagen (1990) argues that the problem is that wehave assumed that the parthood relation is determinate. We haveassumed that it is always determinately true or determinately falsethat one object is a part of another. According to van Inwagen,sometimes neither of these options applies. He thinks that we need toadopt some kind offuzzy logic when we are discussing parts and wholes. It can be true to degree0.7, for example, that one object is part of another. Given theseresources, van Inwagen says, we are free to conclude that there isexactly one cloud in the sky, and that some of the ‘outer’water droplets are part of it to a degree strictly between 0 and 1.This lets us keep the intuition that it is indeterminate whether theseoutlying water droplets are members of the cloud without acceptingthat there are millions of clouds. Note that this is not what vanInwagen would say aboutthis version of the paradox, since heholds that some simples only constitute an object when that object isalive. For van Inwagen, as for Unger, there are no clouds, onlycloud-like swarms of atoms. But van Inwagen recognises that a similarproblem arises for cats, or for people, two kinds of things that hedoes believe exist, and he wields this vague constitution theory tosolve the problems that arise there.

Traditionally, many philosophers thought that such a solution wasdownright incoherent. A tradition stretching back to Bertrand Russell(1923) and Michael Dummett (1975) held that vagueness was always andeverywhere a representational phenomenon. From this perspective, itdidn’t make sense to talk about it being vague or indeterminatewhether a particular droplet was part of a particular cloud. But thistraditional view has come under a lot of pressure in recent years; seeBarnes (2010) for one of the best challenges, and Sorensen (2013,section 8) for a survey of more work. So let us assume here it islegitimate to talk about the possibility that parthood itself, and notjust our representation of it, is vague. As Hudson (2001) notesthough, it is far from clear just how the appeal to fuzzy logic ismeant to helphere. Originally it was clear for each ofn water droplets whether they were members of the cloud todegree 1 or degree 0. So there were 2ⁿ candidateclouds, and the Problem of the Many is finding out how to preserve theintuition when faced with all these objects. It is unclear howincreasing the range of possible relationships between eachparticle and the cloud from 2 to continuum-many should help here, fornow it seems there are at least continuum-many cloud-like objects tochoose between, one for each function from each of thendroplets to [0, 1], and we need a way of saying exactly one of them isa cloud. Assume that some droplet is part of the cloud to degree 0.7.Now consider the object (or perhaps possible object) that is just likethe cloud, except this droplet is only part of it to degree 0.6. Doesthat object exist, and is it a cloud? Van Inwagen says, in a wayreminiscent of Markosian’s brutal composition solution, thatsuch an ‘object’ does not even exist.

A different kind of solution is offered by Mark Johnston (1992) and E.J. Lowe (1982, 1995). Both of them suggest that the key to solving theProblem is to distinguish cloud-constituters from clouds. They say itis a category mistake to identify clouds with any fusion of waterdroplets, because they have different identity conditions. The cloudcould survive the transformation of half its droplets into puddles onthe footpath (or whatever kind of land it happens to be raining over),it would just be a smaller cloud, the fusion could not. As Johnstonsays, “Hence Unger’s insistent and ironic question‘But which ofo₁,o₂,o₃, … is our paradigm cloudc?’ has as its proper answer ‘None’”(1992: 100, numbering slightly altered).

Lewis (1993) listed several objections to this position, and Lowe(1995) responds to them. (Lewis and Lowe discuss a version of theproblem using cats not clouds, and we will sometimes follow thembelow.)

Lewis’s first objection is that positing clouds as well ascloud-constituting fusions of atoms is metaphysically extravagant. AsLowe (and, for separate reasons, Johnston) point out, these extraobjects are arguably needed to solve puzzles to do with persistence.Hence it is no objection to a solution to the Problem of the Many thatit posits such objects. Resolving these debates would take us too farafield, so let us assume (as Lewis does) that we have reason tobelieve that these objects exist.

Secondly, Lewis says that even with this move, we still have a Problemof the Many applied to cloud-constituters, rather than to clouds. Loweresponds that since ‘cloud-constituter’ is not a folkconcept, we don’t really have any philosophically salientintuitions here, so this cannot be a way in which the position isunintuitive.

Finally, Lewis says that each of the constituters is so like theobject it is meant to merely constitute (be it a cloud, or a cat, orwhatever), it satisfies the same sortals as that object. So if we wereoriginally worried that there were 1001 cats (or clouds) where wethought there was one, now we should be worried that there are 1002.But as Lowe points out, this argument seems to assume thatbeing acat, orbeing a cloud, is an intrinsic property. If weassume that it is extrinsic, if it turns on the history of the object,perhaps its future or its possible future, and on which object it isembedded in, then the fact that a cloud-constituter looks, whenconsidered in isolation, to be a cloud is little reason to think itactually is a cloud.

Johnston provides an argument that the distinction between clouds andcloud-constituting fusions of water droplets is crucial to solving theProblem. He thinks that the following principle is sound, and notthreatened by examples like our cloud.

(9′) Ify is a paradigmF, andx isan entity that differs fromy in any respect relevant tobeing anF only very minutely,and x is of the rightcategory, i.e. is not a mere quantity or piece of matter, thenx is anF. (Johnston 1992: 100)

The theorist who thinks that clouds are just fusions of water dropletscannot accept this principle, or they will conclude that everyo_i is a cloud, since for them eacho_i is of the right category. On the other hand,Johnston himself cannot accept it either, unless he denies there canbe another objectc′ which is in a similar position toc, and is of the same category asc, but differswith respect to which water droplets constitute it. It seems that whatis doing the work in Johnston’s solution is not just thedistinction between constitution and identity, but a tacit restrictionon when there is a ‘higher-level’ object constituted bycertain ‘lower-level’ objects. To that extent, his theoryalso resembles Markosian’s brutal composition theory, thoughsince Johnston can accept that every set of atoms has a fusion histheory has different costs and benefits to Markosian’stheory.

A recent version of this kind of view comes from Nicholas K. Jones(2015), though he focusses on constitution, not composition. (Indeed,a distinctive aspect of his view is that he takes constitution to bemetaphysically prior to composition.) Jones rejects the followingprinciple, which is similar to 4 in the original inconsistent set.

• If the water droplets ins_i constituteo_i, and the objects ins_jconstituteo_j, and the setss_iands_k are not identical, then the objectso_i ando_j are notidentical.

He rejects this claim. He argues that some water droplets canconstitute a cloud, and some other water droplets can constitute thevery same cloud. On this view, the predicateconstitute xbehaves a bit like the predicatesurround the building. Itcan be true that the Fs surround the building, and the Gs surround thebuilding, without the Fs being the Gs. And on Jones’s view, itcan be true that the Fs constitutex, and the Gs constitutex, without the Fs being the Gs. This resembles Lewis’ssolution in terms of almost-identity, since both Jones and Lewis saythat there is one cloud, yet boths_i ands_kcan be said to compose it. But for Lewis, thisis possible because he rejects the inference fromThere is onecloud, toIf a and b are clouds, they are identical.Jones accepts this inference, and rejects the inference from thepremise thats_i ands_k aredistinct, and each compose a cloud, to the conclusion that theycompose non-identical clouds.

9. Rethinking Parthood

After concluding that all of these kinds of solutions face seriousdifficulties, Hudson (2001: Chapter 2) outlines a new solution, onewhich rejects so many of the presuppositions of the puzzle that it isbest to count him as rejecting the reasoning, rather than rejectingany particular premise. (Hudson is somewhat tentative aboutendorsing this view, as opposed to merely endorsing the claimthat it looks better than its many rivals, but for expository purposeslet us refer to it here as his view.) To see the motivation behindHudson’s approach, consider a slightly different case, a variantof one discussed in Wiggins 1968. Tibbles is born at midnight Sunday,replete with a splendid tail, called Tail. An unfortunate accidentinvolving a guillotine sees Tibbles lose his tail at midday Monday,though the tail is preserved for posterity. Then midnight Monday,Tibbles dies. Now consider the timeless question, “Is Tail partof Tibbles?” Intuitively, we want to say the question isunderspecified. Outside of Monday, the question does not arise, forTibbles does not exist. Before midday Monday, the answer is“Yes”, and after midday the answer is “No”.This suggests that there is really no proposition that Tail is part ofTibbles. There is a proposition that Tail is part of Tibbles on Mondaymorning (that’s true) and that Tail is part of Tibbles on Mondayafternoon (that’s false), but no proposition involving just theparthood relation and two objects. Parthood is a three-place relationbetween two objects and a time, not a two-place relation between twoobjects.

Hudson suggests that this line of reasoning is potentially on theright track, but that the conclusion is not quite right. Parthood is athree-place relation, but the third place is not filled by a time, butby a region of space-time. To a crude approximation,x ispart ofy ats is true if (as we’d normallysay)x is a part ofy ands is a region ofspace-time containing no region not occupied byy and allregions occupied byx. But this should be taken as aheuristic guide only, not as a reductive definition, since parthood isreally a three-place relation, so the crude approximation does noteven express a proposition according to Hudson.

To see how this applies to the Problem of the Many, let’ssimplify the case a little bit so there are only two water droplets,w₁ andw₂, that are neitherdeterminately part of the cloud nor determinately not a part of it. Aswell there is the core of the cloud, call ita. On anorthodox theory, there are four proto-clouds here,a,a +w₁,a +w₂ anda +w₁ +w ₂. On Hudson’s theory the largest and thesmallest proto-clouds still exist, but in the middle there is a quitedifferent kind of object, which we’ll callc. Letr₁ be the region occupied bya andw₁, andr₂ the region occupiedbya andw₂. Then the following claimsare all true according to Hudson:

• c exactly occupiesr₁;
• c exactly occupiesr₂;
• c does not occupy the region consisting of the union ofr₁ andr₂;
• c hasw₁ as a part atr₁, but not atr₂;
• c hasw₂ as a part atr₂, but not atr₁;
• c has no parts at the region consisting of the union ofr₁ andr₂.

Hudson defines “x exactly occupiess” asfollows:

• x has a part ats,
• there is no region of space-time,s*, such thats* hass as a subregion, whilex has a partats*, and
• for every subregion ofs,s′,xhas a part ats′. (Hudson 2001: 63)

At first, it might look like not much has been accomplished here. Allthat we did was turn a Problem of 4 clouds into a Problem of 3 clouds,replacing the fusionsa +w₁ anda +w₂ with the new, and oddly behaved,c. But that is to overlook a rather important feature of theremaining proto-clouds. The three remaining proto-clouds can bestrictly ordered by the ‘part of’ relation. This was notpreviously possible, since neithera +w₁nora +w₂ were part of the other. If weadopt the principle that ‘cloud’ is a maximal predicate,so no cloud can be a proper part of another cloud, we now get theconclusion that exactly one of the proto-clouds is a cloud, asdesired.

This is a quite ingenious approach, and it deserves some attention inthe future literature. It is hard to say what will emerge as the maincosts and benefits of the view in advance of that literature, but thefollowing two points seem worthy of attention. First, if we areallowed to appeal to the principle that no cloud is a proper part ofanother, why not appeal to the principle that no two clouds massivelyoverlap, and get from 4 proto-clouds to one actual cloud that way?Secondly, why don’t we have an object that is just like the olda +w₁, that is, an object that hasw₁ as a part atr₁, and doesnot havew₂ (or anything else) as a part atr₂? If we get it back, as well asa +w₂, then all of Hudson’s tinkering withmereology will just have converted a problem of 4 clouds into aproblem of 5 clouds.

Neither of these points should be taken to be conclusive refutations.As things stand now, Hudson’s solution joins the ranks of themany and varied proposed solutions to the Problem of the Many. Forsuch a young problem, the variety of these solutions is ratherimpressive. Whether the next few years will see these ranks whittleddown by refutation, or swelled by imaginative theorising, remains tobe seen.

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