## 1. Introduction



In his (1980), Peter Unger introduced the "Problem of the
Many". A similar problem appeared simultaneously in P. T. Geach
(1980), but Unger's presentation has been the most influential
over recent years. The problem initially looks like a special kind of
puzzle about
 vague predicates,
 but that may be misleading. Some of the standard solutions to
 Sorites paradoxes
 do not obviously help here, so perhaps the Problem reveals some
deeper truths involving the metaphysics of material constitution, or
the logic of statements involving identity.



The puzzle arises as soon as there is an object without clearly
demarcated borders. Unger suggested that clouds are paradigms of this
phenomenon, and recent authors such as David Lewis (1993) and Neil
McKinnon (2002) have followed him here. Here is Lewis's
presentation 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. Not
> so. The cloud is a swarm of water droplets. At the outskirts of the
> cloud, the density of the droplets falls off. Eventually they are so
> few and far between that we may hesitate to say that the outlying
> droplets are still part of the cloud at all; perhaps we might better
> say only that they are near the cloud. But the transition is gradual.
> Many surfaces are equally good candidates to be the boundary of the
> cloud. Therefore many aggregates of droplets, some more inclusive and
> some less inclusive (and some inclusive in different ways than
> others), are equally good candidates to be the cloud. Since they have
> equal claim, how can we say that the cloud is one of these aggregates
> rather than another? But if all of them count as clouds, then we have
> many clouds rather than one. And if none of them count, each one being
> ruled out because of the competition from the others, then we have no
> cloud. 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 eight
claims each seem to be true, but they are mutually inconsistent.


0. There are several distinct sets of water droplets
*sk* such that for each such set, it is not clear
whether the water droplets in *sk* form a
cloud.
1. There is a cloud in the sky.
2. There is at most one cloud in the sky.
3. For each set *sk*, there is an object
*ok* that the water droplets in
*sk* compose.
4. If the water droplets in *si* compose
*oi*, and the objects in *sj*
compose *oj*, and the sets *si*
and *sk* are not identical, then the objects
*oi* and *oj* are not
identical.
5. If *oi* is a cloud in the sky, and
*oj* is a cloud in the sky, and
*oi* is not identical with *oj*,
then there are at least two clouds in the sky.
6. If any of these sets *si* is such that its
members compose a cloud, then for any other set
*sj*, if its members compose an object
*oj*, then *oj* 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 cloud
composed of water droplets. Say this cloud is composed of the water
droplets in *si*, and let *sj* be
any 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 water
droplets in *sj* compose an object
*oj*. By 4, *oj* is not identical
to our original cloud. By 6, *oj* is a cloud, and
since 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 inconsistent
with 2. A solution to the paradox must provide a reason for rejecting
one of the premises, or a reason to reject the reasoning that led us
to the contradiction, or the means to live with the contradiction.
Since none of the motivations for believing in the existence of
 dialetheia
 apply here, let us ignore the last possibility. And since 0 follows
directly from the way the story is told, let us ignore that option as
well. That leaves open eight possibilities.



(The classification of the solutions here is *slightly*
different from that in Chapter One of Hud Hudson's "A
Materialist Metaphysics of the Human Person." But it has a deep
debt to Hudson's presentation of the range of solutions, which
should be clear from the discussion that follows.)


## 2. Nihilism



Unger's original solution was to reject 1. The concept of a
cloud involved, he thought, inconsistent presuppositions. Since those
presuppositions were not satisfied, there are no clouds. This is a
rather radical move, since it applies not just to clouds, but to any
kind 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 undergoes
evaporation or erosion or abrasion. Or think of yourself, or any
organism, with parts that gradually come loose in metabolism or
excretion or perspiration or shedding of dead skin" (Lewis 1993:
165).



Despite Lewis's presentation, the Problem of the Many is not a
problem about change. The salient feature of these examples is that,
in practice, change is a slow process. Hence whenever a cathode, or a
human, is changing, be it by shedding electrons, or shedding skin,
there are some things that are not clearly part of the object, nor
clearly not part of it. Hence there are distinct sets that each have a
good 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 no
cathodes, or no humans, and that is probably enough to reject the
position, if any of the alternatives are not also equally
counterintuitive. It also, as Unger noted, creates difficulties for
many views about singular thought and talk. Intuitively, we can pick
out one of the objects composed of water droplets by the phrase
'that cloud'. But if it is not a cloud, then possibly we
cannot. For similar reasons, we may not be able to name any such
object, if we use any kind of reference-fixing description involving
'cloud' to pick it out from other objects composed of
water droplets. If the Problem of the Many applies to humans as well
as clouds, then by similar reasoning we cannot name or demonstrate any
human, 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 they
form a reductio of his theory. Bradley Rettler (2018) argues that the
nihilist has even more problems than this. Nihilism solves some
philosophical problems, such as explaining which of 0-7 is false. But,
he argues, for any problem it solves, there is a parallel problem
which it does not solve, but rival solutions do solve. For instance,
if you think of the problem here as a version of a
 Sorites paradox,
 nihilism does not help with versions of the paradox which concern
predicates applied to simples.



It is interesting that some other theories of vagueness have adopted
positions resembling Unger's in some respects, but without the
extreme conclusions. Matti Eklund (2002) and Roy Sorensen (2001) have
argued that all vague concepts involve inconsistent presuppositions.
Sorensen spells this out by saying that there are some inconsistent
propositions that anyone who possesses a vague concept should believe.
In the case of a vague predicate *F* that is vulnerable to a
Sorites paradox, the inconsistent propositions are that some things
are *F*s, some things are not *F*s, any object that
closely resembles (in a suitable respect) something that is *F*
is itself *F*, and that there are chains of 'suitably
resembling' objects between an *F* and a non-*F*.
Here the inconsistent propositions are that a story like Lewis's
is possible, and in it 0 through 7 are true. Neither Eklund nor
Sorensen conclude from this that nothing satisfies the predicates in
question; rather they conclude that some propositions that we find
compelling merely in virtue of possessing the concepts from which they
are constituted are false. So while they don't adopt
Unger's nihilist conclusions, two contemporary theorists agree
with him that vague concepts are in some sense incoherent.


## 3. Overpopulation



A simple solution to the puzzle is to reject premise 2. Each of the
relevant fusions of water droplets looks and acts like a cloud, so it
is a cloud. As with the first option this leads to some very
counterintuitive results. In any room with at least one person, there
are many millions of people. But this is not as bad as saying that
there are no people. And perhaps we don't even have
to *say* the striking claim. In many circumstances, we quantify
over a restricted domain. We can say, "There's no
beer," even when there is beer in some non-salient locales. With
respect to some restricted quantifier domains, it is true that there
is exactly one person in a particular room. The surprising result is
that with respect to other quantifier domains, there are many millions
of people in that room. The defender of the overpopulation theory will
hold that this shows how unusual it is to use unrestricted
quantifiers, not that there really is only one person in the room.



The overpopulation solution is not popular, but it is not without
defenders. J. Robert G. Williams (2006) endorses it, largely because
of a tension between the supervaluationist solution (that will be
discussed in section 7) and what supervaluationism says about the
Sorites paradox. James Openshaw (2021) and Alexander Sandgren
(forthcoming) argue that the overpopulation solution is true, and each
offer a theory of how singular thought about the cloud is possible
given overpopulation. Sandgren also points out that there might be
multiple sources of overpopulation. Even given a particular set of
water droplets, some metaphysical theories will say that there are
multiple objects those droplets compose, which differ in their
temporal or modal properties. 



Hudson (2001: 39-44) draws out a surprising consequence of the
overpopulation solution as applied to people. Assume that there are
really millions of people just where we'd normally say there was
exactly one. Call that person Charlie. When Charlie raises her arm,
each of the millions must also raise their arms, for the millions
differ only in whether or not they contain some borderline skin cells,
not in whether their arm is raised or lowered. Normally, if two people
are 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 looks
like at most one of the millions of people around Charlie are free.
There are a few possible responses here, though whether a defender of
the overpopulation view will view this consequence as being more
counter-intuitive than other claims to which she is already committed,
and hence whether it needs a special response, is not clear. There are
some other striking, though not always philosophically relevant,
features of this solution. To quote Hudson:



> 
> Among the most troublesome are worries about naming and singular
> reference ... how can any of us ever hope to successfully refer
> to himself without referring to his brothers as well? Or how might we
> have a little private time to tell just one of our sons of our
> affection for him without sharing the moment with uncountably many of
> his brothers? Or how might we follow through on our vow to practice
> monogamy? (Hudson 2001: 39)
> 


## 4. Brutalism



As Unger originally states it, the puzzle relies on a contentious
principle of mereology. In particular, it assumes mereological
Universalism, the view that for any objects, there is an object that
has all of them as its fusion. (That is, it has each of those objects
as parts, and has no parts that do not overlap at least one of the
original objects.) Without this assumption, the Problem of the Many
may have an easy solution. The cloud in the sky is *the* object
up there that is a fusion of water droplets. There are many other sets
of water droplets, other than the set of water droplets that compose
the cloud, but since the members of those sets do not compose an
object, they do not compose a cloud.



There are two kinds of theories that imply that only one of the sets
of water droplets is such that there exists a fusion of its atoms.
First, there are *principled* restrictions on composition,
theories that say that the *x*s compose an object *y*
iff the *x*s are *F*, for some natural property
*F*. Secondly, there are *brutal* theories, which say
it's just a brute fact that in some cases the *x*s
compose an object, and in others they do not. It would be quite hard
to imagine a principled theory solving the Problem of the Many, since
it is hard to see what the principle could be. (For a more detailed
argument for this, set against a somewhat different backdrop, see
McKinnon 2002.) But a brutal theory could work. And such a theory has
been defended. Ned Markosian (1998) argues that not only does
brutalism, the doctrine that there are brute facts about when the
*x*s compose a *y*, solve the Problem of the Many, the
account of composition it implies fits more naturally with our
intuitions about composition.



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



> 
> In particular, a good metaphysical theory or scientific theory should
> avoid positing a plethora of quite specific, disconnected, *sui
> generis*, compositional facts. Such facts would be ontological
> danglers; they would be metaphysically queer. Even though explanation
> presumably must bottom out somewhere, it is just not credible--or
> even intelligible--that it should bottom out with specific
> compositional facts which themselves are utterly unexplainable and
> which do not conform to any systematic general principles. (Horgan
> 1993: 694-5)
> 



On the other hand, this kind of view does provide a particularly
straightforward solution to the Problem of the Many. As Markosian
notes, if we have independent reason to view favourably the idea that
facts about when some things compose an object are brute facts, which
he thinks is provided by our intuitions about cases of composition and
non-composition, the very simplicity of this solution to the Problem
of 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 water
droplets, there is an object those water droplets compose. Since that
object looks for all the world like a cloud, we will say it is a
cloud. The fourth solution accepts those claims, but denies that there
are many clouds. It is true that there are many fusions of atoms, but
these are all the same *cloud*. This view adopts a position
most commonly associated with P. T. Geach (1980), that two things can
be the same *F* but not the same *G*, even though they
are both *G*s. To see the motivation for that position, and a
discussion of its strengths and weaknesses, see the article on
 relative identity.



Here is one objection that many have felt is telling against the
relative identity view: Let *w* be a water droplet that is
in *s*1 but not *s*2. The relative
identity solution says that the droplets in
*s*1 compose an object *o*1, and
the droplets in *s*2 compose an object
*o*2, and though *o*1 and
*o*2 are different fusions of water droplets, they
are the same cloud. Call this cloud *c*. If
*o*1 is the same cloud as *o*2,
then presumably they have the same properties. But
*o*1 has the property of having *w* as a
part, while *o*2 does not. Defenders of the relative
identity theory here deny the principle that if two objects are the
same *F*, they have the same properties. Many theorists find
this denial to amount to a *reductio* of the view.


## 6. Partial Identity



Even if *o*1 and *o*2 exist, and
are clouds, and are not the same cloud, it does not immediately follow
that there are two clouds. If we analyze "There are two
clouds" as "There is an *x* and a *y* such
that *x* is a cloud and *y* is a cloud, and *x*
is not the same cloud as *y*" then the conclusion will
naturally follow. But perhaps that is not the correct analysis of
"There are two clouds." Or, more cautiously, perhaps it is
not the correct analysis in all contexts. Following some suggestions
of D. M. Armstrong's (Armstrong 1978, vol. 2: 37-8), David
Lewis suggests a solution along these lines. The objects
*o*1 and *o*2 are not the same
cloud, but they are *almost* the same cloud. And in everyday
circumstances (AT) is a good-enough analysis of "There is one
cloud"



(AT)
There is a cloud, and all clouds are *almost* identical
with it.


As Lewis puts it, we 'count by almost-identity' rather
than by identity in everyday contexts. And when we do, we get the
correct result that there is one cloud in the sky. Lewis notes that
there are other contexts in which we count by some criteria other than
identity.



> 
> If an infirm man wishes to know how many roads he must cross to reach
> his destination, I will count by identity-along-his-path rather than
> by identity. By crossing the Chester A. Arthur Parkway and Route 137
> at the brief stretch where they have merged, he can cross both by
> crossing 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 by
identity, 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 that
the only reason this seems intuitive is that it is normally intuitive
to say that there is only one cloud in the sky. And that intuition is
respected! More contentiously, it may be argued that it is a
*good* thing to predict that when we count by identity we get
the result that there are millions of clouds. After all, the only time
we'd do this is when we're doing metaphysics, and we have
noted that in the metaphysics classroom, there is some intuitive force
to the argument that there are millions of clouds in the sky. It would
be a brave philosopher to endorse this as a *virtue* of the
theory, but it may offset some of the costs.



Secondly, something like the Problem of the Many can arise even when
the possible objects are not almost identical. Lewis notes this
objection, and provides an illustrative example to back it up. A
similar kind of example can be found in W. V. O. Quine's
*Word and Object* (1960). Lewis's example is of a house
with an attached garage. It is unclear whether the garage is part of
the house or an external attachment to it. So it is unclear whether
the phrase 'Fred's house' denotes the basic house,
call it the home, or the fusion of the home and the garage. What is
clear is that there is exactly one house here. However, the home might
be quite different from the fusion of the home and the garage. It will
probably be smaller and warmer, for example. So the home and the
home-garage fusion are not even almost identical. Quine's
example is of something that looks, at first, to be a mountain with
two peaks. On closer inspection we find that the peaks are not quite
as connected as first appeared, and perhaps they could be properly
construed as two separate mountains. What we could not say is that
there are *three* mountains here, the two peaks and their
fusion, but since neither peak is almost identical to the other, or to
the 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 Lopez de
Sa (2014) argues is central to a Lewisian solution to the problem.



> 
> You draw two diagonals in a square; you ask me how many triangles; I
> say there are four; you deride me for ignoring the four large
> triangles 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 you
> couldn't see it because you insisted on counting by strict
> identity. I meant that, for some *w*, *x*, *y*,
> *z*, (1) *w*, *x*, *y*, and *z* are
> triangles; (2) *w* and *x* are distinct, and ... and so
> are *y* and *z* (six clauses); (3) for any triangle
> *t*, either *t* and *w* are not distinct, or ...
> or *t* and *z* 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 counting
sentences in ordinary language. So *There is exactly one F*
gets interpreted as *There is an F, and no F is wholly distinct
from it*; and *There are exactly two Fs* gets interpreted
as *There are wholly distinct things x and y that are both F, and
no F is wholly distinct from both of them*, and so on. Lewis
writes as if this is an explication of the almost-identity proposal,
but this is at best misleading. A house with solar panels partially
overlaps the city's electrical grid, but it would be very
strange to call them almost-identical. It sounds like a similar, but
distinct, proposed solution.



However we understand the proposal, Lopez de Sa notes that it
has a number of virtues. It seems to account for the puzzle involving
the house, what he calls the Problem of the Two. If in general
counting involves distinctness, then we have a good sense in which
there is one cloud in the sky, and Fred owns one house.



There still remain two challenges for this view. First, one could
still follow Hudson and argue that even if we ordinarily understand
counting sentences this way, we do still know how to count by
identity. And when we do, it seems that there is just one cloud, not
millions of them. Second, it isn't that clear that we always
count by distinctness, in the way Lopez de Sa suggests. If I
say there are three ways to get from my house to my office, I
don't mean to say that these three are completely distinct.
Indeed, they probably all start with going out my door, down my
driveway etc., and end by walking up the stairs into my office. So the
general claim about how to understand counting sentences seems
false.



C. S. Sutton (2015) argued that we can get around something like the
second of these problems if we do two things. First, the rule
isn't that we don't normally quantify over things that
overlap, just that we don't normally quantify over things that
substantially overlap. We can look at a row of townhouses and say that
there are seven houses there even if the walls of the house overlap.
Second, the notion of overlap here is not sensitive to the quantity of
material that the objects have in common, but to their functional
role. If two objects play very different functional roles, she argues
that we will naturally count them as two, even if they have a lot of
material in common. This could account for a version of the getting to
work example where the three different ways of getting to work only
differ 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 the
way to get from A to D is to go A-B-C-D, then there is a good sense in
which there are three ways to get from A to D. Sutton's theory
explains how this could be true even if the B-C leg is a short part of
the trip.



David Liebesman (2020) argued for a different way of implementing this
kind of theory. He argues that the kinds of constraints that Lewis,
Lopez de Sa and Sutton have suggested don't get
incorporated into our theory of counting, but into the proper
interpretation of the nouns involved in counting sentences. It helps
to understand Liebesman's idea with an example.



There is, we'll presumably all agree, just one colour in Yves
Klein's
 *Blue Monochrome*.
 (It says so in the title.) But *Blue Monochrome* has blue in
it, and it has ultramarine in it, and blue doesn't equal
ultramarine. What's gone on? Well, says Liebesman, whenever a
noun occurs under a determiner, it needs an interpretation. That
interpretation will almost never be maximal. When we ask how many
animals a person has in their house, we typically don't mean to
count the insects. When we say every book is on the shelf, we
don't mean every book in the universe. And typically, the
relevant interpretation of the determiner phrase (like 'every
book', or 'one cloud') will exclude overlapping
objects. Typically but not, says Liebesman, always. We can say, for
instance, that every shade of a colour is a colour, and in that
sentence 'colour' includes both blue and ultramarine.



This offers a new solution to the Problem of the Many. He argues that
all of 0 to 7 are true, but they are true for different
interpretations 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' in *Every 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 is
one that we frequently use.


## 7. Vagueness



Step 6 in the initial setup of the problem says that if any of the
*oi* is a cloud, then they all are. There are three
important arguments for this premise, two of them presented explicitly
by Unger, and the other by Geach. Two of the arguments seem to be
faulty, 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 principles
are clearly stated in his version of the argument. (The argument shows
that if an *ok* is a cloud for arbitrary *k*,
we can easily generalize to the claim that for every *i*,
*oi* is a cloud.)




D1.
If all the water droplets not in *sk* did not
exist, then *ok* would be a cloud.
D2.
Whether *ok* is a cloud does not depend on
whether things distinct from it exist.
C.
*ok* is a cloud.



D2 implies that *being a cloud* is an
 intrinsic
 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 a
rather implausible conclusion, that there are millions of clouds where
we think there is one, there is some reason to believe it is false. We
can argue directly for the same conclusion. Assume many more water
droplets coalesce around our original cloud. There is still one cloud
in the sky, but it determinately includes more water droplets than the
original cloud. The fusion of those water droplets exists, and we may
assume that they did not change their intrinsic properties, but they
are now a *part* of a cloud, rather than a cloud. Even if
something looks like a cloud, smells like a cloud and rains like a
cloud, 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.




S1.
For some *j*, *oj* is a typical
cloud.
S2.
Anything that differs minutely from a typical cloud is a
cloud.
S3.
*ok* differs minutely from
*oj*.
C.
*ok* is a cloud.



Since we only care about the conditional *if oj*
*is a cloud, so is ok*, it is clearly acceptable to
assume that *oj* is a cloud for the sake of the
argument. 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, there
appear to be some clear counterexamples to it. The fusion of a cloud
with 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 is
only 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 the
most persuasive force. Unger says that if exactly one of the
*oi* is a cloud, then there must be a
'selection principle' that picks it over the others. But
it is not clear just what kind of selection principle that could be.
The underlying argument seems to be something like this:




M1.
For some *j*, *oj* is a cloud.
M2.
If *oj* is a cloud and the rest of the
*oi* are not, then some principle selects
*oj* to be the unique cloud.
M3.
There is no principle that selects *oj* to be
the unique cloud.
C.
At least one of the other *oi* is a cloud.



The idea behind M2 is that word meanings are not brute facts about
reality. As Jerry Fodor put it, "if aboutness is real, it must
be really something else" (Fodor 1987: 97). Something makes it
the case that *oj* is the unique thing (around here)
that satisfies our term 'heap'. Maybe that could be
because *oj* has some unique properties that make it
suitable to be in the denotation of ordinary terms. Or maybe it is
something about our linguistic practices. Or maybe it is some
combination of these things. But something must determine it, and
whatever it is, we can (in theory) say what that is, by giving some
kind of principled explanation of why *oj* is the
unique cloud.



It is at this point that theories of
 vagueness
 can play a role in the debate. Two of the leading theories of
vagueness, epistemicism and supervaluationism, provide principled
reasons to reject this argument. The epistemicist says that there are
semantic facts that are beyond our possible knowledge. Arguably we can
only know where a semantic boundary lies if that boundary was fixed by
our use or by the fact that one particular property is a natural kind.
But, say the epistemicists, there are many other boundaries that are
not like this, such as the boundary between the heaps and the
non-heaps. Here we have a similar kind of situation. It is vague just
which of the *oi* is a cloud. What that means is
that there is a fact about which of them is a cloud, but we cannot
possibly know it. The epistemicist is naturally read as rejecting the
very last step in the previous paragraph. Even if something (probably
our linguistic practices) makes it the case that
*oj* is the unique cloud, that need not be something
we can know and state.



The supervaluationist response is worth spending more time on here,
both because it engages directly with the intuitions behind this
argument and because two of its leading proponents (Vann McGee and
Brian McLaughlin, in their 2001) have responded directly to this
argument using the supervaluationist framework. Roughly (and for more
detail see the section on supervaluations in the entry on
 vagueness)
 supervaluationists say that whenever some terms are vague, there are
ways of making them more precise consistent with our intuitions on how
the terms behave. So, to use a classic case, 'heap' is
vague, which to the supervaluationist means that there are some piles
of sand that are neither determinately heaps nor determinately
non-heaps, and a sentence saying that that object is a heap is neither
determinately true nor determinately false. However, there are many
ways to *extend* the meaning of 'heap' so it
becomes precise. Each of these ways of making it precise is called a
*precisification*. A precisification is *admissible* iff
every sentence that is determinately true (false) in English is true
(false) in the precisification. So if *a* is determinately a
heap, *b* is determinately not a heap and *c* is neither
determinately a heap nor determinately not a heap, then every
precisification 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 a
first approximation, to be admissible a precisification must assign
all the determinate heaps to the extension of 'heap' and
assign none of the determinate non-heaps to its extension, but it is
free to assign or not assign things in the 'penumbra'
between these groups to the extension of 'heap'. But this
is not quite right. If *d* is a little larger than *c*,
but still not determinately a heap, then the sentence "If
*c* is a heap so is *d*" is intuitively true. As
it is often put, following Kit Fine (1975), a precisification must
respect 'penumbral connections' between the borderline
cases. If *d* has a better case for being a heap than
*c*, then a precisification cannot make *c* a heap but
not *d*. These penumbral connections play a crucial role in the
supervaluationist solution to the Problem of the Many. Finally, a
sentence is determinately true iff it is true on all admissible
precisifications, determinately false iff it is false on all
admissible precisifications.



In the original example, described by Lewis, the sentence "There
is one cloud in the sky" is determinately true. None of the
sentences "*o*1 is a cloud",
"*o*2 is a cloud" and so on are
determinately true. So a precisification can make each of these either
true or false. But, if it is to preserve the fact that "There is
one cloud in the sky" is determinately true, it must make
exactly *one* of those sentences true. McGee and McLaughlin
suggest that this combination of constraints lets us preserve what is
plausible about M2, without accepting that it is true. The term
'cloud' is vague; there is no fact of the matter as to
whether its extension includes *o*1 or
*o*2 or *o*3 or .... If there
were such a fact, there would have to be something that made it the
case that it included *oj* and not
*ok*, and as M3 correctly points out, no such facts
exist. But this is consistent with saying that its extension does
contain exactly one of the *oi*. The beauty of the
supervaluationist solution is that it lets us hold these seemingly
contradictory positions simultaneously. We also get to capture some of
the plausibility of S2--it is consistent with the
supervaluationist position to say that anything similar to a cloud is
not determinately not a cloud.



Penumbral connections also let us explain some other puzzling
situations. Imagine I point cloudwards and say, "That is a
cloud." 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 no
determinate answer as to which of the *oi* it picks
out.) On different precisifications, 'that' picks out
different *oi*. But on every precisification it
picks out the *oi* that is in the extension of
'cloud', so "That is a cloud" comes out true
as desired. Similarly, if I named the cloud 'Edgar', then
a similar trick lets it be true that "Edgar" is vague,
while "Edgar is a cloud" is determinately true. So the
supervaluationist solution lets us preserve many of the intuitions
about the original case, including the intuitions that seemed to
underwrite M2, without conceding that there are millions of clouds.
But there are a few objections to this package.


* *Objection*: There are many telling objections to
supervaluationist theories of vagueness.
	+ *Reply*: This may be true, but it would take us well beyond the
	scope of this entry to outline them all. See the entries on
	 vagueness
	 (the section on supervaluation) and the
	 Sorites Paradox
	 for more detail.
* *Objection*: The supervaluationist solution makes some
existentially quantified sentences, like "There is a cloud in
the sky" determinately true even though no instance of them is
determinately true.
	+ *Reply*: As Lewis says, this is odd, but no odder than things
	that 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 owe
	you."
* *Objection*: The penumbral connections here are not
precisely specified. What is the rule that says how much overlap is
required before two objects cannot both be clouds?
	+ *Reply*: It is true that the connections are not precisely
	specified. It would be quite hard to carefully analyze
	'cloud' to work out exactly what they are. But that we
	cannot say exactly what the rules are is no reason for saying no such
	rules exist, any more than our inability to say exactly what knowledge
	is provides a reason for saying that no one ever knows anything.
	Scepticism can't be proven *that* easily.
* *Objection*: The penumbral connections appealed to here are
left unexplained. At a crucial stage in the explanation, it seems to
be just assumed that the problem can be solved, and that it is a
determinate truth that there is one cloud in the sky.
	+ *Reply 1*: We have to start somewhere in philosophy. This kind
	of reply can be spelled out in two ways. There is a
	'Moorean' move that says that the premise that there is
	one cloud in the sky is more plausible than the premises that would
	have to be used in an argument against supervaluationism.
	Alternatively, it might be claimed that the main argument for
	supervaluationism is an inference to the best explanation. In that
	case, the intuition that there is exactly one cloud in the sky, but it
	is indeterminate just which object it is, is something to be
	explained, not something that has to be *proven*. This is the
	kind of position defended by Rosanna Keefe in her book *Theories of
	Vagueness*. Although Keefe does not apply this *directly*
	to the Problem of the Many, the way to apply her position to the
	Problem seems clear enough.
	+ *Reply 2*: The penumbral connections we find for most words are
	generated by the inferential role provided by the meaning of the
	terms. It is because the inference from "This pile of sand is a
	heap", and "That pile of sand is slightly larger than this
	one, and arranged roughly the same way," to "That pile of
	sand is a heap" is generally acceptable that precisifications
	which make the premises true and the conclusions false are
	inadmissible. (We have to restrict this inferential rule to the case
	where 'this' and 'that' are ordinary
	demonstratives, and not used to pick out arbitrary fusions of grains,
	or else we get bizarre results for reasons that should be familiar by
	this point in the story.) And it isn't too hard to specify the
	inferential rule here. The inference from
	"*oj* is a cloud" and
	"*oj* and *ok* massively
	overlap" to "*ok* is a cloud" is
	just as acceptable as the above inference involving heaps. Indeed, it
	is part of the meaning of 'cloud' that this inference is
	acceptable. (Much of this reply is drawn from the discussion of
	'maximal' predicates in Sider 2001 and 2003, though since
	Sider is no supervaluationist, he would not entirely endorse this way
	of putting things.)
* *Objection*: The second reply cannot explain the existence
of penumbral connections between 'cloud' and
demonstratives like 'that' and names like
'Edgar'. It would only explain the existence of those
penumbral connections if it was part of the meaning of names and
demonstratives that they fill some kind of inferential role. But this
is inconsistent with the widely held view that demonstratives and
names are
 directly referential.
 (For more details on debates about the meanings of names, see the
entries on
 propositional attitude reports
 and
 singular propositions.)
	+ *Reply*: One response to this would be to deny the view that
	names and demonstratives are directly referential. Another would be to
	deny that inferential roles provide the *only* penumbral
	constraints on precisifications. Weatherson 2003b sketches a theory
	that does exactly this. The theory draws on David Lewis's
	response to some quite different work on semantic indeterminacy. As
	many authors (Quine 1960, Putnam 1981, Kripke 1982) showed, the
	dispositions of speakers to use terms are not fine-grained enough to
	make the language as precise as we ordinarily think it is. As far as
	our usage dispositions go, 'rabbit' could mean undetached
	rabbit part, 'vat' could mean vat image, and
	'plus' could mean quus. (Quus is a function defined over
	pairs of numbers that yields the sum of the two numbers when they are
	both small, and 5 when one is sufficiently large.) But intuitively our
	language is not that indeterminate: 'plus' determinately
	does not mean quus.
	
	
	
	Lewis (1983, 1984) suggested the way out here is to posit a notion of
	'naturalness'. Sometimes a term *t* denotes the
	concept *C*1 rather than *C*2 not
	because we are disposed to use *t* as if it meant
	*C*1 rather than *C*2, but simply
	because *C*1 is a more natural concept.
	'Plus' means plus rather than quus simply because plus is
	more natural than quus. Something like the same story applies to names
	and demonstratives. Imagine I point in the direction of Tibbles the
	cat and say, "That is Edgar's favourite cat." There
	is a way of systematically (mis)interpreting all my utterances so
	'that' denotes the Tibbles-shaped region of space-time
	exactly one metre behind Tibbles. (We have to reinterpret what
	'cat' means to make this work, but the discussions of
	semantic indeterminacy in Quine and Kripke make it clear how to do
	this.) So there's nothing in my usage dispositions that makes
	'that' mean Tibbles, rather than the region of space-time
	that 'follows' him around. But because Tibbles is more
	natural than that region of space-time, 'that' does pick
	out Tibbles. It is the very same naturalness that makes
	'cat' denote a property that Tibbles (and not the trailing
	region of space-time) satisfies that makes 'that' denote
	Tibbles, a fact that will become important below.
	
	
	
	The same kind of story can be applied to the cloud. It is because the
	cloud is a more natural object than the region of space-time a mile
	above the cloud that our demonstrative 'that' denotes the
	cloud and not the region. However, none of the *oi*
	are more natural than any other, so there is still no fact of the
	matter as to whether 'that' picks out
	*oj* or *ok*. Lewis's theory
	does not eliminate all semantic indeterminacy; when there are equally
	natural candidates to be the denotation of a term, and each of them is
	consistent with our dispositions to use the term, then the denotation
	of the term is simply indeterminate between those candidates.
	
	
	
	Weatherson's theory is that the role of each precisification is
	to arbitrarily make one of the *oi* more natural
	than the rest. Typically, it is thought that the denotation of a term
	according to a precisification is determined directly. It is a fact
	about a precisification *P* that, according to it,
	'cloud' denotes property *c*1. On
	Weatherson's theory this is not the case. What the
	precisification does is provide a new, and somewhat arbitrary,
	standard of naturalness, and the content of the terms according to the
	precisification is then determined by Lewis's theory of content.
	The denotations of 'cloud' and 'that'
	according to a precisification *P* are those concepts and
	objects that are the most *natural-according-to-P* of the
	concepts and objects that we could be denoting by those terms, for all
	one can tell from the way the terms are used. The coordination between
	the 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): The
supervaluationist account cannot handle speech reports involving vague
names. Imagine that Alex points cloudwards and says, "That is a
cloud." Later Sam points towards the same cloud and says,
"Alex said that that is a cloud." Intuitively, Sam's
utterance is determinately true. But according to the
supervaluationist, it is only determinately true if it is true on
every precisification. So it must be true that Alex said that
*o*1 is a cloud, that Alex said that
*o*2 is a cloud, and so on, since these are all
precisifications of "Alex said that that is a cloud." But
Alex did not say all of those things, for if she did she would be
committed to saying that there are millions of clouds in the sky, and
of course she is not, as the supervaluationists have been arguing.
	+ *Reply*: There is a little logical slip here. Let
	*Pi* be a precisification of Sam's word
	'that' that makes it denote *oi*. All
	the supervaluationist who holds that Sam's utterance is
	determinately true is committed to is that for each *i*,
	according to *Pi*, Alex said that
	*oi* is a cloud. And this will be true if the
	denotation of Alex's word 'that' is also
	*oi* according to *Pi*. So as long
	as there is a penumbral connection between Sam's word
	'that' and Alex's word 'that', the
	supervaluationist avoids the objection. Such a connection may seem
	mysterious at first, but note that Weatherson's theory predicts
	that just such a penumbral connection obtains. So if that theory is
	acceptable, then Schiffer's objection misfires.
* *Objection* (From Neil Mackinnon 2002 and Thomas Sattig
2013): It is part of our notion of a mountain that facts about
mountain-hood are not basic. If something is a mountain, there are
facts in virtue of which it is a mountain, and nearby things are not.
Yet on each precisification of 'mountain', this
won't be true; it will be completely arbitrary which fusion of
rocks is a mountain.
	+ *Reply*: It is true that on any precisification there will be
	no principled reason why this fusion of rocks is a mountain, and
	another is not. And it is true that there should be such a principled
	reason; mountainhood facts are not basic. But that problem can be
	avoided by the theory that denies that "the supervaluationist
	rule [applies] to any statement whatever, never mind that the
	statement makes no sense that way" (Lewis 1993, 173).
	Lewis's idea, or at least the application of Lewis's idea
	to this puzzle, is that we know how to understand the idea that
	mountainhood facts are non-arbitrary: we understand it as a claim that
	there is some non-arbitrary explanation of which precisifications of
	'mountain' are and are not admissible. If we must apply
	the supervaluationist rule to every statement, including the statement
	that it is not arbitrary which things are mountains, this
	understanding is ruled out. Lewis's response is to deny that the
	rule must always be applied. As long as there is some sensible way to
	understand the claim, we don't have to insist on applying the
	supervaluationist machinery to it.
	
	
	
	That said, it does seem like this is likely to be somewhat of a
	problem for everyone (even the theorist like Lewis who uses the
	supervaluationist machinery only when it is helpful). Sattig himself
	claims to avoid the problem by making the mountain be a maximal fusion
	of candidates. But for any plausible mountain, it will be vague and
	somewhat arbitrary what the boundary is between being a
	mountain-candidate and not being one. The lower boundaries of
	mountains are not, in practice, clearly marked. Similarly, there will
	be some arbitrariness in the boundaries between the admissible and
	inadmissible precisifications of 'mountain'. We may have
	to live with some arbitrariness.
* *Objection* (From J. Robert G. Williams 2006): When we look
at an ordinary mountain, there definitely is (at least) one mountain
in front of us. That seems clear. The vagueness solution respects this
fact. But in many contexts, people tend to systematically confuse
"Definitely, there is an *F*" with "There is
a definite *F*." Indeed, the standard explanation of why
Sorites arguments seem, mistakenly, to be attractive is that this
confusion gets made. Yet on the vagueness account, there is no
definite mountain; all the candidates are borderline cases. So by
parity of reasoning, we should expect intuition to deny that there is
definitely a mountain. And intuition does not deny that, it loudly
confirms it. At the very least, this shows a tension between the
standard account of the Sorites paradox, and the vagueness solution to
the Problem of the Many.
	+ *Reply*: This is definitely a problem for the views that many
	philosophers have put forward. As Williams stresses, it isn't on
	its own a problem for the vagueness solution to the Problem of the
	Many, but it is a problem for the conjunction of that solution with a
	widely endorsed, and independently plausible, explanation of the
	Sorites paradox. In his dissertation, Nicholas K. Jones (2010) argues
	that the right response is to give up the idea that speakers typically
	confuse "Definitely, there is an *F*" with
	"There is a definite *F*", and instead use a
	different resolution of the Sorites.



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



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


## 8. Rethinking Parthood



Some theorists have argued that the underlying cause of the problem is
that we have the wrong theory about the relation between parts and
wholes. Peter van Inwagen (1990) argues that the problem is that we
have assumed that the parthood relation is determinate. We have
assumed that it is always determinately true or determinately false
that one object is a part of another. According to van Inwagen,
sometimes neither of these options applies. He thinks that we need to
adopt some kind of
 fuzzy logic
 when we are discussing parts and wholes. It can be true to degree
0.7, for example, that one object is part of another. Given these
resources, van Inwagen says, we are free to conclude that there is
exactly 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 these
outlying water droplets are members of the cloud without accepting
that there are millions of clouds. Note that this is not what van
Inwagen would say about *this* version of the paradox, since he
holds that some simples only constitute an object when that object is
alive. For van Inwagen, as for Unger, there are no clouds, only
cloud-like swarms of atoms. But van Inwagen recognises that a similar
problem arises for cats, or for people, two kinds of things that he
does believe exist, and he wields this vague constitution theory to
solve the problems that arise there.



Traditionally, many philosophers thought that such a solution was
downright incoherent. A tradition stretching back to Bertrand Russell
(1923) and Michael Dummett (1975) held that vagueness was always and
everywhere a representational phenomenon. From this perspective, it
didn't make sense to talk about it being vague or indeterminate
whether a particular droplet was part of a particular cloud. But this
traditional view has come under a lot of pressure in recent years; see
Barnes (2010) for one of the best challenges, and Sorensen (2013,
section 8) for a survey of more work. So let us assume here it is
legitimate to talk about the possibility that parthood itself, and not
just our representation of it, is vague. As Hudson (2001) notes
though, it is far from clear just how the appeal to fuzzy logic is
meant to help *here*. Originally it was clear for each of
*n* water droplets whether they were members of the cloud to
degree 1 or degree 0. So there were 2*n* candidate
clouds, and the Problem of the Many is finding out how to preserve the
intuition when faced with all these objects. It is unclear how
*increasing* the range of possible relationships between each
particle and the cloud from 2 to continuum-many should help here, for
now it seems there are at least continuum-many cloud-like objects to
choose between, one for each function from each of the *n*
droplets to [0, 1], and we need a way of saying exactly one of them is
a 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 like
the cloud, except this droplet is only part of it to degree 0.6. Does
that object exist, and is it a cloud? Van Inwagen says, in a way
reminiscent of Markosian's brutal composition solution, that
such 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 the
Problem is to distinguish cloud-constituters from clouds. They say it
is a category mistake to identify clouds with any fusion of water
droplets, because they have different identity conditions. The cloud
could survive the transformation of half its droplets into puddles on
the footpath (or whatever kind of land it happens to be raining over),
it would just be a smaller cloud, the fusion could not. As Johnston
says, "Hence Unger's insistent and ironic question
'But which of *o*1, *o*2,
*o*3, ... is our paradigm cloud
*c*?' 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 the
problem using cats not clouds, and we will sometimes follow them
below.)



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



Secondly, Lewis says that even with this move, we still have a Problem
of the Many applied to cloud-constituters, rather than to clouds. Lowe
responds that since 'cloud-constituter' is not a folk
concept, we don't really have any philosophically salient
intuitions here, so this cannot be a way in which the position is
unintuitive.



Finally, Lewis says that each of the constituters is so like the
object it is meant to merely constitute (be it a cloud, or a cat, or
whatever), it satisfies the same sortals as that object. So if we were
originally worried that there were 1001 cats (or clouds) where we
thought there was one, now we should be worried that there are 1002.
But as Lowe points out, this argument seems to assume that *being a
cat*, or *being a cloud*, is an intrinsic property. If we
assume 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 is
embedded in, then the fact that a cloud-constituter looks, when
considered in isolation, to be a cloud is little reason to think it
actually is a cloud.



Johnston provides an argument that the distinction between clouds and
cloud-constituting fusions of water droplets is crucial to solving the
Problem. He thinks that the following principle is sound, and not
threatened by examples like our cloud.



> 
> (9') If *y* is a paradigm *F*, and *x* is
> an entity that differs from *y* in any respect relevant to
> being an *F* only very minutely, *and x is of the right
> category, i.e. is not a mere quantity or piece of matter*, then
> *x* is an *F*. (Johnston 1992: 100)
> 



The theorist who thinks that clouds are just fusions of water droplets
cannot accept this principle, or they will conclude that every
*oi* is a cloud, since for them each
*oi* is of the right category. On the other hand,
Johnston himself cannot accept it either, unless he denies there can
be another object *c'* which is in a similar position to
*c*, and is of the same category as *c*, but differs
with respect to which water droplets constitute it. It seems that what
is doing the work in Johnston's solution is not just the
distinction between constitution and identity, but a tacit restriction
on when there is a 'higher-level' object constituted by
certain 'lower-level' objects. To that extent, his theory
also resembles Markosian's brutal composition theory, though
since Johnston can accept that every set of atoms has a fusion his
theory has different costs and benefits to Markosian's
theory.



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 be
metaphysically prior to composition.) Jones rejects the following
principle, which is similar to 4 in the original inconsistent set.


* If the water droplets in *si* constitute
*oi*, and the objects in *sj*
constitute *oj*, and the sets *si*
and *sk* are not identical, then the objects
*oi* and *oj* are not
identical.



He rejects this claim. He argues that some water droplets can
constitute a cloud, and some other water droplets can constitute the
very same cloud. On this view, the predicate *constitute x*
behaves a bit like the predicate *surround the building*. It
can be true that the Fs surround the building, and the Gs surround the
building, without the Fs being the Gs. And on Jones's view, it
can be true that the Fs constitute *x*, and the Gs constitute
*x*, without the Fs being the Gs. This resembles Lewis's
solution in terms of almost-identity, since both Jones and Lewis say
that there is one cloud, yet both *si* and
*sk* can be said to compose it. But for Lewis, this
is possible because he rejects the inference from *There is one
cloud*, to *If a and b are clouds, they are identical*.
Jones accepts this inference, and rejects the inference from the
premise that *si* and *sk* are
distinct, and each compose a cloud, to the conclusion that they
compose non-identical clouds. 


## 9. Rethinking Parthood



After concluding that all of these kinds of solutions face serious
difficulties, Hudson (2001: Chapter 2) outlines a new solution, one
which rejects so many of the presuppositions of the puzzle that it is
best to count him as rejecting the reasoning, rather than rejecting
any particular premise. (Hudson is somewhat tentative about
*endorsing* this view, as opposed to merely endorsing the claim
that it looks better than its many rivals, but for expository purposes
let us refer to it here as his view.) To see the motivation behind
Hudson's approach, consider a slightly different case, a variant
of one discussed in Wiggins 1968. Tibbles is born at midnight Sunday,
replete with a splendid tail, called Tail. An unfortunate accident
involving 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 part
of Tibbles?" Intuitively, we want to say the question is
underspecified. Outside of Monday, the question does not arise, for
Tibbles 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 of
Tibbles. There is a proposition that Tail is part of Tibbles on Monday
morning (that's true) and that Tail is part of Tibbles on Monday
afternoon (that's false), but no proposition involving just the
parthood relation and two objects. Parthood is a three-place relation
between two objects and a time, not a two-place relation between two
objects.



Hudson suggests that this line of reasoning is potentially on the
right track, but that the conclusion is not quite right. Parthood is a
three-place relation, but the third place is not filled by a time, but
by a region of space-time. To a crude approximation, *x* is
part of *y* at *s* is true if (as we'd normally
say) *x* is a part of *y* and *s* is a region of
space-time containing no region not occupied by *y* and all
regions occupied by *x*. But this should be taken as a
heuristic guide only, not as a reductive definition, since parthood is
really a three-place relation, so the crude approximation does not
even express a proposition according to Hudson.



To see how this applies to the Problem of the Many, let's
simplify the case a little bit so there are only two water droplets,
*w*1 and *w*2, that are neither
determinately part of the cloud nor determinately not a part of it. As
well there is the core of the cloud, call it *a*. On an
orthodox theory, there are four proto-clouds here, *a*,
*a* + *w*1, *a* +
*w*2 and *a* + *w*1 +
*w* 2. On Hudson's theory the largest and the
smallest proto-clouds still exist, but in the middle there is a quite
different kind of object, which we'll call *c*. Let
*r*1 be the region occupied by *a* and
*w*1, and *r*2 the region occupied
by *a* and *w*2. Then the following claims
are all true according to Hudson:


* *c* exactly occupies *r*1;
* *c* exactly occupies *r*2;
* *c* does not occupy the region consisting of the union of
*r*1 and *r*2;
* *c* has *w*1 as a part at
*r*1, but not at *r*2;
* *c* has *w*2 as a part at
*r*2, but not at *r*1;
* *c* has no parts at the region consisting of the union of
*r*1 and *r*2.



Hudson defines "*x* exactly occupies *s*" as
follows:


* *x* has a part at *s*,
* there is no region of space-time, *s*\*, such that
*s*\* has *s* as a subregion, while *x* has a part
at *s*\*, and
* for every subregion of *s*, *s'*, *x*
has a part at *s'*. (Hudson 2001: 63)



At first, it might look like not much has been accomplished here. All
that we did was turn a Problem of 4 clouds into a Problem of 3 clouds,
replacing the fusions *a* + *w*1 and
*a* + *w*2 with the new, and oddly behaved,
*c*. But that is to overlook a rather important feature of the
remaining proto-clouds. The three remaining proto-clouds can be
strictly ordered by the 'part of' relation. This was not
previously possible, since neither *a* + *w*1
nor *a* + *w*2 were part of the other. If we
adopt the principle that 'cloud' is a maximal predicate,
so no cloud can be a proper part of another cloud, we now get the
conclusion that exactly one of the proto-clouds is a cloud, as
desired.



This is a quite ingenious approach, and it deserves some attention in
the future literature. It is hard to say what will emerge as the main
costs and benefits of the view in advance of that literature, but the
following two points seem worthy of attention. First, if we are
allowed to appeal to the principle that no cloud is a proper part of
another, why not appeal to the principle that no two clouds massively
overlap, 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 old
*a* + *w*1, that is, an object that has
*w*1 as a part at *r*1, and does
not have *w*2 (or anything else) as a part at
*r*2? If we get it back, as well as *a* +
*w*2, then all of Hudson's tinkering with
mereology will just have converted a problem of 4 clouds into a
problem 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 the
many and varied proposed solutions to the Problem of the Many. For
such a young problem, the variety of these solutions is rather
impressive. Whether the next few years will see these ranks whittled
down by refutation, or swelled by imaginative theorising, remains to
be seen.