Does Vagueness Underlie the Mass/Count
Distinction?
David Liebesman
April 21, 2015
1
Introduction
My answer to the title question is no. I motivate this answer in two ways. First,
I argue against Chierchia’s (2010) attempt to explain the mass/count distinction in terms of vagueness. Second, I argue that, independently of details of
Chierchia’s account, no vagueness-centric account of the mass/count distinction
will succeed.
2
Chierchia’s Proposal
Chierchia claims that by integrating independently motivated semantic
mechanisms–those needed to account for plurality and vagueness– “we can’t
really miss where the mass/count distinction comes from. . . ” (116).1 In order
to understand his approach I’ll first sketch his approaches to plurality, vagueness, and counting.2
2.1
Plural Structures
Following Link (1983),3 Chierchia assumes that the proper domain for understanding plurals is an atomic semilattice closed under a pluralizing (or join)
operation ∪.4 To illustrate this, consider a world with three cats: Spot, Fluffy,
and Whiskers. The idea is that the domain adequate for modelling the meaning
of “cats” consists of Spot, Fluffy, Whiskers, and sums/pluralities composed of
them (via join). Taking “S” to represent Spot, “F” to represent Fluffy, and
1 All
page references to Chierchia’s work are to Chierchia (2010).
relies on a number of assumptions about the nature of plurality and vagueness.
This reliance is largely incidental. Nonetheless, presenting Chierchia’s view complete with his
own assumptions allows for a better understanding of the view.
3 Among many others, including Chierchia (1998a) and (1998b).
4 There is plenty one may worry about with regard to this understanding of plurality.
Philosophically, the most troubling worry is what McKay (2006) calls “singularism”. I’ll
argue that Chierchia’s account fails even given his own understanding of plurality, so I’ll set
aside anti-singularist scruples.
2 Chierchia
1
“W” to represent Whiskers, the domain can be hierarchically represented as
(1). This domain is partially ordered by a relation labeled “≤”, which we may
informally think of as parthood. ≤ is linked to ∪: if b = o ∪ x (for any x 6= b),
then o ≤ b.
(1)
S∪F∪W
S∪F F∪W S∪W
S
F
W
Consider, now, the relationship between the singular “cat” and its pluralization “cats”. “Cat” denotes all and only the bottommost elements of (1):
the elements x such that there does not exist a distinct y such that y≤x. This
captures the fact that “cat” is true of single cats and nothing else: it is not true
of cat pluralities, nor is it true of cat parts. The plural “cats”, on the other
hand, is true of everything in (1): both the individual cats and the pluralities.5
When we count using plurals, we do not count both atoms and pluralities. If
we did, it would be true that there are six cats. Rather, we count only minimal
elements, a.k.a. “atoms”. The AT function extracts minimal elements from a
set. For example, applying AT to (1) yields the set of the cat atoms.6
(2) AT{S, F, W, S∪F, F∪W, S∪W, S∪F∪W} = {S, F, W}
For our purposes, there are three important features of AT. The first is that
AT is an extraction operation: applying AT to a set yields a subset. The second
is that AT is understood in terms of ≤: x∈AT(P) just in case, if x∈P, y∈P and
x6=y then ∼(y≤x) . The third is that AT may yield a particular entity as an
atom relative to one set, though not relative to another. For example, if we
apply the function to the set consisting only of cat pluralities, then we yield
quite a different result:
(3) AT{S∪F, F∪W, S∪W, S∪F∪W} = {S∪F, F∪W, S∪W}
Given its relative nature, AT is not to be understood in terms of mereological
simplicity. Spot is certainly not mereologically simple: he has paws, whiskers,
etc. However, he is a minimal element of the denotation of “cats”, and, relative
to that denotation, he is in the set produced by applying AT.7 As a limit case,
the AT function can be applied to the entire universe. AT(U) yields the set of
5 Taking plurals to be true of atoms is controversial. In earlier work (1998b), Chierchia
rejects this. His change of mind stems from the fact that the sentence “No cats are on the
mat,” is intuitively false if there is just one cat on the mat. This is hard to account for if
single cats are not in the denotation of “cats”.
6 Following Chierchia, I’ll take AT to apply to sets, properties, predicates, and individuals.
If we take the set case as fundamental, then AT applies to properties by applying to the sets
that model them, and AT applies to predicates by applying to the sets that capture their
domains.
7 For simplicity, I am skipping a number of complications and innovations in presenting this
framework. Most notably, Chierchia takes the ‘uppermost’ elements in domains to be kinds.
Following Carlson (1977), among many others, we can then take kinds to be the referents of
a number of occurrences of terms, e.g. bare occurrences of plural count nouns.
2
individual entities.8 Crucially, AT is also context-sensitive. I’ll mark this with
a subscripted “c”.
2.2
Vague Plural Structures and Mass Nouns
In the above sketch of Chierchia’s treatment of plurals, there was no allowance
for vagueness. This is inadequate: the vast majority of natural language count
nouns are vague. If Fluffy gets pregnant and gives birth to five kittens, it is
clear that each of these belongs in the denotation of “cats”. Flash back 25 days:
Fluffy is 30 days into her pregnancy. Do her fetuses count as cats? It seems as
if there is no fact of the matter.9 Therefore, it seems indeterminate whether the
fetus is in the extension of “cat(s)”.
To capture the vagueness of “cat” and “cats”, Chierchia suggests that we
interpret them as partial functions. “Cat”, then, is a partial function from
objects to truth values. When this function takes Fluffy as its argument, the
result is truth. When this function takes my left foot as its argument, the result
is falsity. The function, however, is undefined for Fluffy’s fetuses: they are
neither in the extension of “cat” nor in its anti-extension.10
Just as we can identify objects that neither fall in the extension of “cat” nor
in its anti-extension, we can identify precisifications of “cat” that force those
objects into either its extension or anti-extension. Precisifications all have one
thing in common: they won’t change anything about the initial extension and
anti-extension of “cat”. Rather, precisifications will merely resolve the undefined
cases. Taking contexts to assign extensions and anti-extensions to predicates, we
can consider precisifications of predicates by quantifying over contexts. Contexts
are partially ordered by the ∝ relation: if c∝c0 then predicates in c0 are at least
as precise as predicates in c, and the extensions and anti-extensions of predicates
in c are preserved in c0 .
Using the ∝ relation, we can define a determinateness operator “D”.“ Dφ”
is true relative to a context c just in case φ is true relative to every context c0
such that c∝c0 . Now consider AT and Fluffy’s fetuses. Relative to an ordinary
context, the fetuses will neither be in the extension of AT[[“cats”]] nor in its
anti-extension. Thus, they will not be determinate cat atoms or non-atoms.
To capture this notion, we can define a stable atomicity operator “AT” in the
following way: ATc (P) = λxD[ATc (P)(x)]. In other words, relative to a context
c and a (possibly plural) predicate P, AT provides us with all of the entities that
8 Note that it is problematic to take non-overlap as a constraint on the set of atoms relative
to some property. This is due to the fact that, for instance, we may wish to both count Spot
as an atom, as well as Spot’s left leg.
9 Perhaps this is settled by the existence of a soul, or something of the sort. If so, feel free
to change the example to chairs.
10 There are difficult questions about whether modelling vague predicates as partial functions
suffices for capturing the essence of their vagueness. In particular, we may wish to distinguish
between vagueness, which seems to require that there are borderline cases of the sort that
lead to the Sorites paradox, and indeterminacy, which seems merely to require that there are
cases left open. For a discussion of these issues see Weatherson (2010). I’ll set this aside and
grant Chierchia the adequacy of his model, at least for his purposes.
3
are atoms of P relative to every context at least as precise as c. There will be a
plethora of contexts on this view, only some of which correspond to the actual
contexts that generate the data we are examining. Among these latter contexts
will be what Chierchia calls “ground contexts” which, to a first approximation,
are the sort of ordinary conversational contexts which generate the data that
we are attempting to explain.11 Importantly, note that, according to Chierchia,
for any ground context, AT(CN)=AT(CN), where “CN” is any count noun. In
other words, the atoms are co-extensive with the stable atoms.
Stable atomicity AT is at the heart of Chierchia’s view. According to Chierchia, in every context in which count nouns are non-empty they have stable
atoms. Note that this is compatible with the context-sensitivity of count nouns.
Mass nouns, on the other hand, always lack stable atoms.
The intuitions underlying the picture are familiar. Count nouns denote integral and well-bounded entities, though which (integral, well-bounded) entities
a count noun denotes may be sensitive to context. Mass nouns, on the other
hand, denote amorphous substances. We can resolve all of the indeterminacy
for a given mass noun (e.g. counting barleywine as beer) without it being clear
at all what the most basic elements in the extension of that mass noun are. Is
1cl of beer still in the extension of “beer”? How about half that? No matter
how carefully we resolve the contextual features of “beer”, it still seems that a
principled answer is impossible.
All of this brings us to Chierchia’s purported insight. On his view, nonempty count nouns always have stable atoms, while mass nouns always lack
stable atoms. Something is an unstable atom iff it is neither a determinate atom
nor a determinate sum. Vagueness, then, is a the root of the distinction. It is
the vagueness of atomicity, combined with the generation of plural structures,
that, according to Chierchia, underlies the mass/count distinction.
2.3
Mass Nouns and Counting
If mass noun denotations are generated from unstable atoms while count noun
denotations are generated from stable atoms, and if counting requires stable
atoms, then we’ll be able to count with count nouns and not with mass nouns.
One of Chierchia’s hypotheses is that counting does require stable atoms. To
formalize this idea, Chierchia follows many, e.g. Landman (2004), in assuming
that numerals are basically adjectival.12 For Chierchia, they are functions from
predicates to predicates, or type <<e,t>,<e,t>>. In some contexts, e.g. when
they are sentence-initial, they first compose with predicates of type <e,t> then
11 Chierchia (119) does give a technical definition of “ground context”: they are contexts,
in a given model, that are minimal with respect to ∝. This definition, in an of itself, doesn’t
determine which contexts are ground without an independent means to determine which
models are under discussion. Since we are attempting to explain the ordinary behaviour of
mass and count nouns, it stands to reason that among the ground contexts are ordinary
conversational contexts. Note, also, that in at least one place he refers to them as “base
contexts”.
12 There is substantial disagreement about this, both in philosophy, e.g. Hodes (1984) and
Hofweber (2005), and linguistics, e.g. Kennedy (2013)
4
they are subject to default existential closure. Consider, for example, (4), and
the truth-conditions which Chierchia hypothesizes for it, which I gloss in (5).
(4) Three cats are purring,
(5) ∃x[three stable “cat” atoms are part of x, and purring (x)]
Counting, then, for Chierchia, proceeds by enumerating stable atoms. The
reason that we can’t count with mass nouns is that mass nouns lack stable
atoms.
3
Against Chierchia’s Proposal
With all of the pieces on the table we can now succinctly present Chierchia’s
proposal. First, he hypothesizes that mass nouns never have stable atoms while
count nouns always do:
CC1: Every mass noun “M” is such that, for every ground context
c, ATc (M) = ∅. Gloss: mass nouns never have stable atoms, though
they may have unstable atoms.
CC2: Every count noun “N” is such that, for every ground context c
in which N is non-empty, ∼(ATc (N) = ∅). Gloss: non-empty count
nouns always have stable atoms (not just atoms).
Note first that CC2 is not vacuous! A predicate can be non-empty by applying
to stable atoms, unstable atoms, or non-atoms (sums). CC2 says that whenever
a count noun is non-empty it applies to at least some stable atoms. Note second
that CC1 and CC2 contain reference to ground contexts. Given that contexts
are mere abstract structures, there are surely some contexts in which mass
nouns have stable atoms–consider, for instance, a completely precise context.
However, in attempting to explain the mass/count distinction, we limit ourselves
to the sorts of ordinary contexts in which mass and count nouns exhibit their
characteristic behaviour, among these are ground contexts.
In the next four subsections (3.1-3.4) I’ll give four arguments against Chierchia’s proposal. They share a common thread: that insofar as there is any
vagueness-based difference between mass and count nouns (which is itself dubious), it cannot explain why we can’t count with mass nouns. In the next
section (4) I’ll argue that rather than being problem with Chierchia’s particular
view, this insight shows that no vagueness-centric approach to the mass/count
distinction will succeed.
3.1
Chierchia’s Proposal is Incompatible with the Most
Popular Solutions to The Problem of the Many
The problem of the many, introduced by Unger (1980), shows that our ordinary counting judgements are in tension with some independently attractive
5
metaphysical principles. I look up at the sky and judge that there is just one
cloud. However, the sky contains myriad collections of particles, each which
seems equally suitable for constituting a cloud. In fact, it seems that no such
collection of particles is uniquely privileged as constituting a cloud: if any does,
they all do. So, given that there is a single cloud, it follows that there are myriad. This reasoning can be generalized for just about any kind designated by a
count noun.
This is a quick and crude presentation of the problem (see Weatherson (2009)
for more care and detail), though it will suffice for our purposes. Most versions of
the problem rely on a tension between a counting judgement—to the effect that
there are some relatively limited number of entities—and a parity principle—to
the effect that if there is one entity of a given kind, there must be myriad. We
can articulate these as follows:
PP: There are a multitude of equally good cloud candidates, such that if
one is a cloud then they all are.
CD: There is exactly one cloud in the sky.
The two most popular solutions to the problem are compatibilist in nature,
insofar as they attempt to preserve both PP and CD, or at least close relatives.
The first is a supervaluationist solution. Taking “D” to be our determinacy
operator, there are different ways to add D to CD yielding different readings.
If the operator takes wide scope, as in “D (There is exactly one cloud in the
sky)”, given Chierchia’s semantics, all that is required is that, relative to every
precisification, one entity is in the extension of “cloud”, though which entity
this is may vary. If the operator takes merely scopes over “cloud”, as in “There
is exactly one D cloud in the sky”, then it is required that a single entity is in
the extension of “cloud” relative to every precisification. A supervaluationist
solution upholds the wide-scope reading, but not the narrow-scope reading,
as a way to vindicate PP and CD. The idea is that exactly one of the cloudcandidates is a cloud relative to each precisification. Given that which candidate
is a cloud varies depending on the precisification, and that the precisifications
are on par, PP is upheld. Given that, relative to each precisification, exactly
one candidate is in the extension of “cloud”, CD is upheld, as it is super-true
(where super-truth is truth on all precisifications).
The supervaluationist solution is popular. However, it is incompatible
with Chierchia’s view. Recall the definition of stable atomicity: AT(P) = λx
D[AT(P)(x)]. Taking P to be the property of being a cloud, this requires that a
single entity instantiates the property relative to all precisifications. This is exactly what the supervaluationist denies: on the solution, PP is upheld precisely
because which candidate is a cloud varies depending on the precisification. As
McGee puts it
There isn’t, anywhere in the world, anything of which it is determined that it satisfies ‘mountain’. Forget about thinning hair.
6
Nothing is determined to satisfy ‘bald man’, because nothing is determined to satisfy ‘man’. (1997: 145)
Despite McGee’s claim about the pervasiveness of indeterminacy, he upholds
our counting judgments by holding that when there is one cloud in the sky, it
is true on each precisification of ‘cloud’ that one thing is in its extension, even
if this thing varies between precisifications (and thus is not a stable atom).13
To better see exactly how Chierchia’s proposal comes into conflict with supervaluationism, consider the predicate “heap”. Imagine that we have a set
of heap-candidates. It is hard to see what in our usage could single out one of
these candidates as a heap. This is so no matter how broadly we construe usage:
it could include the past, present, and future, as well as all of our utterances,
thoughts, appeal to expertise, etc. According to the supervaluationist, this impression is accurate: nothing in our usage singles out a particular candidate.
However, it may nonetheless be true that there is exactly one heap because, on
every admissible precisification, exactly one entity is in the extension of “heap”.
However, just which entity this is will vary. According to Chierchia, if it is true
that there is one heap, there must be a single object that is classified as a heap
relative to any precisification. This is just what the supervaluationist denies.14
A second compatibilist solution to the problem of the many is a countingbased solution. This is pursued in Lewis (1993). The idea is that in many
contexts, we count by some relation less discriminating than identity. So, for
example, when counting clouds, we count myriad overlapping clouds as a single
cloud. This solution is straightforwardly incompatible with the second component of Chierchia’s proposal. On the Lewis-style view in question, each cloudcandidate is a stable cloud-atom, but we do not count by stable atoms. Rather,
we count equivalence classes of entities (perhaps stable atoms). Thus, this solution is incompatible with the second component of Chierchia’s view: it conflicts
with Chierchia’s truth-conditions for counting sentences.
Thus far, I’ve merely demonstrated that two solutions to the problem of the
many are incompatible with Chierchia’s view. Given the popularity of these
views, many will take this as highly problematic. It is worth nothing, though,
that other solutions to the problem of the many may better cohere with Chierchia’s view.15 However, as I’ll now argue, reflection on the incompatibility
between Chierchia’s view and the most popular solutions to the problem of the
many reveals that, independently of the problem, Chierchia’s view fails.
13 McGee
and McLaughlin present their determinacy-based solution in their (2000).
Chierchia’s discussion of “heap” on pg. 122. These issues are also discussed in
Rothstein (1998) and (2010), as well as Krifka (1992).
15 In particular, a brutalist view as developed by Markosian (1998) may fit better since there
will be the intuitive number of stable atoms for count nouns. However, even Brutalism will
not be compatible with Chierchia’s view if we combine Brutalism with an epistemic view of
vagueness. Furthermore, given Brutalism an epistemic view is particularly plausible.
14 Compare
7
3.2
Chierchia’s Counts are Both Too Precise and not Precise Enough
The reason that we doubted the second component of Chierchia’s view was that
on a counting-based solution to the problem of the many, we don’t count by
stable atoms. Rather, we count by some less discriminating standard (perhaps
overlapping atoms, or something of the sort). However, even if we think that
such a solution to the problem of the many isn’t promising, there is a more
general reason to think that we don’t count by stable atomicity: counting by
stable atomicity yields incorrect truth-conditions.
To see that counting by stable atoms yields incorrect results return to Chierchia’s hypothesized truth-conditions for “Three mountains are snowy.”
(6) ∃x[three stable “mountain” atoms are part of x, and snowy (x)]
Now imagine that we are staring at a range of ten peaks. Two of the peaks
are stable mountain atoms, while the remaining eight are in the extension of
“mountain” relative to some precisifications, but not others.16 According to
Chierchia’s truth-conditions “Three mountains are snowy” is false. Furthermore, according to his account of determinacy, it is determinately false. (This is
due to the fact that whether something is a stable atom (relative to a predicate)
doesn’t itself change from precisification to precisification of that predicate.)
Neither of these results is correct. While we may hesitate in the envisioned
context to affirm that three mountains are snowy, we would equally hesitate to
reject it, and we certainly would deny that it is determinately false.
Another example: imagine we are in a context in which the only animals
(besides ourselves) are Fluffy’s fetuses. It is indeterminate whether these fetuses
are cats. According to Chierchia’s proposal, “there are no cats” is true in the
context; in fact it is determinately true. This is odd. At best, the sentence
seems indeterminate.
There is a straightforward reason that Chierchia’s truth-conditions for counting sentences yields the wrong results: it makes counting overly precise. In many
cases, it is indeterminate how many of a kind are present, due to indeterminacy
about whether objects are members of the kind. Chierchia’s semantics for counting sentences rules out such indeterminacy by allowing only stable atoms into
the count.
A possible reply to this worry distinguishes our judgments from actual truthconditions. The thought is that Chierchia’s truth-conditions are indeed correct,
though we are ignorant of the facts in both of the aforementioned cases. Our unwillingness to endorse a particular cat or mountain count stems from ignorance
of the facts, rather than genuine indeterminacy.
This reply is flawed. I have stipulated that in the aforementioned cases the
relevant peaks and fetuses are neither in the extension nor anti-extension of
“cat”/“mountain”.17 Setting aside the view that indeterminacy is itself epis16 These
remaining peaks are stable universe atoms, but not stable mountain atoms.
clearly allows the possibility of such contexts, as he treats vague predicates as
predicates with entities that are neither in their extension nor ant-extension.
17 Chierchia
8
temic, this entails that there is no fact of the matter of which we are ignorant. If,
on the other hand, we give a epistemic account of vagueness on which the indeterminacy consists precisely in our ignorance, then, likewise, the indeterminacy
of counting statements will be guaranteed when the relevant sort of ignorance
is present. So, even if indeterminacy is epistemic, Chierchia’s truth-conditions
are overly precise.
Chierchia’s counts are also not precise enough. In some contexts true counts
are provided by non-natural numbers. If I slice a bagel in half and put it on
the kitchen table along with two whole bagels, then “Two and a half bagels
are on the table” is true. However, if we count by stable atoms, it is hard to
see how anything besides a natural number could provide the correct count.18
Accommodating for these counts is an open and difficult issue, so we wouldn’t
want to criticize Chierchia too severely for failing to consider them. However,
the possibility of such counts does show that there must be more to counting
than identifying stable atoms.
3.3
Count Nouns and Unstable Atoms: A Dilemma
According to Chierchia, in any ground context in which a count noun is nonempty, it has stable atoms. Mass nouns, on the other hand, never have stable
atoms. In this subsection, I will present a dilemma. On the first horn, nonempty count nouns lack stable atoms in many ground contexts. On the second
horn, mass nouns have stable atoms in many ground contexts.
We’ve already seen one reason that Chierchia may be impaled on the first
horn. On a supervaluationist approach to the problem of the many, “exactly
one cloud is in the sky” is true because, on every precisification, “cloud” has one
object in its extension. However, proponents of that approach deny that there
is one such object that is in the extension of “cloud” relative to every precisification: the object in the extension of “cloud” varies between precisifications. If
this is correct, then it undermines the claim that any ground context in which
count nouns have non-empty extensions, they have stable atoms. This is false
about “cloud”, on the envisioned view.
The key conflict between Chierchia and the supervaluationist arose as follows. The supervaluationist holds that there are ground contexts and count
nouns that (a) give rise to non-zero counts, and (b) lack stable atoms. Chierchia denies that there are every contexts and count nouns such that (a) and
(b) are both true. Note that this conflict can arise even for those who reject
a supervaluationist solution to the problem of the many. All that is required
to reject Chierchia’s view is that there is some count noun, and some context
in which (a) and (b) are both true. One is not required to claim that every
instance of the problem of the many provides us with such a context and noun.
Perhaps one rejects the supervaluationist approach to the problem of the
many because they think in the case of natural kinds, there is a brute fact
18 See Salmon (1997), Fox and Hackl (2006), and Liebesman (2015) for a discussion of these
issues.
9
about which candidate composes a complex object. Nonetheless, they will likely
hold (a) and (b) for non-natural kinds such as puddle. Perhaps one rejects
the supervaluationist approach to the problem of the many because they think
that vagueness is a matter of ignorance, rather than semantic indecision as the
(standard) supervaluationist would have us believe. Nonetheless, they will likely
hold (a) and (b) with terms that give rise to this characteristic sort of ignorance.
The upshot is that many theorists will accept (a) and (b) for some contexts
and count nouns, even if they don’t accept the supervaluationst approach to the
problem of the many. However, the dilemma does not end here: this is merely
the first horn.
On the second horn, we are to imagine a theorist who, with Chierchia, denies
that there are ever contexts and count nouns such that uphold (a) and (b). In
other words, whenever we can give a non-zero count with a count noun, it has
stable atoms in its extensions. The problem is that such a theorist will be forced
to hold that there are stable atoms in the extensions of mass nouns.
According to Chierchia, whenever there is one heap, there is an object that
is a determinate heap atom. This is prima facie puzzling. Which, of the many
heap-candidates, is chosen at the exclusion of others? The claim is puzzling for
at least three reasons. The first is that, in some sense, the meaning of “heap”
depends on our usage. Our usage, in turn, doesn’t discriminate between myriad
heap-candidates. After all, the difference between many of them is perceptually
insignificant. The second is that if there a single determinate heap, it is hard
to see how, even in principle, we could come to know which it is. So, “heap” is
epistemically recalcitrant. The third is that it seems metaphysically arbitrary
to distinguish among the heap candidates. After all, there doesn’t seem to be
any relevant intrinsic or extrinsic difference between them.
So, if one holds that, in any context in which there is one heap, there is
a stable heap atom, one also holds these three puzzling claims: that when it
comes to the extensions of count nouns, there can be stable atoms that don’t
seem to be clearly determined by our usage, of which we’re necessarily ignorant,
and which are metaphysically arbitrary.
Now consider a typical mass noun like “rice”. What was the motivation for
claiming that it lacks stable atoms? Perhaps the motivation was that nothing
in our usage determines which is the stable atom. Given that one rejects this
for count nouns, it can hardly be motivating here. Perhaps the motivation was
that we seem necessarily ignorant of which is the smallest bit of rice. Again,
given that accepts such ignorance for count nouns, it can hardly be motivating
here. Perhaps the motivation was that it seems metaphysically arbitrary to pick
a single bit as a rice atom. Again, given that one accepts such arbitrariness in
the case of count nouns, it can hardly be motivating here.
The second horn of the dilemma is now clear. If one holds that whenever
a count noun gives rise to a non-zero count, it has stable atoms, then one is
left without a reason to think that mass nouns lack stable atoms. Furthermore,
since such a theorist takes count nouns to have stable atoms even in the face of
these puzzling facts, it seems they should take mass nouns to have stable atoms
as well.
10
In presenting the dilemma I’ve focussed squarely on Chierchia’s account.
However, as I’ll discuss in 4.3, the same sort of dilemma can be used to argue
against the idea that any vagueness-centric account of the mass/count distinction will succeed.
3.4
Mass Nouns and Stable Atoms: Possible Bad Luck
According to Chierchia, there is no possible ground context in which a mass
noun designates stable atoms. The intuition behind this claim is clear. Take
any quantity of, say, rice, and it seems that its lower half has a decent claim
to also be rice. This is due to the fact that the lower half seems to have the
relevant features for being rice: it is similar to the entire grain with regard to
texture, color, nutritional properties, etc.
For all we know, this sort of situation may hold in the actual world for every
mass noun. Every mass noun may be such that it seems that there is no nonarbitrary way to identify its atoms. Note, though, that if this generalization
holds there is a sense in which it is lucky. It is lucky because the metaphysical
facts—which are not up to us!—determine that it holds. Whether some subquantity of a quantity of rice has the relevant nutritional value, color, etc. to
count as rice is a matter of the metaphysical nature of that sub-quantity.
Now imagine that we aren’t so lucky. Take a quantity of rice, call it r.
Perhaps, surprisingly, all of its sub-quantities are radically different in terms of
behaviour and nutritional content. We would then have good reason to think
that these sub-quantities are not rice. Given this, r has no sub-quantities that
are rice. According to Chierchia, r would qualify it as a rice atom. Since the
nature of those quantities doesn’t change as we precisify the context, r is a
stable rice atom. Call a situation like this “a bad luck situation”. In bad-luck
situations, mass nouns have stable atoms.
These reflections yield a straightforward argument against Chierchia. According to Chierchia’s view, every mass noun lacks stable atoms relative to every
possible ground context. However, there seem possible (if not actual) ground
contexts–bad luck situations–in which mass nouns have stable atoms. Therefore,
Chierchia’s view is false.
The key premise here is that there are possible bad luck situations. However, there’s another controversial aspect of the argument. I have taken the
quantification over every ground context in CC1 to range over every possible
ground context. Chierchia must intend this, or his explanation of the distinction between mass and count would not extend to other possible languages and
contexts. Furthermore, given that Chierchia is attempting to give an account
of the distinction that explains its stable cross-linguistic features, he cannot
reasonably limit himself to a small class of actual contexts.
Why think that bad-luck situations are possible? I’ll give two reasons. The
first is a direct appeal to intuition. It seems that it is possible for there to be
a situation in which every proper part of a grain of rice is radically different
in terms of behavior than the grain itself. Perhaps proper parts of grains are,
when detached, indigestible poison! In that case, the proper parts would not
11
be rice and the grains–having no proper parts that are rice–would be stable
rice atoms. Since such a situation is possible, it is possible for there to be bad
luck situations. The second provides an underlying reason to believe in bad
luck situations. Stable atomicity is understood partly in terms of parthood.
Our perceptions about parthood can diverge from reality. It could seem to me
that a substance has many proper parts of a similar type, but, as a matter of
fact, I may be wrong. This sort of mismatch between perception of (lack of)
atomicity and genuine atomicity is clearly possible. In such a case, it may be
that the perception of atomicity is all that drives the intuition that mass nouns
lack stable atomicity; however, the intuition simply fails to be true.
Stepping back, Chierchia begins his paper by noting that a remarkable feature of the mass/count distinction is that it lines up (though imperfectly) with
the psychological distinction between objects and substances. Substances are almost always denoted by mass nouns. The worry is that the fact that something
is a substance psychologically speaking doesn’t guarantee that, as a matter of
metaphysics, it has the structure Chierchia requires. If there is a mismatch between our psychological categorization and the structure of the perceived entity,
then the mass noun in question may have stable atoms. Furthermore, such a
mismatch is clearly possible, even if it is not actual.
4
Against Any Vagueness-Centric Account
Even if Chierchia’s account fails, one may hold out hope that some vaguenesscentric account of the mass/count distinction succeeds. I’ll now turn to arguing
against any such account. In order to give the argument, I’ll address three
questions: (1) in accounting for the mass/count distinction, what is the proper
explanandum? (2) what role could vagueness play in an explanation? and (3)
it is plausible that vagueness does play such a role?
4.1
What is the Proper Explanandum?
The mass/count distinction has generated interest in philosophy, linguistics,
and psychology. Theorists in these various disciplines have varied interests and,
given that fact, it may seem that giving anything like a general account of
the mass/count distinction is overambitious. Such skepticism is hasty. While
there may be plenty of reasonable disagreement on what it would take to give
a fully satisfying account of the mass/count distinction, we can give a plausible
necessary condition on any such account. As summarized by Chierchia (section
2), recent research has revealed that the mass/count distinction exhibits more
cross-linguistic variation than we might have expected. From the data, Chierchia
elicits only three features of mass nouns that seem cross-linguistically stable.
Given that there are only three stable properties of the mass/count distinction, a necessary condition on an account of that distinction is that it explains
these properties. It follows, then, that a weaker necessary condition on any
such account is that it explains why mass nouns cannot combine with number
12
words: what Chierchia calls “the signature property”. Question (1) asked for
the explanandum in accounting for the mass/count distinction. We now have
one answer: any account must explain why mass nouns cannot combine with
number words. As I stressed, this is likely not the only thing to be explained
by an account of the mass/count distinction, but explaining it is necessary.
4.2
What Role Could Vagueness Play in an Explanation?
Our explanandum is a distributional fact: that mass nouns cannot combine with
number words. As usual, we can attempt to explain distributional constraints in
syntactic, semantic, or pragmatic terms. All three options have been explored in
this case. Rather than reaching a verdict on the viability of various explanations,
for our purposes it will suffice to explore what role vagueness could play.
On a syntactic explanation, there would be a syntactic feature marking nouns
as mass or count, to which number words are sensitive. An agreement error
would then be generated when a number word is combined with a mass noun, or
when a mass quantifier is combined with a count noun. This type of explanation
is unpopular, though the only important point for our purposes is that it is hard
to see how vagueness would play a primary role in such an explanation.19 To
see why vagueness will not likely play a primary role, focus on two options for
the theorist pursuing a syntactic explanation of the distributional facts. Option
one is to take these syntactic features to be semantically impotent. In that
case, it seems that vagueness would have no role to play at all, given that the
syntactic features explaining the distribution of mass nouns and number words
is explained in purely syntactic terms. Option two is to take these syntactic
features to track (or perhaps be grounded in) independent semantic properties.
Here vagueness may play a significant role. The syntax of mass nouns may be
explained in terms of their semantic properties, perhaps, in fact, their distinctive
vagueness.
For all we’ve said thus far, the second option is viable. However, it does
lead to a natural question: what semantic properties of mass nouns plausibly
give rise to the relevant syntactic features? The ultimate explanation here is
semantic, so we should refocus our energy on the role that vagueness can play
in the semantics of mass nouns. I’ll return to this.
The usual pragmatic explanation of the inadmissibility of “two water” invokes the fact that mass nouns are close to homogeneous. Homogeneity can
be understood as the conjunction of two properties: cumulativity and distributivity. A mass noun or count noun (MN/CN) is cumulative just in case every
instance of the following schema holds, where a and b are singular terms.
(7) If a is MN and b is CN/MN then the fusion of a and b is MN/CN.
Unrestricted distributivity claims that every part of anything in the extension of a mass noun is also in the extension of a mass noun. As is familiar, mass
19 For some criticisms of such a solution, see Pelletier and Schubert (2003) and Rothstein
(2010).
13
nouns with minimal parts fail to be distributive in this strong sense. However,
most mass nouns satisfy some weaker version of distributivity on which most
quantities in the extension of a mass noun are such that many of their parts are
also in that extension.20
The near-homogeneity of mass nouns ensures that just about whenever they
apply to something, they apply to myriad things. Thus, it will always be pragmatically odd to claim that “water”, for example, applies to only two things.21
Like the syntactic explanation, this is unpopular. For our purposes, though,
what’s important is that, again, vagueness doesn’t appear to play any crucial
role. It is the near-homogeneity of mass nouns that plays the crucial explanatory
role on a pragmatic account, and near-homogeneity need not involve vagueness:
it is a mereological notion.
Finally, consider semantic accounts of the distribution of mass nouns and
number words. In such explanations, vagueness plausibly plays a role. This
may not be surprising. After all, on the dominant account of vagueness it is
semantic in nature. Chierchia assumes such an account, taking vagueness to be
best understood using supervaluationist machinery.
In giving a semantic explanation of the inadmissibility of “two water”, two
main options present themselves. The first option is that the inadmissibility
is due to a type-mismatch. This sort of explanation can be used to explain
why, for instance, transitive verbs cannot restrict quantifiers. “Two ate”, the
thought goes, is inadmissible because “Two” must be restricted by a predicate of
type <e,t>. Some have pursued this sort of explanation, either by taking mass
nouns to be singular terms (a view suggested in Parsons (1970)), or by taking
count nouns to be some other more complex type, as in Rothstein (2010). On
a Parsons-inspired account, mass nouns are type <e>, so they are not proper
arguments for number words. On Rothstein’s account, there is another fundamental type–that of a counting principle. Number words, require a complex
type constructed partly from this counting principle, and mass nouns are not of
this more complex type.
Initially it is hard to see how vagueness could play a role in a type-driven
explanation of the relevant distributional facts. However, one could, in principle,
hypothesize that the natural-language type system is sensitive to vagueness.
Perhaps, then, the type-mismatch is generated precisely because the distinctive
vagueness of mass nouns guarantees they are of the wrong type to combine with
number words. Note that on this sort of account, we’d need to understand
exactly what sort of vagueness is reflected in the type system.
The second option is to claim that the inadmissibility is due to a semantic
mismatch between mass nouns and number terms that is not reflected in the
type system. On a natural version of such a view, mass nouns are type <e,t>–
the proper type for combining with number terms–but something about their
meanings prevents the combination.22
20 See
Koslicki (1999) for a discussion of these issues.
Burge (1972) for an example of the pragmatic strategy.
22 It is worth noting that the two envisioned semantic explanations may be notational variants of one another if there are multiple equally good ways to type expressions for various
21 See
14
Stepping back, we’ve identified three potential roles for vagueness. On a
syntax-driven explanation of the distributional facts, syntactic features were
explained in terms of vagueness. On a type-driven explanation, natural language typing was explained partly in terms of vagueness. On a non type-driven
explanation, there is a semantic property of mass nouns that is not reflected in
the type system, but which prevents them from combining with number words.
An important observation: all three roles for vagueness depend on a common
core. The common core is that there is a semantic property of mass nouns
that prevents them from combining with number words. The different accounts
disagree as to whether this semantic property is encoded in a syntactic feature
or in the semantic type system. However, they agree that there is such a feature.
We can abstract away from the three different accounts, then, by considering
this common core.
4.3
Does Vagueness Plausibly Play the Role?
I’ll now turn to our third and final question: it is plausible that vagueness does
play such a role? I’ll argue that it is implausible. The considerations here aren’t
decisive, as I cannot survey every possible account, but they do speak strongly
against a vagueness-centric account.
We’re in search of an account of (a) the semantics of number words, and (b)
the semantics of mass nouns that explains, using vagueness, why the two can’t
combine. We’ll begin with (a). Unfortunately for our purposes, the semantics
of number words is highly controversial. (Most of the controversy concerns
issues involving scalar implicature which are orthogonal to our concerns here.)
So, let me consider two different proposals. This first, presented in (8), is an
adjectival theory of number words on which they are type <<e,t><e,t>>. This
is the view adopted by Chierchia, which is similar to a view defended by Ionin
and Matushansky (2006).23 Chierchia defines “u(AT(P))(x)” as the number
of stable P-atoms, which are parts of x. The second, presented in (9), is the
classic theory due to Barwise and Cooper (1981) on which number words are
quantificational determiners.
(8) [[three]] = λPλx[u(AT(P))(x)=3]
(9) [[three]] = λPλQ.|P∩Q| = 3
These accounts have something in common: they both require enumeration
of the members of a set. On Chierchia’s, view the members of the set of stable
atoms that are also parts of x (a plurality) are enumerated. To this account of
number words, Chierchia adds a hypothesis about the semantics of mass nouns
(b) that they don’t designate stable atoms. Combining these, he predicts the
distributional facts.
purposes. On the other hand, if our typing system is not so conventional, then the sorts of
explanations may be genuinely distinct.
23 Landman (2004) also pursues an adjectival theory, though he takes adjectives to be firstorder predicates.
15
On the Barwise and Cooper view, the members of the intersection of P
and Q are enumerated. Notice that this, in and of itself, won’t help with the
mass/count distinction unless mass nouns don’t have extensions that can be
modelled as sets. Let’s set this aside for a moment–I’ll return to it towards
the end of this subsection–and make the plausible assumption that they can be
(perhaps because they are type-identical to plurals). It follows, then, that the
Barwise and Cooper semantic value for “three” would predict that “Three water
are in the bottle” is perfectly interpretable. However, we can see what would
have to be added to the Barwise and Cooper account to distinguish between
mass and count nouns. Rather than merely invoking the absolute value function
in their semantics, they’d have to invoke some function that is ill-defined on
properties designated by mass nouns. A Chierchia sympathizer, in fact, could
easily rejigger (9) to be sensitive to stable atomicity rather than absolute value.
An iteration of the Barwise/Cooper view that accounts for the distributional
facts would contain a modification of (9) and a semantics of mass nouns, such
that the former is ill-defined given the latter.
Stepping back, whatever semantics we give for number words, we’ll want
it to involve some function that can help us to distinguish between count and
mass nouns. This point is independent of whether we analyze number words
as adjectival, quantificational, or whatever else. Thus, a general shape for explaining the mass/count distinction emerges: find such a function that (i) yields
the proper counting results, and (ii) distinguishes count and mass nouns. If we
are going to pursue a vagueness-centric explanation we can gloss the second
requirement as follows (ii0 ) the function distinguishes count and mass nouns on
basis of their differing sorts of vagueness. I have argued that AT fails to satisfy
both (i) and (ii0 ), but it is worth asking whether some other function may do
the trick for the theorist who wishes to explain the mass/count distinction in
terms of vagueness.
So, the attempt to account for the mass/count distinction in terms of vagueness has reduced to the following: the attempt to identify a function that satisfies (i) and (ii0 ), and an account of mass nouns that helps to explain why the
function satisfies (ii0 ). There are at least two reasons to think that there is no
such account: one familiar and the other novel.
The familiar reason stems from an oft-cited observation: that there are nearsynonyms, both intralinguistically and interlinguistically, such that one is mass
and the other is count. Intralinguistically, we can consider pairs like “coins” and
“change”, the former of which is count and the latter is mass. Interlinguistically
we can consider pairs like “spaghetti” in English, which is mass, and “spaghetti”
in Italian, which is count.
The reason that near-synonyms make it implausible that there is a function
satisfying both (i) and (ii0 ) is that it is hard to see how these synonyms differ in
terms of vagueness. Focus on U.S. currency: it is plausible that these terms have
the very same extension. Even somewhat controversial cases like the dollar coin
hardly seem to make a substantial difference as far as explaining the mass/count
distinction. Perhaps one may invoke the intensionality of the distinction: even
if the actual extensions of “coins” and “change” are similar enough to make it
16
implausible that they are distinguished on basis of vagueness, one may have the
sense that, across possible worlds, it becomes vague what the minimal elements
of “change” are, while it remains relatively determinate for “coins”. It only
takes a little imagination to undermine this impression. Imagine a society in
which the monetary currency consists partly of circular metal objects worth one
dollar, along with the following addition: any circular proper part of monetary
currency has a worth exactly in proportion its size relative to the coin of which
it is part. So, we can take a dollar coin and extract from it a 50 cent coin simply
by extracting a proper part with exactly half of initial coin’s size. Of course,
intuitively, we can create such coins ad infinitum, leading to the observation
that there don’t seem to be any minimal coins. The lesson is simple: given
enough imagination, it is not hard to conjure a possibility on which “coins” fails
to have minimal elements suitable for counting. Putting our lessons together,
the existence of near-synonyms is problematic because they make it implausible
that we can find a function satisfying both (i) and (ii0 ) by either focussing on
extensions or intensions.
The familiar reason may convince many. However, there are worries. One
may hold that mass nouns like “change” (“furniture”, etc.) are non-canonical.
The inability of mass nouns to combine with number words, in turn, could
be explained first for canonical mass nouns like “water”, and then the noncanonical cases could be dismissed as arising due to copy-cat effects (Chierchia
139), rather than genuine semantic factors.
The novel reason sidesteps this worry, by directly targeting the connection
between determinacy and counting. Recall that, in 3.3, I outlined a dilemma
for Chierchia. One horn was that some (non-empty) count nouns lack stable
atoms, the other was that some mass nouns have stable atoms. Either of these
is clearly unacceptable for Chierchia. Importantly, the same sort of dilemma
can be used to target any view on which the signature property is explained in
terms of vagueness.
One horn of the more general dilemma is that counting doesn’t require determinacy. The other horn of the dilemma is that, if counting required determinacy,
then we’re left without a means to distinguish mass and count nouns in terms
of vagueness.
According to the first horn, counting does not require determinacy. More
precisely, a sentence of the form “N Ps are Q” can be true, even if there is
nothing that is such that it is determinately in the extension of “P”. Why hold
this view? Well, if one finds the supervaluationist view about the problem of
the many appealing, one will hold the view.
If counting does not require determinacy, then no determinacy-sensitive function will be able to distinguish mass from count nouns. So, vagueness (determinacy) could not play a role in explaining the signature property.
According to the second horn, counting does require determinacy. More
precisely, if a sentence of the form “N Ps are Q” is true, then there must be
something such that it is determinately in the extension of “P”. Consider, now,
a true sentence like “One cloud is in the sky”. On this horn of the dilemma,
one thing must be such that it is determinately a cloud. However, as stressed
17
in 3.3, this gives rise to a trio of puzzles. The first is linguistic: nothing in our
use of “cloud” seems to single out a cloud candidate. The second is epistemic:
it seems hard to see how we could come to know which candidate is the cloud.
The third is metaphysical: it is hard to see what objective differences ground
the fact that one candidate is a cloud, while the others are not.
These claims are puzzling, and they demonstrate why many prefer the first
horn. However, in this context I want to set aside the fact that they are puzzling
and focus on a conditional: if one thinks counting requires determinacy, we’re
left without a means to distinguish mass and count nouns in terms of vagueness.
The main motivations to try and distinguish mass nouns from count nouns
in terms of vagueness stem from the idea that mass nouns seem to designate
amorphous substances. In particular, it seems implausible that (a) mass nouns
have atomic extensions, (b) we can know what the atoms are in the extensions
of mass nouns, and (c) our usage determines atoms in the extensions of mass
nouns. Let’s assume that (a)-(c) are correct. On the second horn we can’t use
(a)-(c) to distinguish mass from count! This is due to the fact that those who
hold the second horn are forced to hold analogs of (a)-(c) for count nouns. These
analogs are just the aforementioned puzzling claims.
So, the dilemma is complete. On the one hand, counting doesn’t require
determinacy. On the other hand, if it does it seems that mass nouns and count
nouns can’t be distinguished with regard to determinacy. Importantly, nothing
in this reasoning requires invoking of near-synonyms across mass and count, or
non-canonical mass (or count) nouns. So, the reasoning is not only genuinely
novel but also avoids worries about arguing from non-canonical cases. Furthermore, the dilemma directly targets the link between vagueness and counting.
These two reasons constitute my main reasons for thinking that no account of
the mass/count distinction in terms of vagueness will succeed. However, there
is one loose end that needs to be tied. In my discussion I assumed that the
semantic values of mass nouns can be modelled with sets. If they can’t, then
the Barwise/Cooper semantics may already explain why number words can’t
combine with mass nouns. Perhaps such an explanation could work, though I
am skeptical. Here, let me argue for a more limited claim: vagueness does not
prohibit set-theoretic semantics. The more limited claim is relevant because
even if the inability to model mass nouns semantics with sets explains why they
cannot combine with number words, the fact that this inability doesn’t stem
from vagueness would show that vagueness doesn’t underlie the distinction.
It doesn’t take much creativity to show that the fact that a term is vague
doesn’t prohibit set-theoretic semantics: we have actual well-established machinery that allows us to set-theoretically model vague terms, e.g. supervaluations. Furthermore, even if the vagueness in question were ontic, we could model
the semantics of vague predicates with sets of (ontically vague) objects. The
loose end is now tied: any purported inability to model mass noun semantics
with sets doesn’t plausibly stem from vagueness.
18
5
Conclusion
Though initially appealing, the thought that vagueness underlies the mass/count
distinction doesn’t bear scrutiny. I made this case by, first, arguing against
Chierchia’s attempt explain the distinction in terms of vagueness. His attempt
is foiled by the fact that vagueness permeates the count realm just as it does
the mass realm. I then gave a more general argument that vagueness doesn’t
underlie the distinction. The argument proceeded as follows. I first gave a
necessary condition on any adequate account of the mass/count distinction: that
it explains the fact that we can’t combine mass nouns with number words. I then
elicited a common core of any such explanation that count involve vagueness.
The common core would involve (a) a semantics for number words and (b) a
semantics for mass nouns, such that (a) and (b) can be used to explain the
distributional facts in a way that crucially invokes vagueness. Focussing on the
semantics of number words I sketched what the common core would look like
and, finally, argued that vagueness can’t play the role it would need to play.
The lesson is that vagueness doesn’t underlie the mass/count distinction.
References
[1] C. Barker. Individuation and Quantification. Linguistic Inquiry, 30(4):68391, 1999.
[2] J. Barwise and R. Cooper. Generalized Quantifiers and Natural Language.
Linguistics and Philosophy, 4(2):159-219, 1981.
[3] T. Burge. Truth and Mass Terms. The Journal of Philosophy, 64(1):263282, 1972.
[4] G. Carlson. A Unified Analysis of the English Bare Plural. Linguistics and
Philosophy, 1(3):413-57, 1977.
[5] G. Chierchia. Plurality of Mass Nouns and the Notion of Semantic Parameter. in Events and grammar, ed. Rothstein, Dordrecht: Kluwer 53-103,
1998.
[6] G. Chierchia. Reference to Kinds Across Language. Natural Language Semantics, 6(4):339-405, 1998.
[7] G. Chierchia. Mass Nouns, Vagueness and Semantic variation. Synthese,
174(1): 99-149, 2010.
[8] D. Fox and M. Hackl. The Universal Density of Measurement. Linguistics
and Philosophy, 29(5): 537-586, 2006.
[9] H. Hodes. Logicism and the Ontological Commitments of Arithmetic. The
Journal of Philosophy, 123-149, 1984.
19
[10] T. Hofweber. Number Determiners, Numbers, and Arithmetic. The Philosophical Review, 179-225, 2005.
[11] T. Ionin and O. Matushansky. The Composition of Complex Cardinals.
Journal of Semantics, 23(4): 315-360, 2006.
[12] K. Koslicki. The Semantics of Mass-Predicates. Noûs, 33(1): 46-91, 2002.
[13] F. Landman. Events and Plurality. Kluwer Academic Publishers, 2000.
[14] D. Lewis. Many, but Almost One. in Cambell, Bacon and Reinhardt, eds.
Ontology, Causality, and Mind, Cambridge University Press pages 23-38,
1993.
[15] D. Liebesman. We Do not Count by Identity. Australasian Journal of Philosophy, 93(1): 123-42, 2015.
[16] G. Link. The Logical Analysis of Plurals and Mass terms. In C. S. R.
Bauerle and A. von Stechow, eds. Meaning, Use, and the Interpretation of
Language, pages 302-323. Mouton de Gruyter, 1983.
[17] N. Markosian. Brutal composition. Philosophical Studies, 92(3):211249,1998.
[18] V. McGee. Kilimanjaro. Canadian Journal of Philosophy, 27(sup1):141163, 1997.
[19] V. McGee and B. McLaughlin. The Lessons of the Many, PhilosophicalTopics, 28(1):129-51, 2000.
[20] T. McKay. Plural Predication. Oxford University Press, 2006.
[21] T. Parsons. An Analysis of Mass Terms and Amount Terms. in Pelletier,
ed. Mass Terms: Some Philosophical Problems, pages 137-166, 1979.
[22] F. Pelletier and L. Schubert. Mass Expressions. Handbook of Philosophical
Logic, 4: 327-407, 1989.
[23] S. Rothstein. Counting and the Mass/Count Distinction. Journal of Semantics, 27(3):343-397, 2010.
[24] N. Salmon. Wholes, Parts, and Numbers. In J. E. Tomberlin, ed.,
Philosophical Perspectives 11: Mind, Causation, and World, pages 1-15.
Ridgeview, 1997.
[25] N. Soja, S. Carey, and E. Spelke. Ontological Categories Guide Young
Children’s Inductions of Word Meaning: Object terms and substance terms.
Cognition, 38(2):179-211, 1991.
[26] P. Unger. The Problem of the Many. Midwest Studies in Philosophy, 5(1):
411-468, 2008.
20
[27] B. Weatherson. The Problem of the Many. The Stanford Encyclopedia of
Philosophy. The Metaphysics Research Lab, Stanford University, 2005.
[28] B. Weatherson. Vagueness as Indeterminacy. In R. Dietz and S. Moruzzi,
editors, Cuts and Clouds: Vagueness, Its Nature, and Its logic. Oxford
University Press, 2010.
21