1. automotive and vehicles
2. auto parts

Composition Is Identity1
Megan B. Wallace
1. The Hand Puzzle and CI
1.1.
The Hand Puzzle
Look at your hand. It is a material object most of us will grant exists. What‟s more, your
hand occupies a certain region, r, at a certain time, t. As such, most of us endorse:
HAND
Your hand, h, exactly occupies r at t.
Yet most of us will also be willing to grant that your hand has parts: it is made up of (at least)
bones, muscle, tissue, and millions of molecules. These millions of parts are also material
objects, and, taken together, are spread out over a hand-shaped area. We can say, then, that
the many parts of your hand exactly occupy region r at time t. This does not mean, of course,
that each of the parts of your hand are in a hand-shaped region, or that each of the parts
occupies r at t. Rather, the parts of your hand taken together occupy r at t.2 So most of us
endorse:
PARTS
The parts, the ps, exactly occupy r at t.
However, most of us think that distinct material objects cannot completely overlap in the
exact same place at the exact same time; material objects cannot compete for the same space
with other material things. We therefore endorse the following material Non-Collocation
Principle:
NCP
Distinct material objects cannot be in the exact same place
at the exact same time.
But now we have a problem.
1
I would like to thank that following for insightful discussion and comments: Dorit Bar-On, Einar Bohn,
Jason Bowers, Matt Kotzen, Bill Lycan, Ram Neta, Ted Parent, Laurie Paul, David Reeve, John Roberts, Adam
Sennet, Ted Sider, and Keith Simmons.
Below, we will see that we can formally disambiguate a sentence such as “The parts occupy a hand-shaped
region of space” such that on one reading, it is true and on another it is false, depending on whether the
predicate “occupying a hand-shaped region” modifies the parts collectively or distributively, respectively.
2
1
By HAND and PARTS, the ps and h are located at exactly the same place and at the
exact same time. By assumption, the ps and h are material objects, and are located in r at t.
There is no place that h is that the ps aren‟t; and there is no place that the ps are that h isn‟t.
But by NCP, the ps and h cannot both be in r at t. And the problem generalizes; what goes
for h and the ps goes for any ordinary object and its parts. Let us call this the Hand Puzzle.
Perhaps, you might think, the above line of reasoning assumes more than just
HAND, PARTS, and NCP. And perhaps it is one or more of these hidden, suppressed
premises that are at fault. To begin with, I have assumed that the parts of your hand, the ps,
are non-identical to your hand, h. If the ps are simply identical to your hand, h, then there is
no problem. For suppose we endorse ID:
ID
the ps = h
Then if HAND and PARTS are true, we are not in violation of NCP, since the ps occupy r at
t, and h occupies r at t, and the ps = h, which is consistent with NCP. There is not both your
hand and all of the molecules occupying the same place at the same time, for your hand just
is the molecules; the ps and h are not distinct, if ID is true. And, indeed, this is the line I am
going to be pushing below.
However, many think that ID is false (for reasons that will be detailed more fully
below). Briefly, many of us maintain that ID is false because the parts of your hand
seemingly have many features that your entire hand does not. The parts of your hand are
many in number, for example, while your whole hand is one; the parts of your hand are
relatively small, as another example, while your whole hand is not. Etc. But then by Leibniz‟s
Law, ID must be false. So even though the Hand Puzzle does assume that ID is false, this
isn‟t a usual place of resistance, since most people agree that ID is false.
Another assumption the Hand Puzzle seemingly makes is not that the ps and h are
distinct material objects, but that the ps are an object at all. One might object that NCP
2
implicitly assumes that a single material object cannot be collocated with another single
material object. And so when we apply NCP to PARTS, we are illegitimately assuming that
the ps are a single thing that cannot be collocated with another, distinct single thing. But, one
might continue, to treat the ps as a single material object is a mistake, which accounts for our
misapplication of NCP to PARTS.
To remedy this seeming confusion, we might more precisely rephrase NCP as:
NCP-S:
For any material object, x, and any material object, y, if x exactly
occupies region r at t and y exactly occupies r at t, then x =y.3
One might argue that NCP-S is the intuitive principle that I was appealing to when I
presented the more loosely stated NCP. Furthermore, if NCP-S is the principle at work in
the Hand Puzzle, then there is simply no puzzle at all. For PARTS involves a plural term, the
ps, and HAND involves a singular term, h. NCP-S only tells us that that if a single item is in
a certain place at a certain time, then no other (distinct) single item can be exactly collocated
where the first one is. Your hand and its parts—and indeed, parts and wholes in general—do
not involve two distinct objects; parts and wholes are cases of collocation that we
wholeheartedly accept. So, one might argue, I have implicitly assumed that the ps are a
singular thing—which they are not—and so the Hand Puzzle is no puzzle at all!
It is important that I have not assumed that the ps are a singular thing; on the
contrary—and as we shall see below—it is important that we think of “the ps” as an
irreducibly plural term, that refers to many (plural) objects. As for NCP, it is true that we are
all accustomed to thinking in terms of NCP-S, and the traditional puzzles that it creates. And
it is true that we usually dismiss worries about parts and wholes being collocated. But this
seems to be an unprincipled prejudice. It is simply irrelevant that the loosely stated NCP, or
any variation thereof, involves only singular terms. The spirit of NCP-S—what is integral to
I call this NCP-S because this non-collocation principle is phrased in terms of singular terms, x and y. This is
the familiar principle that generates metaphysical puzzles such as the statue and the clay, or Tib and Tibbles,
etc.
3
3
it—is the idea that material objects “muscle” each other out for space. This intuition is about
material objects crowding each other out, irrespective of whether we are talking about many
objects or just one or two. If there is an object or objects in a certain location at a certain
time, then no other object or objects can occupy this exact location at this exact time. If
there are a bunch of stars in an Orion-shaped region, O, at time t, then a bunch of
spaceships cannot also be in the exact Orion-shape region, O, at t. If NCP is in need of
tightening, we can rephrase it so that it is neutral between plural and singular terms:
NCP-N:
For any material object(s), , and any material object(s), , if
exactly occupies region r at t and exactly occupies r at t,
then =h .
It may be that we have, until now, simply dismissed worries about parts and wholes being
collocated. But this dismissal is metaphysically irresponsible. If NCP-S is intuitive, NCP-N is
as well. At the very least, the onus is on someone who thinks that NCP-S is intuitive, but
NCP-N is not, to give a non-ad-hoc explanation of why this is so. Yet if NCP-N is intuitive,
then we are off and running with our hand puzzle—and, indeed, off and running with a very
deep metaphysical puzzle about parts and wholes in general. The Hand Puzzle is a new
challenge, one similar to but distinct from traditional metaphysical worries, such as the statue
and the clay. It forces us to abandon one of either: HAND, PARTS, NCP-N, or the
rejection of ID. We cannot maintain all of them consistently; one of them must go.
1.2.
CI
In what follows, I will argue that we should reject the rejection of ID; .i.e., we should
embrace ID. As a more general claim, I am going to endorse the view that whenever we
have some parts, the as, which compose some whole, w, then the as are identical to w. I am
going to endorse:
4
Composition is Identity (CI)
Let O be any composite object, and let O1,
…, On be its parts. Then O is identical to O1,
…, On, taken collectively.4
To be clear, this is not the view that the whole is identical to each of the parts. Rather, it is the
idea that the whole is identical to the parts collectively, or taken together. CI claims that the
whole is just these parts and nothing more.
As explained above, many think that ID is false. Many think that the ps
h. There
are at least three reasons why many think this is so.5 First, claiming that one thing is identical
to many seems to directly violate the Indiscernibility of Identicals:
Indiscernibility of Identicals
x y (x = y
P (Px
Py))
If h is one, then it is not many, and if the ps are many then they are not one. But then h
cannot be identical to the ps for the simple fact that h is one and the ps are many. Let us call
this the Indiscernibility of Identicals Objection. David Lewis (1991) gives just such an
argument against CI, as does McKay (2006), among others. If there is any attribute that the
whole has that the parts do not, then this will be a (seemingly) decisive refutation of CI.6
Second, h cannot be identical to the ps for reasons involving existential quantification
and counting. Suppose for simplicity that your hand is made up of only three parts. Then
you have at least three things in front of you. We can represent the statement “there are (at
least) three things in front of you” as (1), where „P‟ stands for the predicate is a thing in front of
you:
(1)
x y z (Px & Py & Pz & x
y&x≠z&y
z)
This is a rough, first-pass definition of Composition is Identity. As will be shown in the following sections,
this definition will be amended so as to incorporate plural terms and what I call “hybrid identity”, which will be
discussed in detail below.
4
There are many arguments against CI. I address the three most intuitive ones (to my mind) here; I deal with
others elsewhere.
5
As we will see below, there are other worries involving the Indiscernibility of Identicals that do not concern
the number of wholes and the number of parts.
6
5
This says that there are (at least) three things in front of you, x, y, and z, none of which is
identical to any of the others. And we‟ve already granted that your hand is in front of you. So
we can quantify over h as well. And since h is not identical to any one of the ps, then we get
statement:
(2)
x y z w (Px & Py & Pz & Pw & x
y≠w & z ≠ w)
y&x≠z&y
z&x≠w&
But by counting up all of the things we have quantified over—the three parts of your hand,
the ps, and your entire hand, h,—we see that you have at least four things in front of you.
This means that the whole, h, is something distinct from the parts, the ps, since it is an entity
in addition to the three parts. Whenever we have an object, o, that is identical to another
object, p, it is never the case that having p in our ontology adds to the number of things in
our world if we already have o. So assuming that we have already admitted your hand‟s parts
into our ontology, then if admitting your hand adds to our total count of all the things that
there are, then one thing we know is that your hand, h, is not identical to its molecular parts,
the ps.7 Let us call this the Counting Objection.
Finally, many think that CI is false because of the varying modal properties between
your hand and its parts: your hand could survive losing a few molecules here and there, but
the molecules could not. Conversely, your hand could not survive being thrown in a blender,
but the molecules composing your hand presumably could.8 Yet if your hand has a property
that the molecules do not—viz., could survive a loss of molecules—and if the molecules
have a property that your hand does not—viz., could survive being thrown in a blender—
This is a modified version of Van Inwagen‟s argument that he gives against Composition is Identity in Van
Inwagen (1994).
7
I am modifying Wiggins‟ Tree and Cellulose example here, and his arguments against the claim that the tree
just is the cellulose molecules. The modification? Wiggins considers only the aggregate of the cellulose molecules,
a singular item, not the molecules, plural. See Wiggins (1968).
8
6
then by the Indiscernibility of Identicals, your hand is not identical the molecules. Let us call
this the Modal Objection.
In this paper, I will defend CI against two of these objections. Space prohibits me
from adequately addressing the third, the Modal Objection, here.9 I will, however, give a
skeleton sketch of this defense after I have addressed the first two objections. I will begin by
adopting a plural language, which allows us to talk about objects plurally rather than merely
singularly, as traditional first order logic does. We‟ve already adopted a kind of plural
language by allowing a term such as “ps” to range collectively over the parts of your hand.
We will see that there are many advantages to adopting a plural language, one of which is
providing important resources for a proponent of CI in defending her view. Then I will
investigate our methods of counting, and will introduce what I shall call “Plural Counting”,
which is a proposal about how we should—and in fact do—count material objects in the
world. As is obvious, the Counting Objection depends crucially on a particular method of
counting, in which one uses singular existential quantifiers and singular identity and nonidentity claims. If there is another type of counting available—namely, Plural Counting—
then a CI theorist can easily avoid this objection against her view. So armed with these
formal tools, a plural language and Plural Counting, I will defend CI against the
Indiscernibility of Identicals Objection and the Counting Objection, thereby allowing us to
appeal to CI to solve the Hand Puzzle.
2. Plural Language
In classical logic, we have singular terms, singular predicates, singular quantifiers,
etc., such that a sentence such as (3) can be represented by (4) (where „t‟ stands for Ted,
„Mxy‟ stands for the relation x moved y, „c‟ stands for the coffin):
9
Chapter 4 of my dissertation is dedicated to modal objections against CI.
7
(3)
(4)
Ted moved the coffin.
Mtc
However, things get a bit more complicated when we have a sentence such as (5):
(5)
Jason and Lucy moved the coffin.
For (5) could mean either that Jason and Lucy moved the coffin together, or that each of
(the very strong) Jason and Lucy moved the coffin individually. If we intend the latter, then
we may symbolize the sentence such as (6) (where „j‟ stands for Jason, „l‟ stands for Lucy):
(6)
Mjc & Mlc
Yet if we intend the former, then it is not clear how we could treat the subject term „Jason
and Lucy‟ in a logic that only allows subject terms to be singular. Moreover, as some have
argued,10 it is not merely difficult to come up with a proper way to symbolize sentences such
as (5) in a classical, singular logic, it is impossible to symbolize certain other sentences. The
Kaplan-Geach sentence (7), for example, is inexpressible with a singular logic alone:
(7)
Some critics admired only one another.11
It is argued that the only way we can fully express many of our expressions in English is by
adopting some sort of plural logic—a logic that allows us to talk of plural subjects,
predicates, etc. 12 The sort of plural language I have in mind has (at least) the following
features:13
(i)
(ii)
singular and plural variables, constants, and quantifiers.
plural predicates, distributive and non-distributive predicates (i.e., predicates
10
Kaplan and Geach, e.g.
11
See Boolos 1984; Rayo 2002.
See for example: Boolos (1984), Link (1987), McKay (2006), Schien (1993), Sider (2007) Yi (2005), (2006),
etc.
12
I am assuming that our plural language is irreducible—that is, plural terms, quantifiers, etc., cannot be
reduced to singular terms, quantifiers, etc. Also, this language is obviously more developed in the literature, as
well as in my thesis, where a detail-rich language is needed to answer other objections to CI.
13
8
(iii)
that distribute over the subject terms that they modify, like sneezes, and
predicates that do not, like meets.)
a hybrid singular/plural identity predicate.
An example of a plural term would be the term „Jason and Lucy‟ in (5),
(5) Jason and Lucy moved the coffin.
where Jason and Lucy moved the coffin together. We want a language that will treat terms
such as „Jason and Lucy‟ in (5) as a plural unit, such that we cannot infer from (5), (8):
(8) Jason moved the coffin and Lucy moved the coffin.
Assuming that the coffin is so heavy that neither Jason nor Lucy could move it by
themselves (but that they are strong enough to move it together), (5) is true and (8) is false.14
In the first part of this paper, I had suggested the term „the ps‟ to stand for „the parts‟
(of your hand). I intended that the term „the ps‟ would refer to all of the parts (of your hand)
all at once, just as we might want „Jason and Lucy‟ in (5) to refer to both Jason and Lucy all
at once. So in addition to the usual singular variables, x, y, z, x1, etc., our plural language will
also have (irreducibly) plural variables, X, Y, Z, X1, etc.; in addition to the usual singular
constants, a, b, c, a1, etc., we will also have (irreducibly) plural constants, A, B, C, A1, etc. So,
for example, if we want to represent a statement such as (9):
(9) There are some people surrounding a building.
we would use a sentence such as (10), where „P‟ stands for the predicate are people, „B‟ stands
for the predicate is a building, and „SXy‟ stands for the relation X are surrounding y:
(10) X y(PX & By & SXy)
14
Or, at the very least, (5) is ambiguous, or has more than one reading, or ways of being true, such that (8)
sometimes fails to follow from (5) (on one way of disambiguating or reading (5)). So we want a language that
will allow us to symbolize the situation as such. A classical singular language with only singular terms will not
get the job done; one with plural constants, variables, and quantifiers will.
9
And to represent a statement such as (5),
(5) Jason and Lucy moved the coffin.
where Jason and Lucy moved the coffin together, we could use (11), where „A‟ is an irreducibly
plural term that stands for Jason and Lucy, „c‟ the coffin, and MXy stands for the relation X
moved y:
(11) MAc
Importantly, (10) and (11) do not express the idea that there is a single thing—a
collection, grouping, sum, set, etc.—that is made up of many things—some people, or Jason
and Lucy, etc.—and that this single thing is what surrounded the building or moved a coffin.
Rather, it is important that (10) and (11) express the idea that certain things—taken all at
once or taken together—did something or other. In other words, it is important that the
plural variables and terms are plural.15
Examples of plural predicates include surrounded the building, met for lunch, argued about
philosophy, etc., as was illustrated in some of the sample sentences above. Let us reserve P, Q,
R, …, P1, etc., to represent plural predicates. Plural predicates may be attached to plural
terms, such that they can admit of collective—as opposed to a distributive—reading.
Compare, for example, (12) with (13):
(12) Dan and Eddie met for lunch.
(13) Dan and Eddie sneezed.
The predicate met for lunch in (12) is collective in that it predicates a feature of both Dan and
Eddie together. We wouldn‟t say that Dan met for lunch and Eddie met for lunch; meeting
for lunch seems to be something that one person cannot do by himself. Contrast this with
sneezed in (13). Sneezing is something that each individual does separately; sneezed typically
modifies its subject terms distributively.
Moreover, if a plural language committed us to sets or sums, then it may be seen (by some) that having a
plural language is ontologically costly, which it is not. See McKay (2006) et. al.
15
10
We can represent the distinction between collective and distributive predicates in
(12) and (13) by (14) and (15) respectively, where „M‟ represents the predicate met for lunch,
„A‟ is a plural term that stands for Dan and Eddie, collectively, „S‟ stands for the predicate
sneezed, „d‟ represents Dan, „e‟ stands for Eddie:
(14) PA
(15) Sd & Se
(15) does not include any collective predicates, which is appropriate given that „sneezed‟ is
typically a distributive predicate that applies to objects individually.16 However, (14) does
include the predicate „M‟, which only takes (irreducibly) plural terms or variables as objects
(in this case, „A‟). Thus, we cannot conclude from (14) that Eddie met for lunch and Dan
met for lunch; such an inference is blocked since (14) treats the term „Dan and Eddie‟ as a
plural unit, so to speak.
Notice that in (14) we used the plural variable „A‟, but that we could have also used a
different sort of plural terminology. Let us use „,‟ as a way of concatenating singular terms,
where, for example „x,y‟ means “x and y, taken together.”. So, for example, we can also
represent (12) as (16):
(16) M(d,e)17
16I
say “typically”; on my view, the story gets a bit more complicated. For I will claim that ordinary objects such
as people just are the (mereological) sums of lots and lots of different parts—e.g., bodily parts, molecular parts,
temporal parts, modal parts, etc. Yet if this is true, then sneeze won‟t be purely distributive. That is, it won‟t be
true that if a person sneezes, then all of the parts that compose that person sneeze. Sneeze is not a predicate that
applies to individual objects all the way down. So while sneeze is a distributive predicate in the above example, it is
only partially distributive on my account, given my metaphysics of persons. But, since we may separate my
claims about language and composition, from my claims about what ordinary objects (such as people) are, I will
leave this issue aside for now. I take up such issues further in my thesis, and will allude to these views at the
end of this paper.
17
Let us reserve parentheses for concatenated plural terms. So, for example, „Px‟ would represent a one-place
predicate with the single term „x‟ in the subject slot; „Pxy‟ would represent a two-place relation with the singular
term „x‟ and „y‟ in their respective slots; „P(x,y)‟ would represent a one-place predicate with the plural term „x,y‟
in the subject position; „P(x,y)z‟ would represent a two-place relation with the plural term „x,y‟ in the first
subject slot, and the singular term „z‟ in the second, etc.
11
Having both ways of referring to more than one object at once will endow our plural
language with greater expressive power. For example, if we only had the irreducibly plural
terms A, B, C, A1, etc., then a sentence such as (19) may be difficult to express:
(17) Dan and Eddie met for lunch, and Dan sneezed.
For the natural way to symbolize this if we don’t have recourse to our concatenated terms
such as „a,b‟ etc., and have only plural terms such as „A‟, „B‟, etc., is (18):
(18) MA & Sd
Yet (18) leaves opaque information that seems transparent in the natural language sentence
(17)—namely, that Dan is one of (or among, or part of) the plural things that met for lunch.
If we adopt the plural terminology that was introduced above, however, then we could
represent (17) by the more structure-revealing (19):
(19) M(d,e) & Sd
It is transparent in (19) that the constant „d‟ in the second conjunct is related to the plural
terms „d,e‟ in the first conjunct. In this way, it will increase the expressive power of our
plural language if we adopt both ways of referring to many objects at once.
In addition to plural predicates, however, we will also want a plural language that has
an identity predicate that allows both singular and plural terms in its scope. Notice that it is
not uncommon to allow the identity predicate to modify only singular terms or only plural
terms, as in (20) and (21):
(20) Superman is (identical to) Clark Kent.
(21) Locke, Berkeley, and Hume are (identical to) the British Empiricists.
But it is quite another matter to allow the identity predicate to be flanked by a mixture of
plural and singular terms as in (22) and (23):
(22) Rod, Todd, and Maud are (identical to) the comedy troupe.
(23) The comedy troupe is (identical to) Rod, Todd, and Maud.
12
Yet, as we shall see, a singular/plural hybrid identity predicate is particularly important to the
defender of CI since the primary radical claim of her view is that many things can be one.
We can symbolize this two-place, singular/plural hybrid identity predicate as „=h‟, which
takes either plurals or singulars as argument places:
=h , where
and
can be either plural or singular terms.
I intend for hybrid identity to be the classical identity relation, with only one exception.
Hybrid identity is transitive, reflexive, symmetric, and it obeys Leibniz‟s Law. The exception
is that the hybrid identity relation allows us to claim that many things can be identical to a
singular thing.18
The adoption of the hybrid identity predicate, =h, will not force us to abandon the
singular identity predicate in any way, as used in traditional first-order logic. For singular
identity statements are just a special case of hybrid identity statements. We incorporate
singular identity as follows:
=
≡df
=h , where
and
are singular terms.
There are independent reasons—reasons apart from ontology—to have a plural logic
and language. We can say more with a plural language than we can without one. Moreover,
many who propose a plural logic and language insist that such a language carries with it no
ontological burdens. So not only can we say more with a plural language than without one,
but we can do so at no cost in ontology.19 The defender of CI will welcome these advantages
of adopting a plural language, as well as have her own, metaphysical-based reasons for
wanting to adopt such a language. Once we have a way of talking about plural objects—of
quantifying over things plurally, with a plural quantifier, rather than being restricted only to a
And as I will show below, letting many things be identical to one will not itself be a violation of Leibniz‟s
Law.
18
19
See Boolos 1984; Lewis 1991; Yi 2005-6.
13
singular quantifier—then we will be better able to address some of the objections against CI
that I‟ve briefly summarized above. However, in addition to a plural language, we will also
need Plural Counting, which I will discuss in the following section.
3. Logic Book Counting vs. Plural Counting
Van Inwagen gives the following argument against CI. He asks us to imagine that
there is one big parcel of land, divided neatly into six smaller-parcel parts:20
Suppose that we have a batch of sentences containing quantifiers, and that we want
to determine their truth values: x y z(y is a part of x & z is a part of x & y is not
the same size as z)‟; that sort of thing. How many items are in our domain of
quantification? Seven, right? That is, there are seven objects, and not six objects or
one object, that are possible values of our variables, and which we must take account
of when we are determining the truth value of our sentences.21
The idea is that given how we usually quantify over objects in the world—i.e., with a singular
existential quantifier—then there will be no way to quantify over the whole, big parcel of
land without adding to the number of things in our ontology. And if we are adding to the
things in our ontology when we accept wholes, then the relation between a whole and its
parts is not identity.22
And as we saw with the Hand Puzzle, if we suppose that your hand has only three
parts, and granting that (at least) these parts, the ps, are in front of you, then we might
represent this as follows, where „P‟ stands for the predicate is in front of you:
(1) x y z (Px & Py & Pz & x
y&x≠z&y
z)
But then when we quantify over your hand, h, as well as the ps, we get the following:
Van Inwagen assumes that the smaller parcels are simple (i.e., the smaller parcels are not composed of even
smaller parts or parcels), and ignores (for brevity‟s sake) many of the overlapping parts.
20
21
Van Inwagen, Peter 1994: 213.
This argument is slightly modified from van Inwagen‟s: he is concerned with ontological innocence, rather
than identity, per se. But clearly these two topics are closely related.
22
14
(2) x y z w (Px & Py & Pz & Pw & x
& z ≠ w)
y&x≠z&y
z&x≠w&y≠w
By counting up all of the things we have quantified over—the three parts of your hand, the
ps, and your entire hand, h,—we see that you have four things in front of you, not three. This
means that the whole, h, is something distinct from the parts, the ps, since it is one entity in
addition to the three parts. Thus, it seems that your hand, h, is not identical to its parts, the ps.
We can put this Counting Objection more succinctly as follows:
(A) Sentence (2) is true.
(B) If (A), then there are four things in front of us.
(C) If there are four things in front of us, then the hand is not identical to its parts.
(D) So, the hand is not identical to its parts.
The above argument relies on the truth of (B). But (B) assumes that using a singular
existential quantifier to quantify over all of the (singular!) entities in our ontology, and
tallying up the (singular!) identity and non-identity claims regarding those objects, will yield a
flat-out count of all of the objects that there are. Since this way of counting is not
unfamiliar—indeed, since most of us have been taught by our logic books to count this
way—let us call this Logic Book Counting. Logic Book Counting (LBC) is no doubt a
standard way of counting. But it is doubtful that it is the correct way to count.
For one thing, in accepting a plural logic, we have already seen how we can refer to
more than one object at once. Notice that LBC only uses singular quantification. There is no
way—in standard LBC—to refer to more than one object at once. And, as we have seen,
having plural terms that allow us to refer to more than one object at once gives us greater
expressive power. So having a method of counting that allows us to incorporate plural terms
and variables may also inherit greater expressive power. Second, LBC only allows the identity
predicate to be flanked by singular terms. This means that someone who endorses the view
that Composition is Identity cannot even express her view, if LBC is accepted. Notice that it
15
is one thing to think that a view is false; it is another to claim that such a view is incoherent
or inexpressible. Moreover, there is no need to resort to claiming that CI is inexpressible,
since by introducing a plural language, as we did in the previous section, we will thereby have
many of the tools we need to express the CI position.
Our plural language already contains a singular/plural hybrid two-place identity
predicate, „=h‟, that takes either plurals or singulars as argument places—i.e.,
and
=h , where
can be either plural or singular terms. This plural language also allows the
concatenation of singular terms into plural terms, with the use of commas (e.g., “x, y, z”).
Using the hybrid identity predicate, we can re-interpret (1) to yield (1h), where, again, „P‟
stands for the predicate is in front of you:
(1h) x y z (Px & Py & Pz & x ≠hy & x ≠hz & y ≠hz)
And we can likewise re-interpret (2) to yield (2h):
(2h) x y z w (Px & Py & Pz & Pw & x
& z ≠h w)
h
y & x ≠h z & y
h
z & x ≠h w & y ≠h w
The CI theorist doesn‟t think that your hand, h, is identical any one of its parts; she thinks
that h is identical to all of the parts, taken together. We can now represent this using the hybrid
identity predicate and concatenated plural terms as follows:
(24) x y z w (Px & Py & Pz & Pw & x
& z ≠h w & w =h x,y,z)
h
y & x ≠h z & y
h
z & x ≠h w & y ≠h w
Since the singular identity relation is just a special case of the hybrid identity relation, we
should think of (24) as involving the singular non-identity statements of first-order logic
together with the hybrid identity statement that a defender of CI would endorse—viz., that
the parts of your hand (taken together) are identical to the whole hand, or w =h x,y,z. (24)
importantly includes the concatenated plural term “x,y,z”, as well as the singular/plural
16
identity predicate, “=h”. Such statements will be essential in understanding what I call Plural
Counting.
Yet, one might wonder, how are we supposed to interpret (24) as far as counting is
concerned? How many things are in front of you, if we grant the truth of (24)? More
pointedly: just how, exactly, if Plural Counting utilizes sentences such as (24), is Plural
Counting supposed to yield a count?
We will be better able to answer these questions if we examine one more kind of
counting: Relative Counting. Relative Counting (RC) claims that we cannot determine how
many things there are until we have been given a sortal or concept or kind under which to
count by. This view of counting is suggested by Frege in The Foundations of Arithmetic where
he claims:
The Illiad, for example, can be thought of as one poem, or as twenty-four Books, or
as some large Number of verses; and a pile of cards can be thought of as one pack or
as fifty-two cards (§22). One pair of boots can be thought of as two boots (§25).23
What seems to be suggested here is that we can think of thing(s) in various different ways—
e.g., as cards, decks, complete sets of suits, etc.—and depending on these various ways of
thinking about thing(s), we can yield different numbers or counts in answer to the question
how many? One way to interpret this: there are multiple modes or senses a denotation or
reference can have. So, for example, in the way that „Samuel Clemmons‟ and „Mark Twain‟
are two different senses for the same individual, so, too, would „52 cards‟ and „1 deck‟ be
different senses for the same “portion of reality” in front of you. No one of these numerical
23
Emphasis Frege‟s. In §46, Frege continues,
“…it will help to consider number in the context of a judgment that brings out its
ordinary use. If, in looking at the same external phenomenon, I can say with equal truth „This is a
copse‟ and „These are five trees‟, or „Here are four companies‟ and „Here are 500 men‟, then what
changes here is neither the individual nor the whole, the aggregate, but rather my terminology. But
that is only the sign of the replacement of one concept by another. This suggests…that a statement of
number contains an assertion about a concept.”
17
senses is privileged, and so there is no unique, non-sortalized answer to the question how
many things are in front of you?24
In this way, then, it is an ill-formed question to ask how many things there are.
Rather, we need to ask how many Fs or Gs are there, where F and G stand in for specific
sortals, concepts, or kinds. According to this view, since one can only take a count relative to
these sortals, concepts, or kinds, but never a count tout court, this view is called Relative
Counting.
I am leaving the exact details of Relative Counting intentionally vague, since I can
imagine many variations on the Fregean theme suggested above. All that matters for my
purposes, however, is that a theory of counting qualifies as Relative Counting if it claims (i)
that there cannot be a unique numerical answer (e.g., „52‟) to the question how many things are
there?, and (ii) that there can only be a unique numerical answer to questions that include a
legitimate sortal, concept, or kind term (e.g., „how many cards are there?‟).
Perhaps if an answer to non-relativized questions such as how many things are there? or
how many? is demanded, a Relative Counter could give an answer such as: “well, there are
many molecules, and one hand, and the many molecules are identical to the one hand,” etc.
The relativity implicit in the question can be flagged in the answer by having various
numbers of things there are depending on the sortal, together with an inclusion of all of the
hybrid identity claims that hold between the various relevant sortals.
That we can never give a unique numerical value to un-relativized questions such as
how many? is certainly intuitive. Imagine: Someone asks you to count all of the things in your
room right now. Clearly, you would be confused. You might ask, “Do you mean how many
24
This is just one interpretation of Frege; I acknowledge that there are others. For my purposes, it is not
important whether I have read Frege correctly or not. I am interested in relative counting as it is suggested
above insofar as it can help support the Strong Composition Thesis; it is of no importance here that Frege
might not have actually endorsed the idea himself.
18
things in general?” And suppose the person nods. You are to count all of the things in your
room. Whatever objects count as things in your ontology, and are in your room, you have to
count them up, one by one. This isn‟t just a difficult task, says the Relative Counter, it is
impossible. And it is not impossible because, say, there are trillions and trillions of molecules
and it would take you longer to count them than you have years on this earth. It is
impossible because, on reflection, we always count with a sortal, concept, or kind in mind.
Once it has been pointed out to us, for example, that the thing(s) in front of us are
cards, a deck of cards, sets of suits, etc., we are then seemingly unable to give a unique
numerical value to the question how many?. We cannot simply count up all the cards (fiftytwo) and then count up all the decks (one) and then count up all the complete sets of suits
(four), etc., and then add all of these up to get a unique numerical value (e.g., fifty-seven).
For intuitively it is just plain wrong that there are fifty-seven things in front of us! It‟s just one
deck of cards, after all! So the error must have been in thinking that there is a unique
numerical answer to un-relativized how many? questions, or in thinking that such a unique
answer could be acquired by getting the sum total of the number of Fs, plus the number of
Gs, etc. That we sometimes do give answers to unqualified how many? questions can be
explained, perhaps, by the fact that the sortals we are interested in are often implicit or
pragmatically understood. But a bit of reflection reveals that we seem to always have some
sortal or concept or kind in mind when we answer a seemingly unrelativized counting
question. Thus, RC is appealing because, on reflection, that‟s how it seems we do in fact
count.
Despite its intuitiveness, however, I have several worries about RC, which I do not
have the space to address adequately at this time.25 So let me address just one here, briefly.
25
I take this point up extensively in Chapter 2 of my thesis.
19
RC seems to prohibit us from using well-accepted inferences of first order logic. Typically,
we are allowed to infer (26) from (25):
(25) Bottles of beer are in the fridge.
(26) Some things are in the fridge.
We should always be able to infer from a statement about particulars—i.e., that beer is in the
fridge—something more general—i.e., that something is in the fridge. Yet if RC is correct,
then it seems that as soon as numerical quantifiers or determiners are introduced, our ability
to make seemingly acceptable inferences is somehow blocked. For the relative counter will
not want to infer (28) from (27):
(27) There are (at least) two bottles of beer in the fridge.
(28) There are (at least) two things in the fridge.
Yet such an inference is certainly legitimate (where „B‟ stands for the predicate is a bottle of
beer, „F‟ stands for the predicate is in the fridge):
(P1) x y (Bx & By & Fx & Fy & x
(P2) Ba & Bb & Fa & Fb & a b
(P3) Fa & Fb & a b
(C) x y (Fx & Fy & x y)
y)
Premise (sentence (29))
Instantiation
&Elimination
Introduction
If the relative counter allows the above inference, then this would undermine her claim that
all counting is relative, since (C)—i.e., (28)—is an unrelativized count statement. The
predicate “in the fridge” is not modifying the count in any way. And because “thing in the
fridge” does not discern bulky produce from imperceptible molecules, such a predicate will
not count as a legitimate sortal. Indeed, (28) always follows from (27). Sentence (27) is
adequately captured by P1 above. P1 contains the sortal “bottle of beer” via the predicate
“is a bottle of beer”, which is represented by “B”. By the rules Instantiation, &Elimination,
and Introduction, we can then get from (27) to (28), independent of any context or sortals. Yet if
the Relative Counter prohibits the above inference, then it seems that she will either have to
20
prohibit such inferences across the board (i.e., disallowing it even when counting predicates are
not involved), or she will only disallow it when counting predicates are involved. But the
former seems implausible, and the latter ad hoc.26 Best to leave all talk of sortals and kinds
out of it, and see if there is a better alternative.
And, indeed, I think there is. I think that Plural Counting can do everything that RC
can do, without even having to bother with the worry I‟ve raised above. But what is Plural
Counting?
Plural Counting will borrow a bit from each of LBC and RC. From LBC, a plural
counter will embrace the intuitive idea behind it—namely, that we generate our count
statements by quantifying over the objects in our domain, together with the identity and
non-identity claims that are true of these objects. Unlike LBC, however, Plural Counting
recognizes that we can quantify over objects plurally rather than merely singly, and allows
that the identity relation can hold between many and one things. The only way in which LBC
will be helpful for the Plural Counter is when it comes to counting the single variables that
represent singular items in the domain.
As an example, suppose you have a quarter in your pocket, which is made up of the
head side of the coin and the tail side of the coin. The defender of CI maintains that the coin
is not identical to any one of the sides of the coin; the coin is not identical to the head side,
nor is it identical to the tail side. Rather, the coin identical to both of the sides, taken together.
She can express this by sentence (29), where „P‟ stands for the predicate is in your pocket:
(29) x y z(Px & Py & Pz & x ≠hy & x ≠hz & y ≠hz & z =h x,y)
There are certainly moves to be made on behalf of the Relative Counter here. And I have my responses
(which I discuss in Chapter 2 of my dissertation). For the purposes of this paper, however, it is enough if I can
show that Plural Counting can do everything that Relative Counting can do. If my objection against Relative
Counting here does not seem conclusive, then no matter. It is my defense of CI that is my primary concern,
and that can be accomplished with either Plural Counting or Relative Counting.
26
21
The distinguishing identity formula that falls out of (29) is:
x y z(z =h x,y). Notice that we
can take such a statement and count—i.e., Logic Book Count—all of the variables on either
side of the identity predicate.27 In the first case we get a count of one, and on the other we get
a count of two. (It is important to note that, in this particular example, we never get a count
of three.)
In this way, then, the Plural Counter is utilizing our method of LBC, but only at the
level of variables (which represent objects in our ontology). We still have yet to show how a
count of variables could yield a count simpliciter of objects in our domain. For this, the Plural
Counter borrows a technique used by the Relative Counter. The Plural Counter embraces
the intuitive procedure of allowing complex answers to questions such as how many?
For example, we can logic book count all of the variables on either side the hybrid
identity statement in (29), and produce a count such as: “there is one thing and two things,
and the one thing is identical to the two things.” The Plural Counter grants (in this case) that
there is at least one thing, and also that there is at most two things. But she also endorses an
identity claim that cannot be ignored in our count. Thus, similar to the relative counter, she will deny
that there is a flat-out, singular numerical value. Rather, she will claim that there is one of
something, and two of some other things, but that in addition, the one thing is identical to
the two things. Thus, her answer to the question how many? will reflect this, and (in this case)
will be: there is one thing and two things, and the one thing is identical to the two things.
One way to do this: imagine that all of the variables on the left-hand side of the symbol „=h‟ are one domain,
and that the variables on the right-hand side of the hybrid identity symbol are another domain. So then let us
Logic Book Count all of the variables on first one side, and then the other, using „VL‟ and „VR‟ for “is a lefthand variable” and “is a right-hand variable” respectively:
Left-hand-side Domain:
x (VLx & x y(VLx & VLy  x=y))
Right-hand-side Domain: x y (VRx & VRy & x ≠ y) & x y z(VRx & VRy & VRz 
(z=y) v (z=x))
Now, due to the simplicity of the example, counting the variables is a relatively uncomplicated matter. But as
the upper-bound limit of the number of things increase, then so too would the hybrid identity claims that hold
between these things and some others. But then we can just Logic Book Count all of the variables on first the
left-hand side of the domain, and then the right, just as we have in the above example. While the statements
may get more complicated, the strategy remains the same. So this is how LBC is helpful in counting variables
(as opposed to material objects).
27
22
According to a Plural Counter, there is rarely a single numerical answer such as „one‟ or „two‟
in answer to the question how many? Rather, such questions have complex answers that
involve our hybrid identity claims. There are two things before us (e.g., two sides of the
coin), and there is one thing in front of us (e.g., the coin) and the two are identical to the
one. Or: there are many things before us (e.g., some molecules), and there is one thing
before us (e.g., a hand), and the many are identical to the one. It is never the case that we add
up (e.g.) the many and the one to get a sum total of all of the things that there are. If it were,
then we would be ignoring the relevant hybrid identity claim and double counting things in
our ontology.
To show that this is not unintuitive, imagine that you are tempted by the Relative
Counter and understand what she means when she says something like “one deck is identical
to fifty-two cards.” You understand how it could be that one thing is identical to many when
you affix sortals such as „deck‟ and „cards‟ to numerical predicates such as „one‟ and „fiftytwo‟, respectively. But you also understand existential generalization: if there is one deck in
front of you, then there is one thing in front of you; if there are fifty-two cards in front of
you, then there are fifty-two things in front of you. Likewise, if you understand the identity
statement “one deck is identical to fifty-two cards,” then you understand, via existential
generalization, how one thing can be identical to fifty-two things.
In this way, Plural Counting can borrow the Relative Counter‟s strategy of having
non-brute count answers that include the hybrid identity claims she accepts. Yet the Plural
Counter has the added advantage of avoiding a reliance on sortals or kinds, or any of the
complications that a reliance on sortals or kinds brings.28
A Plural Counter is motivated by the desire to be able to coherently express a
statement such as “one hand is identical to many millions of molecules” or “one coin is
28
Which, again, is a point I take up in some detail in Chapter 2 of my thesis.
23
identical to two sides (or faces) of the coin.” When we count up all of the things in the
world, the Plural Counter insists, we want to be able to express that sometimes, many things
can be identical to one (two boots are identical to one pair, fifty-two cards are identical to
one deck, six beers are one six-pack, etc.). Yet in order to makes sense of such locutions—in
order to symbolize them adequately—we need (at least) two things: the two-place hybrid
identity predicate, =h, and concatenated plural terms such as “x,y”. Thus we can generate
sentences such as (30)
(30)
x y z (z =h x,y)
which is read as: “there is something, z, that is identical to something(s), x, y, collectively.”
4. Responding to the Objections
Indiscernibility of Identicals Objection
The first argument against Composition as Identity (CI) appeals to the
Indiscernibility of Identicals. “The parts are many (and not one), while the whole is one (and
not many). Therefore, the parts cannot be identical to the whole; Composition as Identity is
false.”29
I think we can keep our ordinary intuitions about identity, and in particular the
Indiscernibility of Identicals, without this leading to a rejection of CI. This is because I think
that our methods of counting are more complicated than may have first been supposed. It is
initially assumed that we can take a brute count of something and have, e.g., the parts be
many, and the whole be one. But what we have failed to realize is that counts are never taken
simpliciter. And in this respect, Relative Counting has got it right. But we can improve upon
Relative Counting. We can abstract away from sortals, concepts, and kinds and generate
sortal-free counts.
29
Again, see Lewis (1991), McKay (2006), et. al.
24
Plural Counting shows that we almost always have a complicated answer to
questions such as how many?. If so, then it is not true that, given Leibniz‟s Law, the parts are
many and not one, or that the whole is one and not many. Rather, it is this: we have some
thing(s) in front of us. Whatever it(they) is(are) is many, and one, and the many are identical
to the one. The claim the many are identical to the one is assumed false by the claim the things in
front of us are many and not one. Put in terms of Relative Counting, the parts are many parts and
the whole is one whole; but there is not both the many parts and the one whole, added
together. Rather, there are many parts, and one whole, and the parts are identical to the
whole. Or—and now we can just leave the sortals out of it—there are many things and one
thing, and the one is identical to the many. Including the (all-important) hybrid identity claim
in our count statements shows how it is that there is many and one, where the many is
identical to the one. But it is never the case that there are many and not many, or one and not
one. Hence, there is no outright contradiction.
So my quick answer to the first objection against CI is that Plural Counting will show
us that such an objection is misguided. First, it assumes a method of counting (LBC) that the
CI theorist only accepts at the level of variables. Second, once Plural Counting is adopted,
the objection fails to go through. It is no objection to say to the CI theorist: “But wait! The
parts are many and the whole is one, so the parts can‟t be identical to the whole!” For the CI
theorist will say: “Our counts of things are more complicated than we had first supposed.
There are many things in front of us (the parts) and there is one (the whole), and the many
are hybrid identical to the one. So we have something(s) in front of us that is many and one,
without this resulting in an outright contradiction.” Thus, our CI defender can easily address
the first objection.
Yet perhaps one might push the objection as follows: Look. The above argument
against CI has been carefully chosen. Just because Lewis, McKay, et. al., appeal to the number
25
of parts (many) and the number of wholes (one) as the distinguishing difference-making
feature between the parts and the whole, this need not be the only difference-making
feature. Invoking plural counting will only address worries that concentrate on counting up the
number of parts and wholes. Yet many other arguments can be crafted using the
Indiscernibility of Identicals that do not rely on counting.
Suppose we have a cat, Nacho, over here and a mug, Mug, over there. And let us
consider the sum of Mug and Nacho—call this Muggo. That is, let us think of Muggo as the
whole that has Nacho and Mug as parts. 30 Now place Nacho and Mug next to each other.
Ok, so here‟s something that is now true of the parts: they are beside one another. But it is not
true that the whole, Muggo, is beside one another. It is not even grammatical to say that
Muggo is beside one another! So here is a property that the parts have that the whole does
not—being beside each other. So, the parts are not identical to the whole. Thus, CI is false.
My answer to this sort of worry using the Indiscernibility of Identicals will not rely
on plural counting. But it will rely on a robust plural language. For example, I take it that a
statement such as (31):
(31) Nacho and Mug are beside one another.
is best represented by (32), where „n‟ stands for Nacho, „m‟ stands for Mug, „Bxy‟ stands for
the predicate x is beside y, where „Bxy‟ expresses a symmetric relation:
(32) Bnm
The being beside each other relation is a two-place, distributive relation. It applies to Nacho and
Mug individually, albeit in a two-place fashion. Still, it is not the case that Nacho and Mug
I intend for the „sum‟ here to be somewhat ontologically innocuous, like mereological sums. However, it is
probably unnecessary to bring in mereology—i.e., the study of parts and wholes—in this paper, although many
may see that my work here is motivated by, and applies to, ontological issues in mereology. If it will simplify
matters, just read „sum of Nacho and Mug‟ as the grouping together of Nacho and Mug. Or take „Muggo‟ as a
name that refers plurally to Nacho and Mug, collectively, as we have been doing with other plural terms
throughout this paper.
30
26
are, taken together, beside….what? being beside is undeniably a two-place relation. Some
object(s) taken plurally cannot have this attribute simpliciter. That is, after all, what the “one
another” in (31) is doing—it is an ellipsis that indicates which two things instantiate a twoplace (symmetric) relation. Contrast this, for example, with (33)
(33) Nacho and an army of ants are surrounding the building.
Supposing that surrounding a building is not something that a cat and an army of ants can do
(each) by themselves31, (33) is best represented by (34), where „n‟ again stands for Nacho, „b‟
stands for the building, „a‟ stands for the army of ants, „S
‟ stand for the predicate
are
surrounding , where „ ‟ and „ ‟ can be either singular or plural terms:
(34) S(n,a)b
In this case, „S
‟ is representing a two-place relation that holds between some things
(referred to plurally) and another thing (singular).32 The being beside one another relation is best
represented by „Sxy‟, where this is a symmetrical relation (and so it entails „Pyx‟). If this is
right, then given our example above, it is true that objects such as Mug and Nacho are beside
one another, but this just amounts to “Mug is beside Nacho” and “Nacho is beside Mug.”
But then „being beside something‟ is a feature that they each have. It is a distributive (albeit twoplace) relation. In order to refute CI, however, it needs to be shown that the whole has a
Let‟s assume that the army of ants is just too small to surround the building without some help from at least
one substantially larger co-conspirator; and that my cat isn‟t fat enough to surround a building all by himself.
31
Note: surrounding need not be a relation that holds between many things and one; it is not a metaphysical
limitation on how many things can hold this relation to however many other things. One mereological sum, for
example, can surround the metalheads (in which case, we have one thing surrounding many); one piece of
string can surround the flagpole (in which case, we have one thing surrounding one thing), etc. So this isn‟t a
metaphysical point about what kind of things and how many can hold a certain relation to certain other kinds
of singular or plural things, etc. Rather, this is a point about when a relation is distributive or collective, not
whether the relation holds between thing(s) singularly or plurally.
32
27
feature that the parts (taken collectively) do not have, or vice versa. (31) is not an example of
this, since it involves a distributive relation, not a collective one.
To be more explicit, the following claims are endorsed by CI, given the Mug and
Nacho example, where the symbolization is the same as above, and „u‟ stands for Muggo:
(35) Bnm
(36) u =h n,m
Given that „Bxy‟ is a two-place relation, we cannot simply swap “u” in for “n” and “m” in
(35) using something like the Substitutivity of Identicals and the truth of (36). First, this is
because the plural term “n,m” in (36) is distinct from the singular terms “n” and “m” in (35).
In (35) the terms “n” and “m” are referring to Nacho and Mug individually, and it is claimed
that each of them has a certain relation to the other. In (36), the plural term “n,m” is
referring to Nacho and Mug collectively, and claiming that they, taken together, are identical to
Muggo. So the Substitutivity of Identicals doesn‟t apply here. Muggo is not identical to
Nacho, nor to Mug. Muggo is identical to Nacho and Mug. Second, the predicate in (35) is
irreducibly two-placed, and so we cannot merely swap one singular term in for two. In this
way, the above objection is conflating the distinction between distributive and collective
predicates (or relations).
In this way, then, it will be no objection to the CI theorist that there are some parts
of certain wholes that bear a relation to each other (yet which the whole—or the parts taken
together—does not bear to itself). It is no objection to claim that there is property that the
parts (distributively) have that the whole does not, such as being beside each other, to prove CI
false. This is because certain relations that the parts bear to each other are still distributive,
even though they may not seem that way (because the relation is multi-placed, for example).
28
Importantly, the distinction between distributive and collective predicates is going to be
transparent in the plural language I have proposed in this paper. 33
In sum, then, the arguments against CI using the Indiscernibility of Identicals seem
to make the mistake of either (i) using only LBC (as opposed to Plural Counting), to
generate a seeming counterexample to CI, or (ii) confusing the application of distributive and
collective predicates or relations. Both of these mistakes are remedied when we adopt a rich
plural language, and allow our counts to include hybrid identity claims, as Plural Counting
does.
Counting Objection
Van Inwagen‟s objection against CI involves imagining that there is one big parcel of
land, divided neatly into six smaller-parcel parts. He then argues that given how we usually
quantify over objects in the world—i.e., with a singular existential quantifier—there will be
no way to quantify over the large parcel of land without adding to the number of things in
our ontology. And if we are adding to the things in our ontology when we accept a large
parcel of land, then this large parcel must be something distinct from and in addition to the
smaller parcel of land. And what goes for parcels of land, goes for hands and the molecules
that make them up, coins and the sides that make them up, and for wholes and their parts in
general. Thus, wholes are distinct from their parts in general, and hence, CI is false. Or so
goes the argument, anyway.
33
And, as I show in my thesis, having such a language will tout other advantages as well. For example, contrary
to traditional taxonomy, the Fallacy of Composition and the Fallacy of Division can be shown to be formal
rather than informal fallacies, involving structural ambiguities that are apparent when we have a robust plural
language that can accurately distinguish between collective and distributive properties and relations. We can see
this as well in one of the example I brought up at the beginning of this paper. I had said that one might think
that CI is false because “the parts of your hand are relatively small…while your whole hand is not.” We can see
now that the parts of your hand are distributively relatively small, but not collectively. And so this is a fallacious
argument against CI, which can be easily refuted. These points (and related ones) are discussed extensively in
Chapter 3 of my thesis.
29
But we can now see that an appeal to Plural Counting will deflect this objection. For
the Plural Counter does not count by singular existential statements and singular identity and
non-identity claims. She counts by plural terms and variables, uses a hybrid identity
predicate, „=h‟, and Logic Book counts at the level of variables. So, for instance, in van
Inwagen‟s (or Lewis‟s or Baxter‟s) land parcel example, the Plural Counter would
existentially quantify over all six parcels of land, and the one large parcel to yield:
(37)
x y z w v u t(x ≠hy & x ≠hz & x ≠hw &…& t =hx,y,z,w,v,u)34
(37) expresses exactly what the CI theorist thinks is going on with the parcels of land. There
are six smaller parcels, x, y, z, w, v, u, there is one larger parcel, t, and the six are identical to
the one, which is expressed by “t =hx,y,z,w,v,u”. In order to count these objects, the Plural
Counter takes as her domain the variables on right hand side of this identity claim, and then
the variables on left hand side of this identity claim, and then logic book counts all of the
variables in these domains, separately.35, 36 In the first case we get a count of one, and in the
second, six. So in answer to the question how many parcels of land are there?, the Plural Counter
claims: “there is one parcel of land, and there are six, and the one is identical to the six.”
Thus, contrary to what Van Inwagen claims, we never get a count of seven.
The “…” represents all of the many non-identity statements. (37) written out in full is the
cumbersome:
x y z w v u t(x ≠hy & x ≠hz & x ≠hw & x ≠h v & x ≠hu & x ≠ht & y ≠hz & y ≠hw & y
≠hv & y ≠hu & y ≠h t & z ≠hw & z ≠hv & z ≠hu & z ≠ht & w ≠hv & w ≠h u & w ≠ht & v
≠hu & v ≠ht & u ≠ht & t =hx,y,z,w,v,u).
34
35
It will look this:
Left-hand-side Domain:
Right-hand-side Domain:
x (VLx & x y(VLx & VLy  x =h y))
x y z w v u (VRx & VRy &VRz & VRw VRv & VRu & x ≠ y & x ≠hz &
x ≠hw & x ≠h v & x ≠hu & y ≠hz & y ≠hw & y ≠hv & y ≠hu & z ≠hw & z
≠hv & z ≠hu & w ≠hv & w ≠h u & v ≠hu) & x y z w v u s(VRx &
VRy & VRz & VRw & VRv VRu & VRs  (s =h y) v (s =h x) v (s =h z) v (s
=h w) v (s =h v) v (s =h u))
We are counting variables because the variables exactly correspond with objects in our ontology. As such,
counts of variables will accurately represent counts of things.
36
30
And if the story gets more complicated—if we are counting all of the sub-parcels of
land, where a sub-parcel is any (non-overlapping) pair of any of the smaller parcels, for
example—then the Plural Counter can express this as well. For she will claim that there are
some identity claims such as x,y =h t and z,w =h r, etc. Thus, there would be (e.g.) six things
and three things and one thing, and the six things would be identical to the three things, and
the three things would be identical to the one thing, etc. But in such a case, we still never get
a count of more than six (or a count of less than one). And so the Plural Counter can avoid
van Inwagen‟s objection.
To reinforce the point, let‟s go back to the original argument, (A)-(D).
(A) Sentence (2) is true.37
(B) If (A), then there are four things in front of us.
(C) If there are four things in front of us, then the hand is not identical to its parts.
(D) So, the hand is not identical to its parts.
Van Inwagen endorses (A)-(D), which is how he gets the conclusion that CI is false. I reject
premise (B). I claim that we may think that B is true because of LBC. But LBC is not the
correct way to count; Plural Counting (PC) is. But if PC is true, then B is false. And so the
argument against CI doesn‟t work.
Now one might argue against me as follows: “Van Inwagen assumes that LBC is
correct in order to endorse argument (A)-(D) and show that CI is false. You claim that LBC
is incorrect, and argue for PC instead. But in your argument against LBC, you claim that it
does not accurately reflect the world because (e.g.) there are some many-one identity
relations that it does not account for. In other words, you seem to be saying that LBC is
incorrect because of its failure to capture the world if CI is true. So it seems that you are
presupposing CI in your defense of a theory of counting, which will ultimately factor in an
37
Where, again, sentence (2) is:
(2)
x y z w (Px & Py & Pz & Pw & x
31
y&x≠z&y
z & x ≠ w & y ≠ w & z ≠ w)
argument for the falsity of a premise (premise (B)), which is an argument against CI. So aren‟t
you just begging the question against yourself?”
I have two responses. First, Plural Counting (PC) does not presume anything about
the composition relation. Sure, one reason to endorse PC may be that there are some manyone identity relations out in the world, and that our theory of counting should reflect this.
But this does not say that these many-one relations are the composition relation, or that
composition is identity, or anything about parts and wholes, etc. PC is silent about which
many-one identity statements are true, if any. Indeed, PC doesn‟t even claim that there are in
fact any many-one relations at all. But if there are, then PC can represent them and LBC
cannot. Much like adopting a plural logic, PC‟s advantage is mostly about its expressive
power, not its commitment to or presupposition of a particular view of ontology. Second,
while PC does not presuppose CI, van Inwagen‟s assumption that LBC is the correct way to
count does presuppose the falsity of CI. So his argument against CI has no traction, and CI
can meet the Counting Objection.
5. Concluding Thoughts and Promissory Notes
So now we have seen how CI can be defended against two objections, one involving
the Indiscernibility of Identicals, and the other involving existential quantification and
counting. In addition to deflecting these objections, however, there are (at least) two
advantages of CI. First, it solves our initial Hand Puzzle.
HAND
Your hand, h, exactly occupies r at t.
PARTS
The parts, ps, exactly occupy r at t.
NCP-N
For any material object(s), , and any material object(s), , if
exactly occupies region r at t and exactly occupies r at t, then
32
=h .
By HANDS and PARTS, there is no place that h is that the ps are not, and there is no place
that the ps are that h isn‟t. The ps and h exactly occupy r at t. Yet assuming that your hand
and its parts are material objects, then we are in seemingly violation of NCP, since we also
tend to assume that the ps
h
h. So it seems we are in a dilemma, and must give up one of the
three very intuitive principles. Yet CI easily solves this dilemma because we give up the
assumption that the ps
ID
h
h. That is, we maintain that the ps =h h. We endorse ID:
the ps =h h
In this way, we get to maintain all three principles, without violating NCP-N. Our puzzle
about your hand and its parts, and about parts and wholes in general, has a very
straightforward solution if we accept CI. And now that we have seen how a CI theorist can
defend herself against two (of the most intuitive) objections, there should be few
reservations about doing so.
CI also aids in issues concerning Mereology—the study of parts and wholes—and
whether or not Mereology can be accepted as an ontologically innocent system. Obviously,
by accepting CI, mereology is ontologically innocent, since identity—aside from being a
relation that can hold between singular and plural items—is the classic relation that we have
all been brought up to believe. If
is identical to
singular or plural terms), then in committing to
(where
and
are neutral here between
one has already committed to . So if
mereological sums are identical to their parts, then a commitment to sums would not be a
commitment over and above the parts. But such a discussion will have to wait for another
time.38
There is much more to be said here, of course. In particular, there are many more
objections to CI than I have dealt with here. One objection to CI that is especially poignant,
38
And is taken up extensively throughout my thesis.
33
and that I mentioned at the outset of the paper, involves modality. In our token example
throughout this paper, I have imagined that your hand is composed (and identical to!)
millions of molecules. But one might think that there is a clear difference-making feature
between these things—namely, the modal properties of your hand and the molecules. Your
hand, I take it, could survive losing a few molecules here and there, but the molecules could
not. Moreover, your hand could not survive being thrown in a blender, but the molecules
composing your hand presumably could (assuming that we are dealing with a very precise
and discerning blender).39 Yet if your hand has a property that the molecules do not—viz.,
could survive a loss of molecules—and if the molecules have a property that your hand does
not—viz., could survive being thrown in a blender—then by the Indiscernibility of
Identicals, your hand is not identical the molecules.
This objection is related to the charge that CI entails Mereological Essentialism—the
view that all objects have their parts necessarily. That objects have their parts essentially is
typically regarded as so radically false that it forms a direct modus tollens on CI, if it is true
that CI entails Mereological Essentialism.
I take up both of these objections in the same way, and with special care. And I use
these sorts of objections to fuel my metaphysics of objects, where I claim that objects are
trans-space, trans-time, and trans-world mereological sums, or lumps.40 It is an extension of
the 4-dimensional view of objects41: I claim that objects are trans-world sums that have a
part in each world (in which these parts exist). Many objects overlap in various worlds, but it
is never the case that more than one object completely overlaps another, such that „they‟ are
in all of the same worlds, at all of the same times, and all of the same places, etc. So, for
39
See Wiggins (1968).
40
This terminology and idea of objects is borrowed from Weatherson (ms).
41
See Sider (2003), et. al.
34
example, the modal claim „the lump of clay could have been molded into an ashtray‟ is made
true by the trans-world, trans-spatio-temporal object having a part in a possible world that is
an ashtray. A claim such as „the statue could not have been molded into an ashtray‟ is made
true by a trans-world, trans-spatio-temporal object having no parts in any world that are an
ashtray. Embracing this metaphysics of objects will complement CI nicely, and will show
how both Mereological Essentialism and the modal objection against CI can be easily
handled. It should perhaps be mentioned that one need not adopt my metaphysics of objects
in order to address the modal objection. But it is the solution I favor. Unfortunately,
discussion of these points will have to wait for another time.
References
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-------- 1988b. Many-One Identity Philosophical Papers XVII No. 3.
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35
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-------- "The Logic and Meaning of Plurals, Part I", Journal of Philosophical Logic, 34 (5-6)
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--------- "The Logic and Meaning of Plurals, Part II", Journal of Philosophical Logic, 35 (3)
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36