The Identity of Indiscernibles (and the Bundle
Theory)
Jean-David Lafrance
FQRSC Postdoctoral Fellow, Corpus Christi College, University of Oxford
Fonds Québécois de la Recherche sur la Société et la Culture (FQRSC)
Introduction
The Principle of Identity of Indiscernibles
I
(PII)
Necessarily, for any x, any y , and any property U, if (x has U
if and only if y has U), then x is identical to y .
Introduction
The Principle of Identity of Indiscernibles
I
(PII)
Necessarily, for any x, any y , and any property U, if (x has U
if and only if y has U), then x is identical to y .
I
There cannot be distinct indiscernibles.
Introduction
The Principle of Identity of Indiscernibles
I
(PII)
Necessarily, for any x, any y , and any property U, if (x has U
if and only if y has U), then x is identical to y .
I
I
There cannot be distinct indiscernibles.
The strength of (PII) depends on the properties included in
the domain of property quantification.
Introduction
The Principle of Identity of Indiscernibles
I
(PII)
Necessarily, for any x, any y , and any property U, if (x has U
if and only if y has U), then x is identical to y .
I
There cannot be distinct indiscernibles.
I
The strength of (PII) depends on the properties included in
the domain of property quantification.
I
We can distinguish: non-relational properties, relational
properties, and relations.
Introduction
The Principle of Identity of Indiscernibles
I
(PII)
Necessarily, for any x, any y , and any property U, if (x has U
if and only if y has U), then x is identical to y .
I
There cannot be distinct indiscernibles.
I
The strength of (PII) depends on the properties included in
the domain of property quantification.
I
We can distinguish: non-relational properties, relational
properties, and relations.
I
I take non-relational properties to be included in the domain
of property quantification.
Introduction
The Principle of Identity of Indiscernibles
I
The strength of (PII) differ in yet another way.
I
A trope theorist could insist that tropes are instantaneous,
and could claim that no trope is located at more than one
place at any one time.
Introduction
The Principle of Identity of Indiscernibles
I
The strength of (PII) differ in yet another way.
I
A trope theorist could insist that tropes are instantaneous,
and could claim that no trope is located at more than one
place at any one time.
I
So a version of (PII) congenial to this trope-theoretic view
ranges over instantaneous tropes.
Introduction
The Principle of Identity of Indiscernibles
I
The strength of (PII) differ in yet another way.
I
A trope theorist could insist that tropes are instantaneous,
and could claim that no trope is located at more than one
place at any one time.
I
So a version of (PII) congenial to this trope-theoretic view
ranges over instantaneous tropes.
I
So (PII) makes a rather weak claim: it is false just in case
there could be distinct indiscernibles all located at the very
same region of space at the same time.
Introduction
The Principle of Identity of Indiscernibles
I
A stronger version of (PII) quantifies over immanent
universals.
Introduction
The Principle of Identity of Indiscernibles
I
I
A stronger version of (PII) quantifies over immanent
universals.
Immanent universals are:
1. ‘in’ the objects that instantiate them;
2. can be at many places at once.
Introduction
The Principle of Identity of Indiscernibles
I
I
A stronger version of (PII) quantifies over immanent
universals.
Immanent universals are:
1. ‘in’ the objects that instantiate them;
2. can be at many places at once.
I
Then, (PII) is false provided that it is possible for distinct
objects, regardless of their locations in space, to instantiate all
and only the same universals.
Introduction
The Principle of Identity of Indiscernibles
I
In this talk, I am interested in the strongest version of (PII) in
both of the ways introduced. I suppose that non-relational
universals are the only properties included in the domain of
property quantification.
Introduction
The Principle of Identity of Indiscernibles
I
In this talk, I am interested in the strongest version of (PII) in
both of the ways introduced. I suppose that non-relational
universals are the only properties included in the domain of
property quantification.
I
(PII) offers what I will call, for the purpose of the discussion,
sufficient identity conditions for objects. It specifies sufficient
conditions for the identity of objects in terms of universals.
Introduction
The Principle of Identity of Indiscernibles
I
In this talk, I am interested in the strongest version of (PII) in
both of the ways introduced. I suppose that non-relational
universals are the only properties included in the domain of
property quantification.
I
(PII) offers what I will call, for the purpose of the discussion,
sufficient identity conditions for objects. It specifies sufficient
conditions for the identity of objects in terms of universals.
I
This version of the principle is implausible if not paradoxical.
Introduction
(PII) paradoxical?
I
I
I
Suppose a series of worlds, the first one of which is inhabited
only by two spheres exactly alike in all respects except that
one is much bigger than the other, and the last one of which
is inhabited by what we would regard as two indiscernible
spheres.
Any world that lies between the first and last in the series
differ from the preceding one in that the difference in the size
of the spheres is minutely closer to 0.
An advocate of (PII) needs to say that there is a sharp cut-off
in the series—that a minute difference in the size of one
sphere has consequences for identity (i.e., for the number of
objects inhabiting a world). And I take this to be absurd.
Introduction
(PII) paradoxical?
I
In fact (PII) sanctions all the worlds in the series, except the
last. But the cut-off need not be in between the penultimate
and the ultimate worlds.
Introduction
(PII) paradoxical?
I
In fact (PII) sanctions all the worlds in the series, except the
last. But the cut-off need not be in between the penultimate
and the ultimate worlds.
I
It is consistent with the truth of (PII) that one object, a
sphere, could be located at distinct, spatially separated
regions of space, while having different properties at each of
these regions.
Introduction
(PII) paradoxical?
I
In fact (PII) sanctions all the worlds in the series, except the
last. But the cut-off need not be in between the penultimate
and the ultimate worlds.
I
It is consistent with the truth of (PII) that one object, a
sphere, could be located at distinct, spatially separated
regions of space, while having different properties at each of
these regions.
I
As we will shortly see, I accept the possibility of multilocation.
And so I accept this move on behalf of (PII)’s advocates.
Introduction
Plan for the talk
I
We have reasons to deny the truth of (PII).
Introduction
Plan for the talk
I
We have reasons to deny the truth of (PII).
I
Without (PII), can we still give sufficient identity conditions
for objects in terms of universals?
I
And if we reject (PII), does that mean that identity facts are
brute—is it the case that “[w]hat individuates the objects is
simply the objects themselves?” (Della Rocca, 2005, 484)?
Introduction
Plan for the talk
I
We have reasons to deny the truth of (PII).
I
Without (PII), can we still give sufficient identity conditions
for objects in terms of universals?
I
And if we reject (PII), does that mean that identity facts are
brute—is it the case that “[w]hat individuates the objects is
simply the objects themselves?” (Della Rocca, 2005, 484)?
I
In this talk, I give a positive answer to the first question, and I
give a negative, qualified answer to the last.
Introduction
Plan for the talk
I
We have reasons to deny the truth of (PII).
I
Without (PII), can we still give sufficient identity conditions
for objects in terms of universals?
I
And if we reject (PII), does that mean that identity facts are
brute—is it the case that “[w]hat individuates the objects is
simply the objects themselves?” (Della Rocca, 2005, 484)?
I
In this talk, I give a positive answer to the first question, and I
give a negative, qualified answer to the last.
So
I
I
theories of material objects that purport to give sufficient
identity conditions for objects in terms of universals need not
hold on to (PII);
Introduction
Plan for the talk
I
We have reasons to deny the truth of (PII).
I
Without (PII), can we still give sufficient identity conditions
for objects in terms of universals?
I
And if we reject (PII), does that mean that identity facts are
brute—is it the case that “[w]hat individuates the objects is
simply the objects themselves?” (Della Rocca, 2005, 484)?
I
In this talk, I give a positive answer to the first question, and I
give a negative, qualified answer to the last.
So
I
I
I
theories of material objects that purport to give sufficient
identity conditions for objects in terms of universals need not
hold on to (PII);
a rejection of (PII) need not amount to taking identity facts as
brute.
Methodology and Prospect of a Solution
Introducing the Bundle Theory: First advantage
I
I will conduct my analysis in the context of the bundle of
universals theory of material objects. The theory says, roughly,
that objects are mereological fusions of universals. Two
considerations speak in favor of assuming the bundle theory.
Methodology and Prospect of a Solution
Introducing the Bundle Theory: First advantage
I
I
I will conduct my analysis in the context of the bundle of
universals theory of material objects. The theory says, roughly,
that objects are mereological fusions of universals. Two
considerations speak in favor of assuming the bundle theory.
First. In order to reach my conclusions, it suffices only to
show that there is at least one theory of material objects that
gives sufficient identity conditions for objects in terms of
universals, but that does not subscribe to (PII). The bundle
theory, at least the version I shall offer, does just that.
I
Furthermore, the bundle theory is often said to entail (PII). So
it’s an important point in its own right to show that some
version of the theory doesn’t—though it will only be a
corollary of the arguments I give here.
Methodology
Introducing the Bundle Theory: Second advantage
I
Second. Let us say that worlds w1 and w2 qualitatively differ
just in case the spatial distributions of universals in each of
these worlds is not the same. Otherwise, they are qualitatively
similar.
Methodology
Introducing the Bundle Theory: Second advantage
I
Second. Let us say that worlds w1 and w2 qualitatively differ
just in case the spatial distributions of universals in each of
these worlds is not the same. Otherwise, they are qualitatively
similar.
I
Example. Suppose that only a red cube and a blue square
inhabit w1 , and that only a blue cube and a red square are in
w2 . Then, both worlds qualitatively differ. In w1 , redness is
where cubic is and blueness where squareness is. In w2 ,
blueness is with cubic, while redness with squareness.
Methodology
Introducing the Bundle Theory: Second advantage
I
I
I
An inquiry into (PII) often appeals to qualitative differences
between worlds (or the lack thereof).
But in this talk, I restrict my attention to a certain
argumentative strategy deployed on behalf both of the
advocates of (PII) and of the bundle theorists.
Suppose that multilocation is possible, i.e., that an object can
be located at more than one region of space at once (where
the regions are spatially separated).
Methodology
Introducing the Bundle Theory: Second advantage
I
I
I
I
An inquiry into (PII) often appeals to qualitative differences
between worlds (or the lack thereof).
But in this talk, I restrict my attention to a certain
argumentative strategy deployed on behalf both of the
advocates of (PII) and of the bundle theorists.
Suppose that multilocation is possible, i.e., that an object can
be located at more than one region of space at once (where
the regions are spatially separated).
We will be concerned with two worlds. One, w1 , contains
distinct indiscernibles and is a counterexample to (PII). The
other, w2 , is inhabited only by a single, but multilocated,
object. Both worlds are qualitatively similar. So, (PII)’s
advocates can say that w1 does not represent a genuine
possibility.
Methodology
Introducing the Bundle Theory: Second advantage
I
I
I
I
I
An inquiry into (PII) often appeals to qualitative differences
between worlds (or the lack thereof).
But in this talk, I restrict my attention to a certain
argumentative strategy deployed on behalf both of the
advocates of (PII) and of the bundle theorists.
Suppose that multilocation is possible, i.e., that an object can
be located at more than one region of space at once (where
the regions are spatially separated).
We will be concerned with two worlds. One, w1 , contains
distinct indiscernibles and is a counterexample to (PII). The
other, w2 , is inhabited only by a single, but multilocated,
object. Both worlds are qualitatively similar. So, (PII)’s
advocates can say that w1 does not represent a genuine
possibility.
So it appears that multilocation comes at the rescue of (PII)’s
advocates.
Methodology
Introducing the Bundle Theory: Second advantage
I
The second advantage of focusing on the bundle theory is
that it specifies a spatial distribution of universals for any
world (in which there are material objects). It tells us what
universals are in a world, and where. So we can investigate
qualitative differences between worlds from what the theory
says about the objects that inhabit these worlds.
Methodology
Introducing the Bundle Theory: Second advantage
I
The second advantage of focusing on the bundle theory is
that it specifies a spatial distribution of universals for any
world (in which there are material objects). It tells us what
universals are in a world, and where. So we can investigate
qualitative differences between worlds from what the theory
says about the objects that inhabit these worlds.
I
For any two worlds such that one is inhabited (only) by
distinct indiscernibles while the other contains only one object
(multilocated or not), the bundle theory entails that there is a
qualitative difference between these. Or so I will argue.
The Bundle Theory
I
Let’s first introduce the relation of exact location that holds
between an object (such as me) and a region of space that it
completely fills and that has the same shape and size.
The Bundle Theory
I
Let’s first introduce the relation of exact location that holds
between an object (such as me) and a region of space that it
completely fills and that has the same shape and size.
I
My left hand is exactly located at the region it completely fills
and that shares my hand’s shape and size.
The Bundle Theory
I
Let’s first introduce the relation of exact location that holds
between an object (such as me) and a region of space that it
completely fills and that has the same shape and size.
I
My left hand is exactly located at the region it completely fills
and that shares my hand’s shape and size.
I
Of course, my left hand is a part of me. Where is it a part of
me? I stipulate that it is at its exact location. More generally,
parts are part of their wholes at the parts’ exact locations.
And the part-whole relation here is taken to be a ternary one.
In the example just introduced, it holds between material
objects and a region of space.
The Bundle Theory
I
Let’s first introduce the relation of exact location that holds
between an object (such as me) and a region of space that it
completely fills and that has the same shape and size.
I
My left hand is exactly located at the region it completely fills
and that shares my hand’s shape and size.
I
Of course, my left hand is a part of me. Where is it a part of
me? I stipulate that it is at its exact location. More generally,
parts are part of their wholes at the parts’ exact locations.
And the part-whole relation here is taken to be a ternary one.
In the example just introduced, it holds between material
objects and a region of space.
I
The bundle theory takes this ternary part-whole relation to
hold between a universal, a bundle of universals (an object),
and a region of space. More precisely,
An object o is identical to the fusion at its exact location of
(immanent) universals.
The Bundle Theory
I
My hand is identical to the fusion f1 of universals S1 and m1
at region r1 . And I am identical to the fusion f2 of universals
S2 and m2 at region r2 .
The Bundle Theory
I
My hand is identical to the fusion f1 of universals S1 and m1
at region r1 . And I am identical to the fusion f2 of universals
S2 and m2 at region r2 .
I
Universal S1 is a part of f1 (my hand) at r1 . And S2 is a part
of f2 (me) at r2 .
The Bundle Theory
I
So the bundle theory, as I introduce it, explains the two
characteristics of universals:
I
I
I
they are ‘in’ the objects since they are literally parts of them;
they can be at many places at once in virtue of being parts of
objects that are exactly located at distinct places.
Note finally that the location of a universal is derivative of the
exact location of the objects of which it is a part.
The Bundle Theory
Instantiation
I
Typically, the bundle theory claims that objects are identical
to bundles of the universals they instantiate. But I did not
appeal to the instantiation relation when introducing the
bundle theory. Instead, the instantiation relation can and
should be captured within the theory.
The Bundle Theory
Instantiation
I
I
Typically, the bundle theory claims that objects are identical
to bundles of the universals they instantiate. But I did not
appeal to the instantiation relation when introducing the
bundle theory. Instead, the instantiation relation can and
should be captured within the theory.
Instantiation
Object o instantiates universal U iff
1. o is identical to a fusion of compatible universals at a region r
(at o’s exact location);
2. U is a part at r of the fusion.
The Bundle Theory
Instantiation
I
I
Typically, the bundle theory claims that objects are identical
to bundles of the universals they instantiate. But I did not
appeal to the instantiation relation when introducing the
bundle theory. Instead, the instantiation relation can and
should be captured within the theory.
Instantiation
Object o instantiates universal U iff
1. o is identical to a fusion of compatible universals at a region r
(at o’s exact location);
2. U is a part at r of the fusion.
I
I
The first clause is there because I do not wish to restrict the
operation of fusion. And no objects instantiate incompatible
universals (at any one time, I should add, though I leave times
out of the picture for the purpose of the discussion).
And so, being a fusion and being a material object are two
distinct properties. For the purpose of this talk, I assume that
no fusion of universals that fails to instantiate a universal is a
The Bundle Theory
Instantiation
Instantiation
Object o instantiates universal U iff
1. o is identical to a fusion of compatible universals at a region r ;
2. U is a part at r of the fusion.
The Bundle Theory
Resulting View
I
We start with the objects and the universals they instantiate.
I
Then, we identify the objects with fusions of universals at the
exact locations supplied by the objects.
I
We end up with universals distributed over space, and that are
located at some select regions of space.
I
I do not suppose that space is a bundle of properties.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
The typical counterexample to (PII) is introduced by Max
Black (Black, 1952). It is metaphysically possible that two
indiscernible spheres are at some distance from one another in
a symmetric universe. Because the universe is symmetric, the
spheres share both their relational and non-relational
properties, and they enter into the same relations. Therefore,
(PII) is false. (I focus here on non-relational universals.)
I
What ensures that there are more than one sphere is that they
are spatially separated.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
I
John Hawthorne (O’Leary-Hawthorne, 1995) notes that the
bundle of universals theory of objects allows for the possibility
of the multilocation of objects in space.
An object is multilocated in space just in case it is exactly
located (to use my terminology) at more than one spatially
separated region of space (at once).
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
I
I
I
John Hawthorne (O’Leary-Hawthorne, 1995) notes that the
bundle of universals theory of objects allows for the possibility
of the multilocation of objects in space.
An object is multilocated in space just in case it is exactly
located (to use my terminology) at more than one spatially
separated region of space (at once).
In Hawthorne’s view, this is so because universals, and so
bundles of them, can be at more than one region of space at
once.
But on my view, the location of universals is derivative of that
of the objects of which they are parts. And so Hawthorne’s
reason to think that the bundle theory allows for multilocation
is not a good reason to think that the version of the bundle
theory I offered also allows for the possibility of multilocation.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
Yet, I accept the possibility of multilocation for material
objects. There is nothing blatantly incoherent with such a
possibility, and it is relevant to our discussion of (PII).
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
So we have two worlds:
1. Black’s world (wB ): two spatially separated, indiscernibles
spheres.
2. Hawthorne’s world (wH ): one sphere at some distance from
itself. (That is, the sphere is multilocated in space.)
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
So we have two worlds:
1. Black’s world (wB ): two spatially separated, indiscernibles
spheres.
2. Hawthorne’s world (wH ): one sphere at some distance from
itself. (That is, the sphere is multilocated in space.)
I
wH is meant to mimic wB . So we cannot say, in the context
of our discussion, that the sphere of wB are distinguishable,
after all, since one is exactly located at r1 (say) and the other
at r2 . It does not help to adjudicate which of the two worlds
represent a genuine possibility. For the sphere of wH enter
into these relations of exact location as well.
I
In any case, I focus on non-relational universals.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
So the dialectic so far: spatial dispersion serves as a criterion
for the distinctness of the spheres only if there is no
multilocation. But since multilocation is a possibility (for
Hawthorne it is allowed by the bundle theory, for me it is a
brute possibility), one can claim that spatial dispersion is no
indication that the spheres are distinct in Black’s world (wH ).
So (PII)’s advocate can (and should) describe Black’s world
as one in which there is only one sphere spatially separated
from itself. And (PII) is unaffected.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Necessarily, for any x, any y , and any property U, if (x has U if
and only if y has U), then x is identical to y .
I
So the dialectic so far: spatial dispersion serves as a criterion
for the distinctness of the spheres only if there is no
multilocation. But since multilocation is a possibility (for
Hawthorne it is allowed by the bundle theory, for me it is a
brute possibility), one can claim that spatial dispersion is no
indication that the spheres are distinct in Black’s world (wH ).
So (PII)’s advocate can (and should) describe Black’s world
as one in which there is only one sphere spatially separated
from itself. And (PII) is unaffected.
I
In other words, since both wB and wH are qualitatively
similar, an advocate of (PII) should deny that wB represents a
genuine possibility.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
The version of the bundle theory I introduced points to a
difference between the two worlds. In both wB and wH , there
is a fusion of the sphere(s). Since the spheres are distinct in
wB , it would appear that the fusion of the spheres is a
scattered object, of which the spheres are proper parts. In wH ,
there is also a fusion of the sphere, but it is the sphere itself.
For the fusion of an object with itself is the object (i.e., any
object is a improper fusion of itself).
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
Of course, this difference between the worlds is relevant to our
discussion of (PII) only if it translates into a qualitative
difference between the two worlds.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
Of course, this difference between the worlds is relevant to our
discussion of (PII) only if it translates into a qualitative
difference between the two worlds.
I
For suppose that that the fusion of the spheres in wB is not a
propertied object. Then there is no bundle of universals that
capture the fusion of the distinct spheres. And so, given the
bundle theory, it is simply not an object.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
Of course, this difference between the worlds is relevant to our
discussion of (PII) only if it translates into a qualitative
difference between the two worlds.
I
For suppose that that the fusion of the spheres in wB is not a
propertied object. Then there is no bundle of universals that
capture the fusion of the distinct spheres. And so, given the
bundle theory, it is simply not an object.
I
Alternatively, suppose that the fusion of the spheres in wB
instantiates the same properties as the spheres themselves
(i.e., suppose that it is indiscernible from the spheres). Then
an advocate of (PII) will argue that the fusion is not distinct
from the spheres, which are for her also identical. She will
argue that wB is not, after all, a genuine possibility, but that
wH is.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
I will argue: it is an advantage of the bundle theory as I
introduced it that it entails that the fusion of the spheres in
wB has properties distinct from any one of the spheres. So
the theory pushes its advocates to recognize wB as a genuine
possibility. And, therefore, to deny (PII).
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Unrestricted Fusion Existence
For any xs, there is an f (the fusion) such that every
one of the xs is a part of f at some region r and
every part of f at some region s overlaps at a
subregion of s at least one of the xs.
I
Let a1 and a2 be the sphere(s) in both Black’s (wB ) or
Hawthorne’s (wH ) worlds. Let a1 be exactly located at r1 , and
a2 at r2 . And let us finally say that both a1 and a2 instantiate
only the universals U and Q.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Unrestricted Fusion Existence
For any xs, there is an f (the fusion) such that every
one of the xs is a part of f at some region r and
every part of f at some region s overlaps at a
subregion of s at least one of the xs.
I
I
Let a1 and a2 be the sphere(s) in both Black’s (wB ) or
Hawthorne’s (wH ) worlds. Let a1 be exactly located at r1 , and
a2 at r2 . And let us finally say that both a1 and a2 instantiate
only the universals U and Q.
So, in both worlds, the following are true.
1. a1 = fr 1 (U, Q)
2. a2 = fr 2 (U, Q)
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
1. a1 = fr 1 (U, Q)
2. a2 = fr 2 (U, Q)
I
In Hawthorne’s world: a1 = a2 = fr 1 (U, Q) = fr 2 (U, Q)
I
Since the spheres are identical, their fusion is the sphere itself.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
1. a1 = fr 1 (U, Q)
2. a2 = fr 2 (U, Q)
I
In Hawthorne’s world: a1 = a2 = fr 1 (U, Q) = fr 2 (U, Q)
I
Since the spheres are identical, their fusion is the sphere itself.
I
In Black’s world: The fusion of a1 and
a2 = fr [fr 1 (U, Q); fr 2 (U, Q)]
I
Since the spheres are distinct in Black’s world, their fusion, it
would seem, is distinct from them. And since the spheres are
exactly located at, respectively, r1 and r2 , then the fusion is
exactly located at the region r that is itself the fusion of r1
and r2 .
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
1. The fusion of a1 and a2 = fr [fr 1 (U, Q); fr 2 (U, Q)]
2. Instantiation
Object o instantiates universal U iff
2.1 for o to be identical to a fusion of compatible universals at a
region r ;
2.2 for U to be a part at r of the fusion.
I
The right-hand side of the identity sign is the fusion of the
distinct spheres, the existence of which is a consequence of
Unrestricted Fusion Existence.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
1. The fusion of a1 and a2 = fr [fr 1 (U, Q); fr 2 (U, Q)]
2. Instantiation
Object o instantiates universal U iff
2.1 for o to be identical to a fusion of compatible universals at a
region r ;
2.2 for U to be a part at r of the fusion.
I
The right-hand side of the identity sign is the fusion of the
distinct spheres, the existence of which is a consequence of
Unrestricted Fusion Existence.
I
But nothing says that the fusion is a propertied object. We
can only see that it is ‘something’ composed of universals.
Since none of the universals of which it is composed, i.e. U
and Q, is part of the fusion at its exact location, r , then it
does not follow from Instantiation that the fusion
instantiates them.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Instantiation
Object o instantiates universal U iff
1. for o to be identical to a fusion of compatible
universals at a region r ;
2. for U to be a part at r of the fusion.
I
I
Yet it follows from the bundle theory as I introduced it that
the fusion is a propertied object.
The fusion is exactly located at region r , which is the fusion
of r1 and r2 .
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
Instantiation
Object o instantiates universal U iff
1. for o to be identical to a fusion of compatible
universals at a region r ;
2. for U to be a part at r of the fusion.
I
I
I
I
Yet it follows from the bundle theory as I introduced it that
the fusion is a propertied object.
The fusion is exactly located at region r , which is the fusion
of r1 and r2 .
By definition, the exact location of an object is a region that
has the same shape and size as the object—it has the same
geometric (and topological, actually) properties as the object.
From Instantiation, therefore, it follows that the fusion of the
two spheres in Black’s world instantiates these universals of
shape and size.
The Principle of the Identity of Indiscernibles
In the Vicinity of Black’s World
I
So there is a qualitative difference between Black’s and
Hawthorne’s worlds. And so an advocate of the bundle theory
is committed to recognizing Black’s world as a genuine
possibility, and, therefore, committed to accepting it as a
counterexample to (PII).
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
A rejection of (PII) is thought by some (cf. (Della Rocca,
2005)) to commit one to taking identity facts as brute.
I
Suppose that there are 20 indiscernible spheres all exactly
located at the very same region of space. We suppose that
the spheres have all the same parts (whether spatiotemporal
or universals).
I
Clearly, the 20-sphere case should be ruled out by any theory
of material objects.
I
Della Rocca (who is not concerned with the bundle theory)
uses the case to argue that (PII)’s opponents can rule it out
only by seeking an explanation of the identity of the 20
spheres. And that goes against the spirit of their rejection of
(PII), which incites them to recognize identity facts as brute.
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
We can show that the bundle theory introduced above rules
the 20-sphere case out just as naturally as it accepts a
counterexample to (PII). But we need to be a little bit more
explicit about the mereology that lies at the core of the theory.
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
I
We can show that the bundle theory introduced above rules
the 20-sphere case out just as naturally as it accepts a
counterexample to (PII). But we need to be a little bit more
explicit about the mereology that lies at the core of the theory.
The mereology is close to Classical Extensional Mereology, but
it takes a ternary part-whole relation instead of a binary one.
The mereology contains the following three axioms.
1. Unrestricted Fusion Existence
For any xs, there is an f (the fusion) such that every one of the
xs is a part of f at some region r and every part of f at some
region s overlaps at a subregion of s at least one of the xs.
2. Transitivity
If x is a part of y at r and y is a part of z at r , then x is a
part of z at r .
3. Uniqueness of Fusion
If f1 and f2 fuse at the same region r the same xs, then f1 = f2 .
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
Uniqueness of Fusion says that any fusions at the same
region of the same things are identical. It provides sufficient
identity conditions (for objects) in terms of universals and
regions of space.
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
Uniqueness of Fusion says that any fusions at the same
region of the same things are identical. It provides sufficient
identity conditions (for objects) in terms of universals and
regions of space.
I
Together with Instantiation, it entails that there are no
distinct objects that instantiate all and only the same
universals and that are exactly located at the same region.
The Principle of the Identity of Indiscernibles
(PII) and Brute Identity Facts
I
Uniqueness of Fusion says that any fusions at the same
region of the same things are identical. It provides sufficient
identity conditions (for objects) in terms of universals and
regions of space.
I
Together with Instantiation, it entails that there are no
distinct objects that instantiate all and only the same
universals and that are exactly located at the same region.
I
So the 20-sphere case is taken care of.
The Principle of the Identity of Indiscernibles
Conclusion
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Uniqueness of Fusion and Instantiation leave open the
possibility that there could be distinct indiscernibles exactly
located at distinct regions—i.e., distinct fusions of the same
universals at distinct regions.
I
So (PII), even in its non-necessitated version, does not follow
from these two principles.
The Principle of the Identity of Indiscernibles
Conclusion
I
Uniqueness of Fusion and Instantiation leave open the
possibility that there could be distinct indiscernibles exactly
located at distinct regions—i.e., distinct fusions of the same
universals at distinct regions.
I
So (PII), even in its non-necessitated version, does not follow
from these two principles.
I
But, we find in Uniqueness of Fusion an ‘explanation’ (not a
complete one) of the identity of objects couched in terms of
universals and regions of space.
The Principle of the Identity of Indiscernibles
Conclusion
I
Uniqueness of Fusion and Instantiation leave open the
possibility that there could be distinct indiscernibles exactly
located at distinct regions—i.e., distinct fusions of the same
universals at distinct regions.
I
So (PII), even in its non-necessitated version, does not follow
from these two principles.
I
But, we find in Uniqueness of Fusion an ‘explanation’ (not a
complete one) of the identity of objects couched in terms of
universals and regions of space.
I
So, since Uniqueness of Fusion offers sufficient identity
conditions for objects, and since the bundle theory is
compatible with the falsity of (PII), it results that theories of
objects that offer sufficient identity conditions need not hold
on to (PII).
THANK YOU!
Black, M. (1952). The identity of indiscernibles. Mind, 61 (242),
153–164.
Della Rocca, M. (2005). Two spheres, twenty spheres, and the
identity of indiscernibles. Pacific Philosophical Quarterly , 86 ,
480–492.
Hacking, I. (1975). The identity of indiscernibles. The Journal of
Philosophy , 72 (9), 249–256.
O’Leary-Hawthorne, J. (1995). The bundle theory of substance
and the identity of indiscernibles. Analysis, 55 , 191–196.