Identity and Constitution

In this section we will consider a famous metaphysical puzzle. Suppose there is a
clay statue on an otherwise empty table before us. Let’s imagine that it’s a statue of
a horse and rider. Then it’s true that there is a statue on the table, and it’s also true
that there is a piece of clay on the table. Are the statue and the piece of clay just
one physical thing, or are they two? Most people would say they are just one thing
– a thing that (like just about anything) admits of different descriptions. So let’s call
this the ‘Common Sense View.’ It certainly has initial appeal. But we’ll soon see that
there is an important principle that appears to undermine the Common Sense View.
To make matters worse, this principle draws its support directly from common

The claim that the statue and the piece of clay are just one thing is sometimes
expressed by saying that the statue is (or: is identical with) the piece of clay, or that
the statue and the piece of clay are identical. Now, although the words ‘is’ and
‘identical’ are in fact often used in other ways, in this context they are being used
to express a special relation that philosophers call identity. It is a matter of definition
that the identity relation always holds between any entity and itself (no matter what
that entity may be like), and also never holds between any entity and any other
entity. (Mathematicians often express identity by using the familiar symbol ‘=’,
which they call the identity or equality sign.) It should be clear that if a thing x is
identical to a thing y in the present sense of the term, so that the identity relation
holds between x and y, then x and y are just one thing (and never two).

The principle that threatens the Common Sense View concerns the identity
relation. It’s called the principle of the indiscernibility of identicals, and we may
state it as follows: (For any entities x and y) if x is identical to y, then whatever is
true of x is also true of y and whatever is true of y is true of x. Alternatively: if x
is identical to y, then for any property P, x has P if and only if y has P. And yet more
compactly: if x is identical to y, then x and y have exactly the same properties. It is
important to notice that this principle should not be confused with the ‘converse’
principle, that if x and y have exactly the same properties, then x is identical to y.
The principle of the indiscernibility of identicals is sometimes called Leibniz’ Law,
and we will abbreviate it by ‘LL’.

Although most people never have occasion to assert or even to reflect on LL in
this very general form, it is nevertheless deeply embedded in our commonsense way
of thinking about the world, and we rely on it tacitly in everyday life. To see this,
notice first that LL is equivalent to the claim that if a thing x has a property that a
thing y does not have, then x is not identical with y. It is this form of the principle
that often underlies everyday reasoning. For example, suppose you and a friend see
someone in the middle distance who looks a lot like another friend, Sue. You ask,
‘Is that Sue?’ and your friend replies, ‘It can’t be – Sue never wears a hat.’ Your
friend, in effect, has reasoned that since the person you now see has a property –
wearing a hat – that (he believes) Sue doesn’t have, it follows that the person you
see isn’t Sue. Given that he’s right about Sue’s dress habits, this reasoning is
impeccable. In essence, it is an application of LL in a specific case.

This form of reasoning may often have very important consequences. For
example, it might be used to exonerate a defendant in a murder case if his blood
type or eye colour (etc.) didn’t fit with facts presumed to have been established
about the real killer. Given that the various factual claims were beyond dispute,
the reasoning would again be impeccable, and really just a matter of common

We may also argue from common sense directly in favour of LL, as follows.
Suppose the principle is in fact false. Then some x and y are such that x is identical
with y, but x has some property, say P, that y doesn’t have. But since x is identical
with y, x and y are just one thing. Therefore, there is some one thing that both has
and does not have the property P. This is a contradiction. So the assumption that
the principle is false leads directly to a contradiction. It follows that the principle
is true after all. There is nothing even remotely suspicious about this argument. It
is evidently an informal but entirely rigorous proof of the principle – one that relies
only on simple logical steps accepted by common sense.

We have now seen that LL often underlies everyday reasoning, and we’ve also
seen that it admits of a simple proof. We have ample reason to accept the principle
in what follows.

Back to the statue and the piece of clay. Common sense says they’re just one
thing, they’re identical. Common sense also endorses LL. If we accept both, then it
follows that the statue and the piece of clay have exactly the same properties. But
consider this. Suppose the piece of clay had been shaped like a sphere instead of
shaped like a horse and rider. The sculptor might have taken the original blob
of clay and simply worked it into a sphere instead of undertaking the more difficult
task of producing a statue of a horse and rider. Certainly this might have happened.
It just didn’t. But if it had happened, that would be a situation in which the piece
of clay existed but the statue of the horse and rider did not. The piece of clay would
have had a shape different from its actual shape, but that is entirely beside the point,
which is simply that the piece of clay would have existed and the statue would not
have existed.

It therefore appears that something is true about the piece of clay – namely, that
it might have been spherical – that isn’t true about the statue. To put it a little
differently, the piece of clay apparently has a property – being possibly spherical
– that the statue does not have. But then LL should deliver the conclusion that
the piece of clay and the statue aren’t identical after all, that they are two things,
not one.

It now looks very much like our ordinary ways of thinking about ordinary objects
are paradoxical. It looks like we have to choose between Leibniz’ Law and the idea
that the statue and the piece of clay are identical. But we apparently have a clear
proof of Leibniz’ Law. Should we therefore resolve the puzzle by declaring that here
is a case in which there are two physical objects in exactly the same spatiotemporal
location, despite our original thinking?

Let’s take a little time to consider some problems that arise for the two-object
view. We begin with a dilemma. Are the ‘corresponding’ smaller parts of the statue
(S) and the piece of clay (C) identical with each other or not? Neither answer will
seem satisfactory.

Suppose we say yes. Then, for example, any individual molecule that is part of
S is also part of C, and vice versa. (This is already very odd, for how can a single
bunch of specific molecules comprise two different things at the same time?) Also,
the rather large part of S that we would describe as ‘the horse’ is a part of C. Now
let’s fix on a very large part of S, say all of it except for a specific surface molecule,
M. Call this part L. L, of course, is just an aggregation of molecules, related to each
other in a certain specific way. Since L isn’t all of either S or C, it is also a part of
C. Notice that although L isn’t a statue (since there’s only one statue in the general
region) it certainly might have been a statue. The molecule M might have adhered
to the sculptor’s hand instead of becoming an integral part of the statue. Now,
suppose this had happened. Then there would have been a statue, S*, and a piece
of clay, C*. According to the view under discussion, S* would not be identical with C* though their smaller corresponding parts would be identical. Now, as we have
described matters, L is identical with S* and hence not identical with C*. But we
could have described the very same situation in terms that would have led to the
opposite conclusion – that L was identical with C*, not S*! (‘Consider all of C
except for a certain single molecule. It might have been that this molecule adhered
to the sculptor’s hand and hence was absent from the piece of clay he was working
with . . .’)

The idea that the smaller parts of S and C are identical thus has the disconcerting
consequence that if things had been only slightly different, then a certain specific
aggregation of molecules, L, would have existed in precisely the same region as a
certain statue and a certain piece of clay, but would not clearly have been identical
with either one of them. (Yet all of its smaller parts would have been parts of both
the statue and the piece of clay.) We can only conclude that either it is an insoluble
mystery whether L is identical with S* or with C*, or else L is yet a third physical
object sharing their exact spatiotemporal location. (At this point the three-object
view might actually seem the least objectionable since it dodges the mystery of
choosing between S* and C*.)

Two things should be noticed about our predicament. First, it is completely
general. So there is also an aggregation of molecules A in exactly the same region
as both S and C, and we have no good reason to think A is identical with S or
identical with C. Second, if we avoid this mystery by adopting the three-object view,
we will only produce more trouble. For the same sort of reasoning will apply with
respect to the third object, and there will be no limit to the number of physical
objects in exactly the same location. Surely this is too much to believe.

So now suppose we say no to our question, adopting the view that the corresponding
smaller parts of S and C are not identical with each other. For example,
let M be an arbitrary molecule that is part of S. Then there is a distinct molecule,
say M*, in exactly the same location, that is a part of C but not a part of S. And
similar conclusions hold for any parts of S and C, including ones that are much
larger or smaller than individual molecules. For example, there is a horse-shaped
part of C that isn’t part of S, and vice-versa, and so on. This is a very strange view,
and one that may not be attractive even to those initially inclined toward the idea
that S and C are two different entities.

Here is one apparent consequence of the view. Consider a molecule M that is a
part of S. Suppose we detach it from the rest of S (say in a laboratory) so that it’s
now just an isolated molecule. Then another molecule has automatically been
detached from the rest of C as well. (Of course it isn’t M, since M was never a part
of C.) It seems to follow that we now have two molecules in exactly the same place,
even though neither of them is attached to any other molecules in the way they
formerly were. Now, suppose we arrange to ‘vaporise’ the rest of S (and so the rest of C as well). Although it might be consistent, it is very odd to think that this would
have any effect on the molecule M, or on its inseparable partner. For example, it
would be odd to conclude that when the rest of S goes out of existence, so does M,
but that a like molecule comes into existence precisely in the region vacated by M
at precisely the time of annihilation. It’s far more natural to think that M and its
partner still exist. But now consider isolated molecules that have never been bonded
to other physical entities. Do they too come in inseparable pairs? If the histories of
M and its partner are irrelevant, then so should be the histories of eternally isolated
molecules, and indeed of any entities of any description at all. So it seems that the
present view is best seen not just as a view about special entities like statues and
pieces of clay, but about all physical entities in the universe: they just happen come
in inseparable pairs. Many will find this unacceptable.

But nearly all will find it unacceptable if we have to go beyond pairs. The
Common Sense View was that there is just one thing under consideration, a thing
that we might describe as a statue or as a piece of clay. The trouble started when
we noticed that when we describe it one way, some of its features seem to differ
from ones it has when we describe it the other way. But obviously these are just
two among many possible ways of describing this seemingly single entity. And some
of these other descriptions may raise the same sorts of considerations that led us
to take the two-object view seriously, thus raising the spectre of three, four, and
even infinitely many objects in the same place at the same time. If the two-object
view points inevitably in this direction, then we would do well to try to save the
Common Sense View after all. And apparently it does point in this direction, as we
will now see.

Imagine that the object in question was in fact the prototype for a famous Frederic
Remington bronze. Then not only are there a statue and a piece of clay before us,
there is also a statue-prototype. Originally, it seemed that the piece of clay could
have been spherical but the statue could not have been spherical, and this (along
with LL) led us to consider the two-object view. But now imagine that Remington
had instead made the statue with no intention of making a model for a future bronze
work, and in fact never produced any similar work in bronze. That would be a
situation in which the statue existed, but the statue-prototype did not. So something
seems to be true of the statue – it could have existed even though no prototype
existed – that isn’t true of the prototype. So it looks like the prototype and the statue
are two different things. Since it is also clear that the piece of clay has properties
that the prototype lacks (possibly being spherical will do), the original style of
reasoning evidently yields the conclusion that we have three objects, not two.
Discovering a fourth, a fifth, and so on, is now just a matter of finding new descriptions
to support further applications of the two-object reasoning with respect to
each of the objects already conceded to exist.

There is yet a further reason to think that the present view will result in far too
many things. For consider any aggregation of molecules filling the entire location
of S (and C), but some of which are parts of S and some of which are parts of C.
It’s very hard to deny that it’s a physical object. But it can’t be either S or C, since
some of its parts aren’t parts of S and others of its parts aren’t parts of C. Either we
have a new physical object (and hence many, many more, corresponding to the
many different ways of combining parts of S with parts of C), or else we have to
give up the initially compelling idea that any bunch of molecules that are stuck
together in a given such region constitute a legitimate physical object.

We have mentioned only a few of the difficulties that confront the ‘two’-object
view, but they are clearly very serious. So let’s try to save the Common Sense View.
Of course we want to save it without abandoning Leibniz’ Law, but also – if we can
– without abandoning our ordinary beliefs about statues and pieces of clay. Thus
we seek a solution according to which both

(1) The piece of clay could have been spherical.


(2) The statue could not have been spherical.

remain true even though the piece of clay and the statue are just one thing.

Our proposed solution depends on Plato’s bold idea that the various properties
that things have – like being a statue or being made of clay – are ‘abstract’ entities,
entities that genuinely exist but are nevertheless not physical in nature. (Plato’s own
‘theory of forms’ is the cornerstone of his metaphysics and epistemology.)

For now, let’s assume in this Platonic spirit that for an entity ‘to have a property’
is for it to stand in a special relation to a specific such abstract entity. This relation
is usually called instantiation (or exemplification). On this assumption, for example,
for a thing to be a statue is nothing more nor less than for it to instantiate the property
of being a statue. To put it a little differently, the fact that a thing is a statue, and
the fact that it instantiates the property of being a statue, are just one fact.

On this view, to pick a different example, when a thing, x, is red, the basic
metaphysical nature of this state of affairs is relational: it consists in x’s bearing the
instantiation relation to the property being red (or redness). Thus when we say ‘x
is red’, we are really saying that x instantiates being red, that is, that the instantiation
relation holds between x and the property of being red. So the proposition that x
is red is really about two entities, not just one. It’s about the physical entity x and
also about an abstract entity, the property of being red. This metaphysical picture
plays a crucial role in solving the puzzle of the statue.

Because any ordinary entity instantiates a multitude of properties at any given
time, there is no difficulty in supposing that a single thing instantiates both the
property of being the statue and the property of being the piece of clay. The way is
now clear for the possibility of saving the Common Sense View. But first we need
a further ingredient.

Suppose you are holding a red apple. Then you are holding something that has
numerous properties, including being red, being an apple, and being physical. On
our normal conception, these properties are related to each other in certain basic
ways. For example, everyone will agree that something could be red without being
an apple, and vice versa. And also that nothing could be red (or an apple) without
being physical. And, of course, something could be physical though neither red nor
an apple. What is the metaphysical source of these indisputable connections between
these very familiar properties? The Platonic view offers an answer.

In a nutshell, the answer is that these connections are relations between the
properties that depend only on how the properties are in themselves – their so-called
intrinsic properties – and not on any instances that the properties may or may not
have. Relations that hold between any entities strictly as a result of their intrinsic
properties are also sometimes called intrinsic. To illustrate the general idea, suppose
a given jockey is shorter than a given basketball player. Then that is a relation that
holds between these individuals solely as a result of their respective heights. So it
depends only on the athletes’ intrinsic properties, and it is therefore an intrinsic

Analogously, according to the Platonic view, if a property like being red or being
an apple cannot be instantiated by a thing unless that thing also instantiates being
physical, then this is a relation between the properties that depends only on how they
are in themselves. It doesn’t depend on there being any red things or any physical
things even though some things do happen to be both red and physical. The property
of being a unicorn in fact has no instances, but it is nevertheless true that it could
not be instantiated by a thing unless that thing also instantiated being physical. If
there were any unicorns then they would be physical in nature. So being a unicorn
is related to being physical in the same way that being red and being an apple are.
According to the Platonic view, again, to be red is just to instantiate being red.
Now, in keeping with the present idea, instantiating being red automatically involves
instantiating being physical. It’s no accident, it’s something about being red in itself
and being physical in itself that makes it impossible for anything to instantiate the
former property without also instantiating the latter. When this relation holds
between properties x and y, we say that x entails y. So being red, being an apple,
and being a unicorn are three examples of properties that entail being physical.
Obviously, being physical doesn’t entail any of the other three, so entailment
between properties may hold in one direction without holding in the other.

Closely related to entailment is a relation between properties we may call
compatibility. Intuitively, two properties are compatible if there could be something
that instantiated both. Thus being red and being an apple are compatible (though
neither entails the other). And so are being a unicorn and being physical. Being
red and being green, on the other hand, are incompatible, as are being round and
being square. It isn’t hard to see that the compatibility of being red with being an
apple is just the failure of being red to entail not being an apple. Similarly, the
incompatibility of being red and being green consists in the fact that being red entails
not being green. So entailment and compatibility are, in a certain simple way, two
sides of the same coin.

We are now able to state the Platonic solution to the puzzle of the statue and the
piece of clay. Sentences (1) and (2) above do not just assert of a certain physical
object, respectively, that it could, and could not, have been spherical. It is precisely
the assumption that sentences like these merely assert that a given thing could have
been one way or another that leads to all the trouble. The Platonic conception
provides the possibility of a better interpretation. On this interpretation, sentence
(1) is not only about the thing in question and the property of being spherical, but
also about the property of being the piece of clay. Sentence (2), on the other hand,
is not at all about the property of being the piece of clay, but instead is about the
property of being the statue. Here, then, are the suggested Platonic readings of (1)
and (2):

(1*) Something instantiates being the piece of clay, and being the piece of clay is
compatible with being spherical;
(2*) Something instantiates being the statue, and being the statue is incompatible
with being spherical.

Each sentence thus asserts the existence of a physical object with a certain property,
and then goes on to assert either the compatibility or the incompatibility of the
relevant property with being spherical. Intuitively, when we say, ‘The piece of clay
could have been spherical’, we are saying that something could have been spherical
while being the piece of clay. And when we say, ‘The statue could not have been
spherical’, we are saying that nothing could have been spherical while being the
statue. So, if the sculptor had used the clay to make a sphere, then a certain physical
object would have instantiated the properties being the piece of clay and being
spherical, but of course not the property of being the statue.
This solution conforms to the Common Sense View by allowing that the piece
of clay and the statue are just one thing. And our intuition that ‘the piece of clay’
has a property that ‘the statue’ lacks is also accommodated, but with no danger
of some one thing both having and lacking a certain property. The ‘having’ and ‘lacking’ of our intuition are accounted for by the compatibility and incompatibility
of different properties with being spherical. These different properties, of course,
are precisely the ones that are expressed by the different descriptions of what is in
fact one single thing.
It is worth noticing that the present solution and the multiple-object solution
have something important in common. They are both ‘ontological’ positions. They
attempt to resolve a seeming conflict in our ordinary thought by claiming, in effect,
that there are more things in existence than perhaps meet the eye. The multipleobject
solution depends on accepting a multitude of physical objects that happen
to occupy the exact same regions that other physical objects occupy. And the
Platonic solution depends on accepting a multitude of abstract objects that aren’t
located in physical space at all. We now turn to some questions about existence
itself, the fundamental concept of metaphysics and ontology.