Max Black, “The Identity of Indiscernibles”
The principle: if a and b are (numerically) distinct, then there must be at least one property that one of them has that the other doesn’t. (I.e., if they are distinct, they don’t share all their properties, or, alternately, if the do share all their properties, they are not distinct but identical.)
(For your enjoyment: The two characters, A and B, are arguing about two distinct entities, a and b. Question: Are A and B distinct?)
A says the principle is true, i.e., that if a and b are distinct, there must be some property that distinguishes them, that one has but the other lacks.
B says the principle is false, i.e., that it is possible for distinct things to be indistinguishable, for there to be no property that one has which the other lacks.
A’s first argument: assuming a and b are distinct, A tries to prove, by providing an example, there is a property a has that b lacks, specifically, the property of being identical to a. Since a has this property and b doesn’t (and since we have been speaking generally enough that a and b could stand for any two numerically distinct objects whatsoever), it follows, A claims, that distinct objects always have distinct (sets of) properties.
B’s strategy is attack the supposed property of being identical with a as a legitimate property. To say that a has the property of being identical with a, is to say nothing more than “a is a,” which is a “roundabout way of saying nothing.” It is a “useless tautology.”
A doesn’t really buy this, but tries another tack, closely related to the first. Rather than working with being identical with a, A changes strategies and considers being different from b. “a also has, and b does not have, the property of being different from b. So, once again, A claims, we have a property they fail to share, and so (again, since we have been speaking in terms that tell us nothing about a and b except that they are numerically distinct), we have shown that for any distinct objects, there must be some property they fail to share.
B doesn’t like this any better, and for essentially the same reason. What do we gain by learning that a has the property of being distinct from b, other than that a is not identical to b, which we already knew? Remember that what A is trying to show is that, given that a and b are distinct, there must be some property they don’t share. So telling us that a has this property tells us nothing more than if things are numerically distinct, they are numerically distinct, and this is trivially true. If this is all the principle of the identity of indiscernibles tells us, we can accept it, but it is uninteresting. For the principle to be philosophically interesting there must be “properties” other than identity and difference that distinct objects fail to have in common.
A’s second
argument uses relational properties
rather than the properties of identity
and difference. There is some interesting historical
background here regarding the relation (!) between relations and properties:
in our talk about them, one can be defined in terms of the other. In modern logic, properties are treated as 0 place relations. (Recall: being taller than is a 2 place relation,
being between … and ___ is a three
place relation, and so on. So, e.g., the
“property” of being green can be treated as a 0 place relation.) On the other hand, Leibniz denied the
existence of relations over and above
the existence of properties. He
claimed that there are, ultimately, no “extrinsic denominations,” i.e., that
there are no relations over and above the properties (“intrinsic
denominations”) of objects. That a is taller than b, according to Leibniz, is ultimately to be understood as a
property that a has (of being taller than b) and another
property that b has (of being shorter than a). Perhaps because of these historical
questions, A first asks B permission to construct and an
argument involving relational properties (such as the property that I have of being the father of Conrad). B
doesn’t object.
A: (Supposing that a and b are numerically distinct), “The only way that we can discover
that two different things exist is by finding out that one has a quality not
possessed by the other or else that one has a relational characteristic that
the other hasn’t. [But suppose they
share all the same properties.] …. The supposition that in such a case there
might really be two things would be unverifiable in principle. Hence it would
be meaningless.”
This
is really the core of A’s strategy
in all that follows. If we accept
verificationism as an account of meaning, it seems that the identity of
indiscernibles must follow. According to
this principle, the claim that a and b are numerically distinct is only
meaningful if there is, in principle, some experience that could verify this
claim, and that experience would have to involve evidence of some difference
between them, i.e., evidence that there is some property one has that the other
lacks. Without the possibility of such an
experience, the “claim” that they are distinct would be (according to
verificationism) meaningless. Hence it
seems the identity of indiscernibles follows directly from verificationism,
for, according to this principle, to say that things are distinct could only mean that there could in principle be a
way of empirically distinguishing them, which would necessarily involve some
difference in their properties.
B attempts to construct a thought
experiment in which there are numerically distinct objects that have identical
properties, including identical relational properties, thus demonstrating the
logical possibility of indiscernible identicals. There are three versions of this thought
experiment, one involving two qualitatively identical spheres, the second
involving a universe where everything has an “identical” “mirror image,” and
the third involving a universe that is “radically symmetrical,” i.e., a
universe containing a “centre of symmetry,” where “everything that happened
would be exactly duplicated at a place of an equal distance on the opposite
side of the centre of symmetry.”
First
case: two exactly similar spheres. “Then every quality and relational
characteristic of the one would also be a property of the other. If what I am describing is logically
possible, it is not impossible for two things to have all their properties in
common. This seems to me to refute the
principle.”
A’s response: “Your supposition, I repeat, isn’t verifiable
and therefore can’t be regarded as meaningful.
But supposing you have
described a possible world, ….” A discussion follows about how to name the
objects in this world. The point is that
in this described world, there are only these two objects, and so no one to
“name” them. To add someone who could do
this, would change the situation in an important way, in that it would add a
third element into the world, and this would change the relational
characteristics of the two (original) objects, perhaps providing a way of
distinguishing them. So B rejects all attempts to include any
“third thing” in the world described.
A continues to find ways to
establish that there must be some properties that one has that the other lacks,
for example, being at a distance from itself and not from the other. B
rejects this, claiming that, properly described, each will have the same the
properties, e.g., “being at a distance of two miles, say, from the centre of a
sphere one mile in diameter, etc., and (if you want to call it that) of being in the same place as itself.”
(Exercise: Why does express skepticism “(if you want to
call it that”) at the introduction of the property of being in the same place as itself?)
A attempts to avoid this
analysis by claiming that “being in a given place” need not be understood
(only) in terms of a thing’s spatial relation to something else (which is how B implicitly understood it), but in
terms of something inherent within at thing itself, so that it could make sense
to say of a thing, for example, that it had a particular spatial location even
if it were the only object in space.
What
A is presupposing here is an
“absolute” space, i.e., something that exists independently of anything being
in it. On such an absolute view, a thing
could have a specific location in space in virtue of its relation to space itself, independently of any relation
it might (or might not) have to anything else existing in space. This is the view implicit in the works of
Isaac Newton, and explicitly rejected by Leibniz, who held a relational view of
space, i.e., that there is no space apart from the things in it, and so an
object can only have spatial properties in virtue of relations it stands in to
other objects in space. So, if the world
contained only one object, it wouldn’t have any spatial properties, since, for
example, there would be nothing that it would be “above,” “below,” “to the
right of,” or “to the left of.”
Einstein’s theories of general and special relativity both presuppose a
relational understanding of space (and, with more counter-intuitive results, of
time), so it is not as though A is
standing on scientifically respectable grounds by siding with
A’s next attack leads
directly to the second version of B’s
thought experiment attempting to establish the possibility of a possible world
contains numerically distinct objects that are qualitatively identical. A
notes that it is part of B’s
description of the possible world containing two identical spheres that they
are the same size, but in a world with just these two objects, there are no
rulers or other measuring devices, and so this claim has no meaning. If we introduce a ruler into this world (in
order to have something by which we can understand claims about their sizes),
we have introduced some third thing, which, because it can stand in different
relations to each of the two spheres, will provide the spheres with distinct
relational properties. So, without (for
example), a ruler, it makes no sense to say they are the same size, but with a
ruler (or any “third” thing, including a “namer,” there will be properties by
which to distinguish them.
B’s response is to complicate
the thought experiment in a way that allows the existence of rulers and other
measuring devices. Suppose that instead
of a universe with just two objects, we have a universe where everything has
its “twin” on the opposite side of some imaginary plane dividing the universe
in half. So, on both sides of this
plane, there would be rulers by which we could establish measurements of size,
while an object and its “twin” on the opposite side of this plane would be
numerically distinct yet qualitatively identical.
After
some wrangling, A notes that these
“twin” pairs would not be qualitatively identical after all, in that each would
be a mirror image of the other. (Think
about why B’s though experiment
requires them to be mirror images.) But
if they are mirror images, then, for example, on one side of the plane,
Napoleon will have a heart in the right side of his chest, while on the other
side, the twin’s heart will be in the right side. This criticism is effective with respect to B’s second version of the thought
experiment, and so B moves onto the
third version.
The
third version of B’s attempt to
conceive of a logically possible world containing numerically distinct objects
that are qualitatively identical avoids the “mirror image” problem by
postulating a universe that is “radically symmetric,” that is, where there is a
“centre of symmetry” such that everything has an exact duplicate at a place of
the exact same distance as it on the opposite side of this centre. “And to avoid complications, we could suppose
that the centre of symmetry itself was physically inaccessible, so that it
would be impossible for any material body to pass through it.”
I
included B’s remark (about “avoiding
complications”), because I think it is relevant to A’s response: the world B describes here (where everything has
a qualitatively identical duplicate) is empirically indistinguishable from a
world where there are no duplicates.
Once again, according to verificationism, if the claim that there are
two of some object (in this world) is to be meaningful, there must in principle
be some means of empirically confirming this fact, which could only be a means
of finding evidence of some property one has that the other lacks. But the thought experiment is constructed in
such a way as to make this impossible.
In the world B describes,
everything has a numerically distinct duplicate exactly identical to it where
there is no way in principle to distinguish one from the other. In other words, there is no way in principle
to empirically distinguish this world from a world in which there is only one
of everything, and hence the “claim” that they are (even logically possible)
different “worlds” is in principle unverifiable, and hence meaningless.
This
is essentially where the debate ends, and so one’s view on the principle of
identity of indiscernibles seems tied to one’s view on verificationism. B
claims to have described a logically possible world with indiscernible
identicals, while A replies that
their very indiscernibility establishes (given the principle of
verificationism) that the “claim” that they are numerically distinct is (in
Carnap’s words) a meaningless “pseudo-statement.”
So,
what do you think? Can you imagine the
world that B (purports to)
describe(s)? Can you imagine that the
universe has qualities that could never in
principle be empirically verified?
[Just for fun: the “standard” interpretations of quantum mechanics
assumes that it does not make sense
to suppose the universe has qualities that could never be empirically confirmed.
For reasons that go beyond our scope here, physicists have established
that we can never precisely measure both the position and velocity of certain
sub-atomic particles. (This is
Heisenberg’s Uncertainty Principle.) The more precisely we refine one measurement,
the less precise will be the other. This
is not a mere “technological” problem that might be overcome one day if we
invent more precise instruments, because it follows from the very nature of the
theory itself. It is theoretically impossible to get a
precise measurement of both a particle’s location and velocity. Hence, the claim that a particle actually has, at a given time, both a specific,
precise location, and a specific, precise velocity is not capable, in
principle, of verification, and so, according to verificationism, is
meaningless. So, according to quantum
mechanics, it is simply not true of
the ultimate constituents of matter that actually have both a specific location
in space/time and a specific velocity.
It’s not just that we can never determine it. Because
we could never determine it, it doesn’t
have one!)