Max Black, “The Identity of Indiscernibles


Leibniz’ Law:


Two things are identical if and only if they share all the same properties.

Before we look at the claim itself, does anyone note anything funny?  How can two things be identical?


There is something confusing about the way we talk about identity.  If, it seems the “two things” are identical, then there “are” (is) only one of them.  So, what are there two of?  It seems that there must be two names or two referring expressions.  So, when we say that two things are one, what we must mean is that there are two “thing words” (nouns) that have the same referent. 


But, to pick up on some of the terminology we talked about last time, these two expressions have the same referent (extension), but they needn’t have the same meaning (the same sense or intension). 


Consider:  The morning star and the evening star are the same thing.


We have just suggested that the “depth grammar” here is really:


“The morning star” and “the evening star” both refer to the same thing (both have the same extension). 


But note that they do not have the same meaning.  The meaning of the first is “the first, brightest heavenly body visible in the evening sky” while the later means “the last, brightest heavenly body visible in the morning sky.”


Consider also: “rational animal” and ‘featherless biped.”  These are co-extensive but not co-intensive.


Leibniz’ Law is a biconditional, so it can be considered as two hypotheticals:


If two things are identical, they have all their properties in common.

If two things have all their properties in common, they are identical.


The first of these (which we might call the principle of the “Indiscernibility of Identicals”) is trivially true.  If two things are identical, they are not two, but one.  And whatever properties it has are the properties it has.  So, if two expressions have the same referent, then whatever can be said truly of one of them can be said truly of the other.  In logic, this is known as the principle of the “Substitutivity of Identicals.”   (Note: there are certain contexts where, it seems, this principle is not acceptable.)


It is the other direction of the biconditional that expresses a controversial metaphysical principle, namely that if (apparently) two things share all the same properties, then they are the same thing. This is know as the principle of the “Identity of Indiscernibles.”(If things are indiscernible, they are identical.)  We can see the force of this by considering its contrapositive:


            If two things do not share all the same properties (i.e., if they have differing properties), then they are not identical.


Or, if they are distinct, there must be some property that one has which the other lacks.


So, as Leibniz put it: there are no two things alike in nature.


This is (the contrapositive) the claim that is being debated in Black’s dialogue.  


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 entities, a and b, and concerned with whether or not a and b are distinct:  But, are A and B really 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, i.e., 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 the verificationist account of meaning, 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 try 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 A 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 Newton on the matter of absolute space.  It is also an odd position for him to hold, in that the central reason that Einstein had for rejecting an absolute understanding of space and time was none other than (a weaker version of) the principle of verificationism, which is central to A’s position.


A’s next attack leads directly to the second version of B’s thought experiment attempting to establish the possibility of a possible world containing 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 left 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.  (If one could travel “through” this center of symmetry, then one could establish that things “on the other side” of this point were different.  That is why B disallows this possibility.)  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!)