Leibniz's Law,
the Indiscernibility of Identicals,
and the Identity of Indiscernibles

Leibniz's Law is a bi-conditional that claims the following: Necessarily, for anything, x, and anything, y, x is identical to y if and only if for any property x has, y has, and for any property y has, x has. Because this is a bi-conditional, it is comprised of two conditional statements (i) and (ii):

    (i)     If x is identical to y, then for any property x has, y has and for any property y has, x has.
    (ii)    If for any property x has, y has, and for any property y has, x has, then x is identical to y.

The Indiscernibility of Identicals

(i) is called the Indiscernibility of Identicals because it claims that self-identical object(s) must be indiscernible from themselves. It is a fairly uncontroversial thesis. I say "fairly" because there are philosophers who deny this claim. And, time permitting, we will be discussing them at some point in the semester.

Almost everyone else, however, will grant that if something, x, is identical with something, y, then x and y have all of the same properties; x and y are just one thing, after all, merely called by two different names "x" and "y." If Superman is identical to Clark Kent, then Superman wears glasses and Clark Kent has x-ray vision (because Clark Kent wears glasses and Superman has x-ray vision). Since Superman is identical to Clark Kent, there is no property that Superman has that Clark Kent does not have, and there is no property that Clark Kent has that Superman doesn't; "they" are just one guy, after all, not two.

Notice that another way to put the Indiscernibility of Identicals is in terms of qualitative and numerical (or quantitative) identity (which will be discussed in class and on this handout here). We could state the Indiscernibility of Identicals as: If x and y are numerically identical, then x and y are qualitatively identical.

The Identity of Indiscernibles

(ii) is called the Identity of Indiscernibles because it claims that indiscernible objects must be identical. This thesis has raised quite a bit of debate among metaphysicians. You might think it is intuitive because of its practical applications: it seems that we do in fact use this principle when we are trying to determine whether we've got one thing in front of us or two. Suppose we are trying to figure out whether Superman is identical to Clark Kent. We begin: "Well, Superman is 6ft tall, and Clark Kent is 6 ft tall; Superman has dark hair and dark eyes, and Clark Kent has dark hair and dark eyes; Superman can't get injured by fire and--look!--Clark Kent can't get injured by fire, either!", etc. If a certain identity claim such as 'Clark Kent = Superman' is not known, then tallying all of the properties that "each" has and seeing if there is a property that one has that "the other" doesn't, will aid us in determining whether Clark Kent is in fact idenitcal to or distinct from Superman.

We might also think that the Identity of Indiscernibles is true because if there are in fact two, distinct things, then there must be something--some property or feature or other--that makes these two things different. It is explanatorily useful, in other words, to claim that two distinct objects are two distinct objects because they have at least one property that makes them distinct. To put it another way: if the Identity of Indiscernibles were false, then that would mean that we could have numerous qualitatively identical things, but no way to determine whether such 'things' are one or many. Indeed, if distinct objects can have every property in common, then there would be no way to determine whether we have 100 qualitatively identical things in front of us, or ten qualitatively identical things in front of us, or just two, or one, or what. We would just have to stipulate that there are two as opposed to ten or a hundred qualitatively identical things, since, by stipulation, there is no quality or feature that these things have that would explain the difference between two or ten or a hundred qualitatively identical things.

Similar to the Indiscernibility of Identicals, notice that another way to put the Identity of Indiscernibles is in terms of qualitative and numerical identity: If x and y are qualitatively identical, then x and y are numerically identical.

Trouble for the Identity of Indiscernibles

As intuitive as the Identity of Indiscernibles may seem, there are a host of purported counterexamples. Most famously is Max Black's "The Identity of Indiscernibles" [note: this is a link through jstor, so you may need to be on campus or using a university proxy server to follow it].

Roughly, the idea is this: Imagine a possible world that contains just two perfectly round symmetrical spheres. The spheres have the same diameter, they are an equal distance from each other, they are the only inhabitants of a completely symmetrical universe, they are both made of solid iron, etc. Intuitively, such spheres in such a world would have all and only the same properties; they are qualitatively identical. Yet, by stipulation, there is supposedly two of these spheres, not one. But if "they" are qualitatively identical, then by the Identity of Indiscernibles, "they" are numerically identical; "they" are one, not two. Thus, either (i) there cannot be a world where there are two qualitatively identical objects or else (ii) the Identity of Indiscernibles is false. Since it seems that we can imagine a world with two numerically distinct yet qualitatively identical objects, then (by disjunctive elimination) the Identity of Indiscernibles must be false.

Response #1

Perhaps, you think, there is a way to distinguish between the two spheres. One of the spheres, for example, might have the property "being identical to sphere a" or "being distinct from sphere b" or "being this sphere and not that one," etc. The idea would be that there is something special about individual objects--something that makes one object distinct from another, independent of any quality, property, or feature. Such a move would appeal to individual essences or "thisnesses" or "haecceities". And such a move would block the counterexample, but at the cost of inviting even more problems.

Problem 1: Suppose you are a materialist--you believe that there is only physical stuff in the universe. Or, even weaker, suppose that you are a materialist only when it comes to material objects such as rocks and trees and spheres and the like (you are agnostic or a dualist when it comes to people, say). What are you to make of haecceities? If we take away material bits of a  (merely)  material object like a sphere--if we remove one metallic bit of it one by one, and one at a time--when do we get to the haeccity? Where is the haecceity? Is it one of the material bits we remove? It couldn't be, because no one material bit of an object seems essential to the object (we could remove any one material bit from a material object, and that object will still exist). In this way, a commitment to haecceities is skirting dangerously close to a sort of dualism for objects, which might rub more conservative ontologists the wrong way.

Problem 2: If you endorse individual essences or haecceities then you seemingly save the Identity of Indiscernibles, but you have to ultimately bite the bullet and admit that it is impossible for there to be a world with two qualitatively identical spheres. Becuase, ultimately, two spheres that are seemingly qualitatively identical are not entirely qualitatively identical--they differ in their haecceitties. You might want to insist that haeccaties are not strictly speaking qualities or properties, since they can only be had by one individual, and are not repeatable, like redness, and roundness, and bumpiness are. But, intuitively, if you posit individual haecceties, then there is something that makes the difference bewteen two otherwise qualitatively identical objects. If haecceitties are (by stipulation) a difference-making feature, then they are a feature, period. And so you do not think that there can be a world with two completely qualitatively identicial spheres (haecceities included). So you will have to explain why it is that it certainly seems as if we can imagine such a world.

Problem 3: Imagine a world with just one sphere. According to the haecceitist, there is just one haecceity or thisness or individual essence. Now imagine that the sphere morphs like an amoeba and splits itself in half. Each half now morphs into a small sphere, each seemingly qualitatively like the other. No new material is made; no old material disappears. The original sphere is simply changing shape and reforming itself out of the material that was already in the universe. Now we have a world that seemingly has two qualitatively identical spheres. According to the haecceitist, there must now be two haecceities (since distinct individuals are (at the very least) made distinct by their distinct haecceities). But where did the second haecceity come from? By stipulation, the world began with just one! A haecceitist may bite the bullet and say that there was always two, but this seems unmotivated. Consider the counterfactual case:  if the sphere had never morphed, then there would not be a need to posit two haecceities. Does the world just 'know' ahead of time whether objects in it will morph or not, and then supplies the appropriate number of haecceities? That would be wildly ad hoc and implausible. Moreover, anything we can do with a case of fission, we can work in reverse to cases of fusion. If we begin with two distinct shperes that fuse into one, then does the resulting, singular object have one haecceity or two? If one, then where did the other one go? If two, then how can we say there is only one object as a result of the fusion?

Besides, imagine that after the sphere fisses into two spheres, each sphere fisses again into two, making four (seemingly) indiscernible spheres. So now, according to the haecceitist, there are four haecceities to account for the four distinct spheres. But where did these other haecceities come from? If the haecceitist insists that there were always four, then this seems as unmotivated as saying that there were always two haecceities. And we can keep fissing the spheres, ad infinitum, forcing the haecceitist to ultimately say that there are infinitely many haecceities in just one sphere! If we thought a commitment to just one haecceity for one sphere was ontologically burdensome, positing infinitely many is surely worse!

Problem 4: Positing haecceities seems ad hoc. Do we have any other reason to posit these weird, suspicious "thisnesses" except to save the Identity of  Indiscernibles? If the haecceitist can provide no other reason for positing individual essences other than to save the Identity of Indiscernibles, we may wonder why the principle was worth saving in the first place.

Problem 5: Positing haecceities seems to make the Identity of Indiscernibles trivially true. This is a problem if you thought that the principle was supposed to be substantial, metaphysical thesis.

Response #2

Another way we might be able to distinguish the spheres by location. However, the world was stipulated to be a symmetrical one, and the twin spheres were lonely twins--they were the only inhabitants in the entire world. So this means that we cannot specify their location relative to anything else in the universe. So, someone might suggest, the spheres are located at distinct absolute locations of space-time points.

Problem 1: our best scientific theories suggest that that the universe cannot have anything like absolute location.

Problem 2: even if such a suggestion didn't contradict the best scientific theories of the day, it would be odd to think that mere a priori reflection on the identity relation would yield as substantial a metaphysical (and physical!) thesis about whether space was absolute or not.

Problem 3: this response requires that we accept distinctness of space-time points as primitive. Yet if we are going to do this, why not accept distinctness of individuals as primitives as the haecceitist does? What's the real difference, in other words, between individual essences of objects and individual essences of space-time points? If we had problems with haecceities, it seems these problems would repeat themselves at the level of primitively distinct space-time points.

Response #3

A third response is to claim that the counterexample was set up incorrectly. The mistake, you might claim, was in claiming that there are two spheres in the possible world under consideration. Claiming that there are two spheres already begs the questions against someone who accepts the Identity of Indiscernibles. Rather, there are not "two" qualitatively identical spheres, but just one sphere that happens to be bi-located--i.e., one sphere in two places at the same time.

Problem 1: This violates the following intuitive metaphysical principle: no one material object can be at two distinct places at the same time. Let us call this principle P. Accepting that objects can be bi-located would violates P. So we might ask ourselves: which do we think is more intuitive? The Identity of Indiscernibles? Or principle P? Common sense seems to favor principle P; so the Identity of Indiscernibles should be abandoned.

Problem 2: Suppose objects can be bi-located, and that Max Black's supposed counterexample is merely a description of a world where one sphere is bi-located. Now imagine that someone came along and spit in the direction of one of the locations of the bi-located sphere. What happens to the sphere at the opposite location? Does it get spit on, too? How would that be? The person just spit once, in one direction. Yet if there really is just one sphere there (even if it is bi-located) then, by the Indiscernibility of Identicals, it can't have any properties dissimilar from itself. Yet now it does: it (in one location) was just spit upon yet it (in the other location) wasn't. Surely this just goes to show that there are two spheres in this world, not one!


Max Black, "Identity of Indiscernibles," Mind, vol. LXI, no. 242, 1952.
Jason Bowers, personal notes and correspondence 2006-08.
Zimmerman, Dean, "Distinct Indiscernibles and the Bundle Theory" in Mind, New Series, vol. 106, no. 422, April 1997: 305-309
O'Leary-Hawthorne, John, "The Bundle Theory of Substance and the Identity of Indiscernibles" in Analysis, Vol. 55, No. 3, July 1995: 191-96.

Page Last Updated: Sept. 3, 2009
Back to Meg's Teaching Page
Back to Meg's Main Page