Identity of Indiscernibles

Numerical identity and qualitative similarity
Numerical identity is “the relation that everything bears to itself and to no other thing.” It is an equivalence relation, which is to say it is reflexive, symmetric and transitive. This means that for any a, b, c

Reflexivity: a = a  [everything is identical to itself—duh!]

Symmetry: If  a = b then b = a

Transitivity: If a = b and b = c then a = c

Numerical identity is also an indiscernibility relation, that is  Indiscernibility of Identicals (see below) holds for it .

Qualitative similarity is a relation between objects that have various qualities in common. Sometimes when we use words like “identical,” “same” and the like we mean “numerically identical”; sometimes we mean qualitatively similar. “Identical twins” for example are not numerically identical. They are qualitatively similar: if they were numerically identical they would be one person, not twins. We’ll use the word “identity” from now on to mean numerical identity. Sometimes we will slip and use “qualitatively identical” to mean “qualitatively similar in every respect.”

Indiscernibility of Identicals: x = y => (F)(Fx iff Fy)
This says that if x and y are identical then x has the property F if and only if y has F, that is, that x and y have all their properties in common. It is uncontroversial.

Identity of Indiscernibles: "(F)(Fx iff Fy) => x = y
This says that if x and y have all their properties in common then they are identical. It follows contrapositively that if x and y are distinct (not identical) then they don’t have all their properties in common: there must be at least one property that the one has which the other doesn’t have—so this principle is sometimes called the  Dissimilarity of the Diverse. These principles are controversial and they are what Black (“B” in the dialogue) is arguing against.

Stating Identity of Indiscernibles non-trivially: identity as a property
One problem in assessing Identity of Indiscernibles is formulating it in such a way that it’s neither obviously false nor trivially true and uninteresting. That is what Black is negotiating about on the first two pages of his essay. The basic problem is getting clear about what sorts of properties, F, we’re thinking about in formulating Identity of Indiscernibles. If it’s all properties, including properties like being-identical-with-a (where “a” is a name of an individual) then, as Black points out, the principle is true but boring: having the property of being-identical-with-a is just plain being identical with a so, of course if a and b both have the property of being identical with a then a = b [see transitivity and symmetry above!] To formulate the principle in an interesting way we have to distinguish between different kinds of properties.

Purely qualitative properties
Being-identical-with-a is not “purely qualitative,” that is, it has to be analyzed in terms of a relation to some particular individual—in this case a. This is true also of properties like being-2-miles-from-a, being-GeorgeHWBush’s-son, and so on. It is not true for those properties which are purely qualitative, which include being-2-miles-from-an-iron-sphere, being-the-son-of-a-president, being-red, being-fuzzy and so on.

Intrinsic/extrinsic property distinction
We also want to distinguish intrinsic properties, like being-red and being-fuzzy, from extrinsic properties, like being-2-miles-from-an-iron-sphere, being-GeorgeHWBush’s-son and so on. For our purposes, though this is imprecise, extrinsic properties are what Black calls “relational” properties. (But not all relational properties are extrinsic: can you think of counterexamples to the claim that a property is extrinisc iff it’s relational?) The idea is that a property is intrinsic if a thing could have it even if it were the only thing in the universe; otherwise it is extrinsic.

We make this distinction because if Identity of Indiscernibles is construed as saying that having all intrinsic properties in common is enough to make things identical, the principle is just implausible. Given mass production there are surely lots of things that are as close as you please to being exactly similar as regards their intrinsic properties and, even if it is unlikely, it doesn’t seem to be logically impossible that some be exactly similar in this respect. The most plausible version of Identity of Indiscernibles says that having all purely qualitative intrinsic and extrinsic properties in common is enough to make for identity—and that is the version considered in Black’s paper, which he will argue against.

Logical possibility
In any case, we’re not interested in whether there actually are distinct objects that have all their properties in common but whether it is logically possible that there be. Logical possibility is possibility in the broadest sense: something is logically possible if it doesn’t involve a contradiction in the way that, e.g. the existence of married bachelors and round squares does.

Conceivability as a criterion for logical possibility
Traditionally, the test for logical possibility is conceivability: if I can conceive of something then it is logically possible.

Thought experiments
This is why in the literature when the question arises of whether something is logically possible philosophers introduce thought experiments in which the reader is asked to imagine a particular scenario. If it can be imagined, the argument goes, it is logically possible. So, for example, arguing that is logically possible for a person to survive in a disembodied state, Descartes invites us to imagine not having a body. Black’s thought experiment, in which he asks us to imagine a symmetrical universe in which nothing exists except for two exactly similar spheres a mile apart is a thought experiment designed to show that we can imagine two distinct things which have all the same purely qualitative intrinsic and extrinsic properties, hence that it is logically possible that things could have all their intrinsic and extrinsic properties in common and still not be identical.

If we really can conceive of this situation it is a counterexample to Identity of Indiscernibles. A counterexample is a case which shows a general claim to be false, so if I say “all swans are white” and you note that there are black swans in Australia you have produced a counter example to my claim. Similarly if Black has in fact told a story in which there are two distinct objects that have all their intrinsic and extrinsic properties in common he has produced a counterexample to the claim that all objects x and y that have all their purely qualitative properties in common are identical to one another, the most plausible, non-trivial version of Identity of Indiscernibles.