Indiscernible properties

Max Black's argument against the principle of identity of indiscernibles (PII for short) contends that one could conceive of a plurality of objects with the exact same observable properties (for some notion of "observable" and "property"), and proposes the simple example of a universe exclusively composed of two exactly similar spheres some distance apart from each other: in this perfectly symmetrical setup there is no way to tell one sphere from the other except by arbitrarily picking one up, which selection can not be based on their observable properties; so, the spheres are different (there are two of them) but all their salient properties are the same, thus breaking PII.

Let us simplify the two-sphere example even further down to a universe with two points without any intrinsic physical property except their being at some distance of each other. In the 2P universe, formulated as a theory in second-order logic, there are no primitive properties (unary relations) to discuss about and only one primitive binary relation C defined as

C(x, y) := the distance from x to y is zero

(C stands for"colocated") satisfying the following axioms:

∀x C(x, x) (each point is colocated with itself),
∀x∃y ¬C(x, y), (for each point there is another one not colocated with it).

Now, if we accept the axiom scheme of comprehension, we have

∀x∃R∀y(Ry ↔ C(x, y)),

which, in combination with the axioms for C, implies

∀x∃R∃y(Rx ∧ ¬Ry),

that is, for each point x there is a property (namely that of being colocated with x) that x has and some other point y (which, in 2P, is tantamount to saying any other point) does not have. This seems to restitute PII.

To this reasoning Black could have retorted in (at least) two different ways:

• "Being colocated with x" is just a slightly disguised form of "being identical with x", which is trivially true of x itself and trivially false of any other entity, adding nothing to our initial assumption that 2P has a population of two.
• The reasoning involves two ill-defined properties, "being colocated with a" and "being colocated with b", where a and b are one point and the other, or the other way around: without any discernible feature to use, there is no way we can select a particular point out of the two we have in 2P.

The first objection we can easily dispose of: "being colocated with x" is a bound version (via comprehension) of C, which is a perfectly discernible, physical property of pairs of points —if I'm given two points, identical or not, I can certainly inspect whether they are colocated. The second counterargument is more interesting and, in my opinion, allows us to provide a clearer formulation of Black's thesis. The objection could be rephrased as: for the same reasons that naming the two points "a" and "b" requires that we can previously tell one point from the other, their univocally associated properties R_a and R_b are also indiscernible, even if they are different. We can formalize indiscernibility in the following way: 

Definition. A second-order theory T (with or without equality) is said to have indiscernible entities if there exists a formula φ with two free variables such that 

∃x∃y (φ(x, x) ∧ φ(y, y) ∧ ¬φ(x, y))   (1)

is a theorem of T and, for each model M of T and a, b ∈ M satisfying (1), the structure M' obtained by swapping a and b in all the relations of M is also a model of T.

If the set of primitive relations of T is finite, indiscernibility can be expressed as a statement in T itself (sketch of proof: augment (1) with a conjunction of terms expressing swappability of x and y with respect to each position of each primitive relation of T). If T has equality, the statement

∃x∃y (φ(x, x) ∧ φ(y, y) ∧ ¬φ(x, y) ∧ x ≠ y)   (2)

is a theorem of T (proof trivial). 2P has indiscernible entities (proof trivial).

In conclusion, a formal interpretation of PII resists Black's attacks, but the following, stricter version of the principle:

If two entities have the same discernible properties, they are identical,

is false, and probably what Black had originally in mind.