Individuation, Haecceity, and Indiscernibles

What makes two things two? If they differ in some property, that difference answers the question. But suppose they do not — suppose two objects are exactly alike in every respect. Are they then one thing, or two?

The Principle of the Identity of Indiscernibles (PII) says one. It is Leibniz's principle, it is entailed by bundle theory with universals, and it is the subject of the best-designed thought experiment in analytic metaphysics.

Stating the principle

PII must be distinguished from its uncontroversial converse. Write $\forall F$ for quantification over properties:

$$\text{Indiscernibility of identicals:}x=y\to \forall F(Fx\leftrightarrow Fy)$$

$$\text{Identity of indiscernibles (PII):}\forall F(Fx\leftrightarrow Fy)\to x=y$$

The first is a truism — if they are the same thing, there is only one thing to have properties. The second is a substantial metaphysical claim.

Everything turns on which properties $F$ ranges over, and the principle comes in strengths accordingly:

• Trivial PII: $F$ ranges over all properties including being identical to $a$. Then indiscernibility entails identity trivially, since $a$ has the property of being identical to $a$ and nothing else does. This version is true and useless.
• Strong PII: $F$ ranges over pure intrinsic qualitative properties only. The most interesting and most contested version.
• Weak PII: $F$ ranges over qualitative properties including relations. Two objects may then be discerned by standing in different relations.

Black's two spheres

Max Black's 1952 thought experiment is the standard refutation. Imagine a universe containing nothing but two iron spheres, exactly similar in size, shape, composition, and every intrinsic quality, two miles apart. Nothing else exists.

They are two — the description is consistent and we can apparently conceive it. Yet they share every intrinsic property. So strong PII is false.

The natural repair is to invoke relations: sphere $a$ is two miles from $b$, and $b$ is two miles from $a$, so they differ relationally. But this fails, and the reason is instructive. In a perfectly symmetric universe, whatever relational property one sphere has, the other has too: each is two miles from a sphere. There is no relational property of one that the other lacks. Symmetry defeats weak PII along with strong.

Spatial position does not help either, if space is relational — there is no independent fact about which sphere is here. If space is substantival, the spheres differ by occupying different points; but then the burden shifts to individuating the points, which are themselves exactly similar, and the hole argument shows this is not a comfortable place to stand.

Weak discernibility

The most important recent move in the debate comes from Quine's notion of discernibility and was developed by Saunders. It rescues PII in a modified form, and it comes out of physics rather than the armchair.

Distinguish three ways two things can be discerned:

• Absolutely: some monadic formula is satisfied by one and not the other.
• Relatively: some asymmetric relation holds one way — $Rab$ but not $Rba$.
• Weakly: some irreflexive relation holds between them.

Black's spheres are not absolutely or relatively discernible. But they are weakly discernible: the relation is two miles from is irreflexive — nothing is two miles from itself — and it holds between them. So there are two, and what makes them two is a relation, not a qualitative difference.

This is a genuinely new position in a debate that had been static since 1952. It secures objecthood without intrinsic difference and without primitive thisness: the spheres are two because an irreflexive relation obtains, which is a perfectly objective fact requiring no further individuator.

Its provenance matters. The move was developed in response to quantum mechanics, where identical particles present exactly the Black configuration in an unavoidable form: permutation invariance means fermions and bosons share all monadic and relational properties, so French and Redhead concluded they violate PII outright. Saunders' reply is that two fermions in a singlet state stand in the irreflexive relation has opposite spin to, and are thereby two. The philosophical debate and the physical one are here the same debate.

Haecceity

The alternative is to accept that the spheres differ in nothing qualitative and to posit a non-qualitative individuator: a haecceity or primitive thisness. Sphere $a$ has the property being identical to $a$, and that is what makes it the one it is.

The doctrine is Duns Scotus's — haecceitas, "thisness", the individuating principle that contracts a common nature to this individual — and its modern defender is Robert Adams, who argues that primitive thisness is unavoidable and no more mysterious than any other primitive.

The objections are that haecceities are unexplanatory (they name the difference rather than accounting for it) and that they multiply possibilities without empirical difference: haecceitism is the thesis that two possible worlds can differ solely in which individual plays which role, with no qualitative difference whatever. Anti-haecceitists deny that such pairs are genuinely distinct possibilities.

This is not an idle dispute. It is exactly what is at issue in the hole argument: whether two models of general relativity that differ only by a diffeomorphism — a permutation of which spacetime point plays which role — represent different physical situations. The substantivalist who says yes is a haecceitist about points and gets indeterminism; the one who says no denies primitive thisness. Metaphysics and physics turn out to be making the same choice.

Where individuation sits

This is the category question at its finest grain: what, if anything, must be added to the qualitative description of the world to fix which individuals there are. Three answers are now on the table — nothing (PII), an irreflexive relation (weak discernibility), or a primitive thisness (haecceity) — and they line up with three positions in the bundle/substratum debate, since the bundle theorist needs PII, the substratum theorist gets individuation from the bearer, and weak discernibility is the option that needs neither.

The next page takes up the relation the whole discussion presupposes: identity itself.