; Conc is the property of being concurrent, Red is the property of definiteness, and Heavy is the property of vividness.
In the language of modern metaphysics, w and w′ above are qualitatively indiscernible. And anti-haecceitism is the doctrine which says that qualitatively indiscernible worlds are identical. So, we immediately see a problem looming.
But why accept anti-haecceitism? The best reasons focus on physics. Just as the debate between Leibniz and Newton’s followers focused on physics, the strongest arguments still against haecceitism come from physics. Anti-haecceitism as understood here concerns the identity of indiscernible (“isomorphic”) worlds or “situations” or “states”. In many areas of physics, including statistical physics, spacetime physics and quantum theory, the physics tells us that certain “indiscernible situations” are in fact literally identical.
A simple example comes from the statistical physics of “indiscernibile particles”. Consider a box, partitioned into Left-side and Right-side (L and R), and containing two indiscernible particles. One naively thinks this permits four distinct states or situations: i.e., both in L; both in R, and one in L and one in R. However, physics tells us that there are only three states, not four, and we might denote these: S2,0, S1,1, S0,2. The state S1,1, i.e., where “one is L and one is R”, is a single state; there are not two distinct possibilities. The correct description of S1,1 uses existential quantifiers:
∃x ∃y (x ≠ y ∧ Lx ∧ Ry)
One can (syntactically) introduce labels for the particles, say a, b. One can do this in two ways, to obtain:
a ≠ b ∧ La ∧ Rb
b ≠ a ∧ Lb ∧ Ra
But this labelling is purely representational, and not in any way fixed by the physical state S1,1 itself. So, there are distinct indiscernible objects in “situations” or states.
From spacetime physics, consider the principle sometimes called “Leibniz equivalence” (Norton). A formulation (but under a different name) is given in Wald’s monograph General Relativity. Wald’s formulation of Leibniz equivalence is, essentially, this:
isomorphic spacetime models represent the same physical world.
For example, let
S = (M, g, . . . )
be a spacetime model with carrier set |M| of points. (i.e., M is the underlying manifold.) Then Leibniz Equivalence implies:
If π : |M| → |M| is any bijection, then π∗S and S represent
the same world. There are many other examples, including examples from quantum theory. Consequently, independently of our pre-theoretic considerations concerning modality, it seems to me that our best physics – statistical physics, relativity and quantum theory – is telling us that anti-haecceitism is true: given a structure A which represents a world w, any permuted copy π∗A should somehow represent the same world, w.