Saturday, 6 July 2013

Proving Leibniz Equivalence

In work on the foundations of spacetime theories, Leibniz Equivalence is the modal principle:
Leibniz Equivalence
If $\mathcal{A}$ and $\mathcal{B}$ are isomorphic models, then they represent the same world, relative to some fixed interpretation.
where, e.g.,
$\mathcal{A} = (X, \mathcal{C}, g_{ab}, \dots)$
is some spacetime model (similarly for $\mathcal{B}$). Here $X$ is the carrier set of "points", and $\mathcal{C}$ is a maximal atlas on $X$, making $(X, \mathcal{C})$ into a manifold.

But Leibniz Equivalence is not proved. In General Relativity, this principle is assumed. (See Wald 1984, General Relativity, p. 438, for a formulation of Leibniz Equivalence. See here also for what I think is an important point concerning Wald's formulation, sometimes missed by physicists and philosophers.)

In the analytic metaphysics of modality, Leibniz Equivalence corresponds to the principle known as Anti-Haecceitism:
Anti-Haecceitism
Qualitatively indiscernible worlds are identical.
This view, which admittedly is hard to make unambiguous (see, e.g., Sklow 2008 "Haecceitism, Anti-Haecceitism and Possible Worlds"), is defended by several authors, such as David Lewis. But it is not proved. It is added by hand, as it were. In fact, I think that this is unsatisfactory, because I think there is a Hole/Permutation Argument against Lewis's approach.

Here I show that on the propositional diagram conception of worlds ("domainless worlds"), Leibniz Equivalence is a theorem. On the propositional diagram conception of worlds, a world $w$ is a categorical propositional function saturated by relations-in-intension. For example, if $\hat{\Phi}$ is such a propositional function with, say, two argument positions, say one unary and one binary, then $\hat{\Phi}[\mathsf{R}_0,\mathsf{R}_1]$ is the result of "applying" it to the unary relation $\mathsf{R}_0$ and the binary relation $\mathsf{R}_1$. So,
$w = \hat{\Phi}[\mathsf{R}_0,\mathsf{R}_1]$
So, the world $w$ is the result of "saturating" the propositional function $\hat{\Phi}$ with the relations $\mathsf{R}_0$ and $\mathsf{R}_1$.

The propositional diagram always arises from a particular model, $\mathcal{A}$. It is the propositional content of the pure second-order diagram formula $\Phi_{\mathcal{A}}(\vec{X})$ which categorically axiomatizes $\mathcal{A}$: this formula $\Phi_{\mathcal{A}}(\vec{X})$ defines the isomorphism type of $\mathcal{A}$ as follows:
$\mathcal{B} \models \Phi_{\mathcal{A}}(\vec{X})$ if and only if $\mathcal{B} \cong \mathcal{A}$.
In particular, because of the categoricity, we have:
Leibniz Abstraction
$\hat{\Phi}_{\mathcal{A}} = \hat{\Phi}_{\mathcal{B}}$ iff $\mathcal{A} \cong \mathcal{B}$
Next, we can define "represents" as follows:
Definition ("represents")
A model $\mathcal{A}$ represents $w$ with respect to relations $\mathsf{R}_0,\mathsf{R}_1, \dots$ if and only if $w = \hat{\Phi}_{\mathcal{A}}[\mathsf{R}_0,\mathsf{R}_1, \dots]$.
And suppose that we have worlds $w_1$ and $w_2$ such that the antecedent of Leibniz Equivalence holds (equivalently, Anti-Haecceitism):
1. $\mathcal{A} \cong \mathcal{B}$
2. $\mathcal{A}$ represents $w_1$ with respect to $\mathsf{R}_0,\mathsf{R}_1, \dots$
3. $\mathcal{B}$ represents $w_2$ with respect to $\mathsf{R}_0,\mathsf{R}_1, \dots$
We can then prove that $w_1 = w_2$ as follows. From the definition of "represents",
$w_1 = \hat{\Phi}_{\mathcal{A}}[\mathsf{R}_0,\mathsf{R}_1]$
$w_2 = \hat{\Phi}_{\mathcal{B}}[\mathsf{R}_0,\mathsf{R}_1]$
Now, $\mathcal{A} \cong \mathcal{B}$, and hence by Leibniz Abstraction,
$\hat{\Phi}_{\mathcal{A}} = \hat{\Phi}_{\mathcal{B}}$
Hence,
$w_1 = w_2$
as required.

So, in short, we have established:
Theorem (Leibniz Equivalence)
If $\mathcal{A} \cong \mathcal{B}$ and $\mathcal{A}$ represents $w_1$ and $\mathcal{B}$ represents $w_2$, both with respect to $\mathsf{R}_0,\mathsf{R}_1, \dots$, then $w_1 = w_2$.
The central attractiveness of the propositional diagram conception of worlds is that Leibniz Equivalence (Anti-Haecceitism) is a theorem---an automatic consequence. Qualitatively indiscernible worlds are identical, just as Leibniz argued many moons ago.