where, e.g.,Leibniz Equivalence

If $\mathcal{A}$ and $\mathcal{B}$ are isomorphic models, then theyrepresentthe same world, relative to some fixed interpretation.

$\mathcal{A} = (X, \mathcal{C}, g_{ab}, \dots)$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:

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.Anti-Haecceitism

Qualitatively indiscernible worlds are identical.

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:

Next, we can define "Leibniz Abstraction

$\hat{\Phi}_{\mathcal{A}} = \hat{\Phi}_{\mathcal{B}}$ iff $\mathcal{A} \cong \mathcal{B}$

*represents*" as follows:

And suppose that we have worlds $w_1$ and $w_2$ such that the antecedent of Leibniz Equivalence holds (equivalently, Anti-Haecceitism):Definition("represents")

A model $\mathcal{A}$represents$w$ with respect to relations $\mathsf{R}_0,\mathsf{R}_1, \dots$ if and only if $w = \hat{\Phi}_{\mathcal{A}}[\mathsf{R}_0,\mathsf{R}_1, \dots]$.

1. $\mathcal{A} \cong \mathcal{B}$We can then prove that $w_1 = w_2$ as follows. From the definition of "represents",

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 $\mathsf{R}_0,\mathsf{R}_1, \dots$

$w_1 = \hat{\Phi}_{\mathcal{A}}[\mathsf{R}_0,\mathsf{R}_1]$Now, $\mathcal{A} \cong \mathcal{B}$, and hence by Leibniz Abstraction,

$w_2 = \hat{\Phi}_{\mathcal{B}}[\mathsf{R}_0,\mathsf{R}_1]$

$\hat{\Phi}_{\mathcal{A}} = \hat{\Phi}_{\mathcal{B}}$Hence,

$w_1 = w_2$as required.

So, in short, we have established:

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.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 $w_1 = w_2$.

## No comments:

## Post a comment