Eliminating Relata

There's a view in philosophy of science called "ontic structuralism" and it is sometimes expressed by saying it wishes to eliminate relata, or to take "structure" to be primary, or something along those lines.

But the main problem, as I've always seen it, is that a structure (or model) $\mathcal{A}$ in the usual mathematical sense is a mathematical object of the form $(A, R_1, ...)$ with a domain $A$. The domain is some set of objects, and the $R_i$ are relations on $A$, and are called the "distinguished relations" in $\mathcal{A}$. But this isn't structuralism in the required sense because one has a domain. (One kind of structuralism does retain a domain of "nodes": this is Shapiro's ante rem structuralism - or at least I think Shapiro's ante rem structuralism retains a domain.)

So one wants somehow to achieve two goals:

(i) identify something as the "abstract structure" of a given model, or system, etc.;
(ii) while also getting rid of the domain.

The only way I know of that might, in some sense, "eliminate" the domain is to consider a certain (usually very long, possibly infinitely long) sentence which defines $\mathcal{A}$ up to isomorphism. First, consider some first-order language $\mathcal{L}_{\mathcal{A}}$, with identity, whose signature fits that of $\mathcal{A}$ and which also has a unique constant $\underline{c}$ for each element $c \in A$. Suppose that the predicate symbol for $R_i$ is $P_i$, and suppose $P_0$ is $=$. So, $(P_i)^{\mathcal{A}} = R_i$ and $\underline{c}^{\mathcal{A}} = c$. Next take the diagram $D(\mathcal{A})$ of $\mathcal{A}$ in the language $\mathcal{L}_{\mathcal{A}}$. That is the set of all literals true in $\mathcal{A}$. Let $\delta_{\mathcal{A}}$ be (possibly infinitary) conjunction of all elements of $D(\mathcal{A})$ and let $\theta_{\mathcal{A}}$ be the (possibly infinitary) formula:

$\forall x \bigvee_{c \in A} (x = \underline{c})$

Let $\Sigma_{\mathcal{A}}$ be the formula:

$\delta_{\mathcal{A}} \wedge \theta_{\mathcal{A}}$

Then

$\mathcal{B} \models \Sigma_{\mathcal{A}}$ iff $\mathcal{B} \cong \mathcal{A}$.

So, $\Sigma_{\mathcal{A}}$ categorically axiomatizes $\mathcal{A}$.

Example: let $A = \{0,1\}$, let $R = \{(0,0),(0,1)\}$ and let $\mathcal{A} = (A,R)$. Then $\Sigma_{\mathcal{A}}$ is the formula

$\underline{0} \neq \underline{1} \wedge P\underline{0}\underline{0} \wedge P\underline{0}\underline{1} \wedge \neg P\underline{1}\underline{0} \wedge \neg P\underline{1}\underline{1} \wedge \forall x(x = \underline{0} \vee x = \underline{1}).$

As one can intuitively see, this "tells you everything you need to know" about the structure $\mathcal{A}$. All identity facts are in there, and all the atomic truths about $R$ are in there too (with the negated ones).

Next, we want to find something to be the abstract structure of $\mathcal{A}$, as well as "eliminating the relata". To do this, one can just quantify all these individual constants away, by a kind of ramsification. Suppose $(\underline{c}_{\alpha})_{\alpha \in I}$ is a non-repeating enumeration of the constants, and $(y_{\alpha})_{\alpha \in I}$ is an enumeration of distinct new variables with the same index set $I$. Then the new ramsified formula, $\Re(\mathcal{A})$, is:

$\exists y_0 \dots \exists y_{\alpha} \dots \Sigma_{\mathcal{A}}(\underline{c}_0/y_0, \dots \underline{c}_{\alpha}/y_{\alpha}, \dots)$.

(The initial string of quantifiers may be infinitary.) This procedure makes no difference to the result above, and we can now forget the constants.
Example again: $\Re(\Sigma_{\mathcal{A}})$ is the formula:

$\exists y_0 \exists y_1(y_0 \neq y_1 \wedge Py_0 y_0 \wedge P y_0 y_1 \wedge \neg Py_1 y_0 \wedge \neg Py_1 y_1 \wedge \forall x(x = y_0 \vee x = y_1)).$

What one obtains here has the right categoricity property: it determines $\mathcal{A}$ up to isomorphism. Notice that in order for this work, one must keep identity as a primitive (a point made, in this context, in a 2006 paper). Even so, $\Re(\Sigma_{\mathcal{A}})$ is a syntactic entity---a string of symbols---and so will not be invariant under even trivial changes (e.g., relabellings of variables, or switching logical conjuncts). So, one has not found a unique entity to be the abstract structure.

However, one can "quotient this out" by considering the Fregean proposition (the abstract content) expressed by this syntactic string. One can then say that the abstract structure of $\mathcal{A}$ is this proposition: the Fregean proposition expressed by $\Re(\Sigma_{\mathcal{A}})$. This in some important sense has no special individuals associated with it, for the constants used to denote individuals have been quantified away. However, it does in some sense retain all the primitive concepts/relations that one started with.

Example again: let $A$ and $R$ be as before and let $\mathcal{A} = (A,R)$. Then the abstract structure for $\mathcal{A}$ is the proposition that there are exactly two things such that one of them bears $R$ to itself and to the other, but the other does not bear $R$ to itself or the other.

(If one identifies possible worlds with such things, one gets rid of purely haecceistic differences; it's one way of responding to the hole argument and the "Leibniz equivalence" of isomorphic spacetimes.)