Ramsey sentence theorem (two-sorted)

From time to time I'd like to post on the topic of Ramsey sentence structuralism. This post describes a result concerning the truth conditions of Ramsey sentences. It is one way of trying to make precise the Newman objection to Ramsey sentence structuralism about scientific theories (originally going back to Newman 1928 and Demopoulos & Friedman 1985) and summarizes the main idea, and the main result (Theorem 6) from a 2004 BJPS paper "Empirical adequacy and ramsification".

1. Syntax
Suppose $\mathcal{L}$ is a two-sorted first-order language, with variables partitioned into what one might call primary and secondary variables (following the terminology of Burgess & Rosen 1997).
The primary sublanguage, obtained by deleting secondary variables and any secondary and mixed predicates is called ${\mathcal{L}}^{\circ }$.
Let ${\mathcal{L}}_2$ be the result of adding primary, mixed and secondary second-order variables or all arities (and corresponding atomic formulas of the right kind) to $\mathcal{L}$.
Let $({\mathcal{L}}_2{)}^{\circ }$ be the primary restriction of this language (obtained by eliminating secondary variables).
Finally, let $({\mathcal{L}}_2{)}^c$ be the sublanguage of ${\mathcal{L}}_2$ obtained by eliminating all non-logical mixed and secondary predicates.

The language $({\mathcal{L}}_2{)}^c$ is a two-sorted rendition of the mature Carnapian "observational language": it allows observational predicates, and first-order observational variables; in addition, it has first-order variables ranging over unobservable objects; and it has primary, mixed and secondary second-order variables, giving what amounts to a general theory of sets of, and relations amongst, the first-order entities (either observable or unobservable). In principle, one could add third-order, fourth-order, etc., variables, giving type hierarchy. It makes no difference to the result below.

2. Semantics
If $\mathcal{M}$ is an two-sorted $\mathcal{L}$-structure, the primary domain is called ${\mathsf{dom}}^{\circ }(\mathcal{M})$ and the secondary domain is called ${\mathsf{dom}}^{†}(\mathcal{M})$.
Furthermore, the reduct of $\mathcal{M}$ to the primary part (i.e., just the primary domain and the distinguished relations on the primary domain) is called ${\mathcal{M}}^{\circ }$.
Let $\mathcal{I}$ be any full ${\mathcal{L}}_2$-structure. So, $({\mathcal{L}}_2,\mathcal{I})$ is an interpreted language, and $(({\mathcal{L}}_2{)}^{\circ },{\mathcal{I}}^{\circ })$ is the interpreted primary language.
Any full ${\mathcal{L}}_2$-structure $\mathcal{M}$ can be regarded as an $({\mathcal{L}}_2{)}^c$-structure ${\mathcal{M}}^c$, by just forgetting the secondary and mixed relations, but not the secondary domain. So, $(({\mathcal{L}}_2{)}^c,{\mathcal{I}}^c)$ is the interpreted "Carnapian" language.

3. Ramsey sentence
Suppose $\mathrm{\Theta }(M_1,\dots ,M_k,P_1,\dots ,P_n)$ is a finitely axiomatized theory in ${\mathcal{L}}_2$ containing precisely the mixed predicates $M_1,\dots ,M_k$ and the secondary predicates $P_1,\dots ,P_n$. Then the Ramsey sentence of $\mathrm{\Theta }$, written $\mathrm{\Re }(\mathrm{\Theta })$, is:

$\mathrm{\exists }{X}_1\dots X_k\mathrm{\exists }{Y}_1\dots Y_n\mathrm{\Theta }(M_1/X_1,\dots ,M_k/X_k,P_1/Y_1,...,P_n/Y_n)$

where the mixed predicates $M_i$ are replaced by second-order variables $X_i$ (of the right arities) and the secondary predicates $P_i$ are replaced by second-order variables $Y_i$ (of the right arities): we say that the mixed and secondary predicates have been "ramsified".

Note that $\mathrm{\Re }(\mathrm{\Theta })$ is a sentence of the language $({\mathcal{L}}_2{)}^c$, the Carnapian "observational" language, which has first-order variables ranging over observable and unobservable objects, and it has second-order variables ranging over all sets and relations amongst these.

4. Ramsey sentence theorem
Let $\mathcal{I}$ be a full ${\mathcal{L}}_2$-structure. Thus, $(({\mathcal{L}}_2{)}^c,{\mathcal{I}}^c)$ is the corresponding interpreted "Carnapian" language. The Ramsey sentence $\mathrm{\Re }(\mathrm{\Theta })$ is a sentence in this language.

$\mathrm{\Re }(\mathrm{\Theta })$ is true in $(({\mathcal{L}}_2{)}^c,{\mathcal{I}}^c)$
iff
there is a full ${\mathcal{L}}_2$-structure $\mathcal{M}$ such that

i. $\mathcal{M}\models \mathrm{\Theta }$;
ii. $|{\mathsf{dom}}^{†}(\mathcal{M})|=|{\mathsf{dom}}^{†}(\mathcal{I})|$;
iii. ${\mathcal{M}}^{\circ }\cong {\mathcal{I}}^{\circ }$.