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 $\mathsf{dom}^{\dagger}(\mathcal{M})$.

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 $\Theta(M_1, \dots, M_k, P_1, \dots, P_n)$ 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 $\Theta$, written $\Re(\Theta)$, is:

$\exists X_1 \dots X_k \exists Y_1 \dots Y_n \Theta(M_1/X_1, \dots, M_k/X_k, P_1/Y_1, ..., P_n/Y_n)$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 $\Re(\Theta)$ 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 $\Re(\Theta)$ is a sentence in this language.

$\Re(\Theta)$ is true in $((\mathcal{L}_2)^{c}, \mathcal{I}^{c})$

iff

there is a full $\mathcal{L}_2$-structure $\mathcal{M}$ such that

i. $\mathcal{M} \models \Theta$;

ii. $|\mathsf{dom}^{\dagger}(\mathcal{M})| = |\mathsf{dom}^{\dagger}(\mathcal{I})|$;

iii. $\mathcal{M}^{\circ} \cong \mathcal{I}^{\circ}$.

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