Recall ![[Model classes]] >[!info] Definition > Let $\mathcal{L},\mathcal{L^{*}}$ be logics. $\mathcal{L}\leq \mathcal{L}^{*}$ if every class EC in $\mathcal{L}$ is EC in $\mathcal{L}^{*}$. > I.e. if for every $\varphi\in \mathcal{L}$ there is $\varphi'\in\mathcal{L}^{*}$ such that $\mathrm{Mod}_{\mathcal{L}}(\varphi')=\mathrm{Mod}_{\mathcal{L}^{*}}(\varphi)$. > Similarly define $\leq_{PC}, \leq_{RPC}$. > ^e77292 >[!note] Remark > $\mathcal{L}\nleq\mathcal{L}^{*}$: >- If there is $\varphi\in \mathcal{L}$ such that for every $\varphi'\in\mathcal{L}^{*}$, $\mathrm{Mod}_{\mathcal{L}}(\varphi)\ne\mathrm{Mod}_{\mathcal{L}^{*}}(\varphi')$. >- If there is $\varphi\in\mathcal{L}$ such that there are models $M_{1}\vDash_{\mathcal{L}} \varphi , M_{2}\nvDash_{\mathcal{L}} \varphi$ but $M_{1}\equiv_{\mathcal{L}^{*}}M_{2}$. **Proposition**. Assume $\mathcal{L}\leq_{RPC}\mathcal{L}^{*}$ and $\kappa$ to be infinite. Then: (i) If $\mathcal{L}^{*}$ is $\kappa$-compact, then so is $\mathcal{L}$. Hence, if $\mathcal{L}^{*}$ is compact, then so is $\mathcal{L}$. (ii) If $\mathcal{L}^{*}$ has the Löwenheim-Skolem property down to $\kappa$ (every satisfiable sentence has a model of size $\leq \kappa$), then so does $\mathcal{L}$. >[!info] Definition ([[Makowsky, Shelah - The theorems of Beth and Craig in abstract model theory. I. The abstract setting|Makowsky and Shelah]]) >$\mathcal{L}\leq_{\mathrm{Th}} \mathcal{L'}$ if whenever $\mathfrak{A} \equiv_{\mathcal{L'}} \mathfrak{B}$ then also $\mathfrak{A} \equiv_{\mathcal{L}} \mathfrak{B}$.