# Definitions Let $L^*$ and $L^{\dagger}$ be two logics, $L^* \leq L^{\dagger}$ (see [[Comparing logics]]). For a class of structures $K$ denote by $\bar{K}$ its complement (with respect to structures of the same language). ## Interpolation properties ### Craig interpolation >[!info] Definition >We say that *$L^{\dagger}$ has the (Craig) interpolation property for $L^{\ast}$*, or that *$\operatorname{CRAIG}\left(L^{\ast}, L^{\dagger}\right)$ holds*, if whenever $K_1, K_2 \in \operatorname{(R)PC}\left(L^{\ast}\right)$ and $K_1 \cap K_2=\varnothing$ then there is $K_3 \in \mathrm{EC}\left(L^{\dagger}\right)$ such that $K_1 \subseteq K_3 \subseteq \bar{K}_2$ . ^craig-interpolation This generalizes the following: >[!info] Definition >$\mathscr{L}$ has the *Craig* or *interpolation property* iff for all $\tau_0, \tau_1:$ if $\varphi_i \in \mathscr{L}\left[\tau_i\right]$ $(i=0,1)$ and $\varphi_0 \vDash \varphi_1$, then there is an *interpolant*, that is, a sentence $\psi \in \mathscr{L}\left[\tau_0 \cap \tau_1\right]$ such that $\varphi_0 \vDash \psi$ and $\psi \vDash \varphi_1$ (provided - in the manysorted case - that $\tau_0 \cap \tau_1$ contains at least one sort symbol). ### $\Delta$-interpolation An important weakening of interpolation is the following: >[!info] Definition >We say that *$L^{\dagger}$ has the $\Delta$-interpolation property for $L^{\ast}$*, or that *$\Delta\operatorname{-Int}(L^{\ast}, L^{\dagger})$ holds* if for every class $K$, $K, \bar{K} \in \mathrm{PC}(L^{\ast})$ ($K$ is *$\Delta$ in $L^{\ast}$*) implies $K \in \mathrm{EC}(L^{\dagger})$. >$\Delta(L^{\ast})$ ([[Delta-extension]]) is the smallest extension $L^{\dagger}$ of $L^{\ast}$ such that $\Delta\operatorname{-Int}(L^{\ast}, L^{\dagger})$ holds. ^delta-interpolation This generalizes the following: >[!info] Definition >A logic $\mathcal{L}$ satisfies the *Souslin-Kleene* property if every class of structures $K$ which is $\Delta$ in $\mathcal{L}$ is definable in $\mathcal{L}$. ### Failures of interpolation **Theorem (Keisler 1971).** $\mathcal{L}(Q_{1})$ doesn't have the Souslin-Kleene property *Proof idea.* Let $\varphi_0(E, R)$ express that $E$ is an equivalence relation with only uncountable equivalence classes and that $R$ is a countable set of representatives. Let $\varphi_1(E, S)$ express a similar statement with $S$ being an uncountable set of representatives. $ \varphi_0(E, R) \vDash \neg \varphi_1(E, S) $ holds, but Keisler shows there is no $\mathcal{L}\left(Q_1\right)$-interpolant. This also gives a counterexample to $\Delta$-interpolation. ## Definability properties. ### Implicit and explicit definability, Beth property >[!info] Definition >Let $\tau$ be a vocabulary, $R \in \tau$ a relation symbol, $\varphi \in L^{\ast}[\tau]$ >1. $R$ is *(strongly) implicitly defined in $L^{\ast}$ by $\varphi$*, if every $\tau \backslash\{R\}$-structure has at most (exactly) one expansion to a $\tau$-structure satisfying $\varphi$. >2. $R$ is *explicitly definable in $L^{\dagger}$ relative to $\varphi$* if >$ >\{(\mathfrak{A},\bar{a}) \mid \mathfrak{A} \vDash \varphi \land \bar{a}\in R^{\mathfrak{A}}\} \in \operatorname{EC}(L^{\dagger}[\tau]) >$ >3. We say that $L^{\dagger}$ has the *(weak) Beth property* for $L^{\ast}$, or that $\operatorname{BETH}\left(L^{\ast}, L^{\dagger}\right)$ ($\operatorname{WBETH}\left(L^{\ast}, L^{\dagger}\right)$) holds, if every relation which is (strongly) implicitly definable in $L^{\ast}$ is explicitly definable in $L^{\dagger}$. ### Joint consistency, Robinson property Joint consistency. $T \subseteq L^{\ast}$ is $L^{\ast}$-complete if whenever $T$ is a set, $\mathfrak{X} \neq T$ and $\mathfrak{O} \vDash T$ then $\mathfrak{U} \equiv \mathfrak{B}\left(L^{\ast}\right)$. We say that $\operatorname{ROB}\left(L^{\ast}, L^{\dagger}\right)\left(\operatorname{WROB}\left(L^{\ast}, L^{\dagger}\right)\right)$ holds if whenever $T \subseteq L^{\dagger}$ is $L^{\dagger}$-complete, $L_1=L \cup\{P\}, P$ an $n$-ary predicate symbol, and $\varphi, \psi \in L_1^{\ast}$ are such that $T \cup\{\varphi(P)\}$ and $T \cup\{\psi(P)\}$ have a model, then $T \cup\left(\left\{\varphi(P), \psi\left(P^{\prime}\right)\right\}\right)\left(\left\{\varphi(P), \psi\left(P^{\prime}\right)\right\}\right)$ has a model (where $P^{\prime}$ is a new $n$-ary predicate symbol not in $L_1$ and $\psi\left(P^{\prime}\right)$ is the result of substituting $P^{\prime}$ for $P$ in $\psi$ ). # Sources [[Barwise, Feferman (eds.) - Model-theoretic logics#Chapter II. Extended Logics The General Framework]] section 7. [[Makowsky, Shelah - The theorems of Beth and Craig in abstract model theory. I. The abstract setting]]