>[!info] Definition >A (finite set of) $n$-ary relation(s) $\mathcal{R}$ and a logic $\mathcal{L}^*$ are *symbiotic* if the following conditions are satisfied: (1) Every $\mathcal{L}^*$-definable model class is $\Delta_1(\mathcal{R})$-definable. (2) Every $\Delta_1(\mathcal{R})$-definable model class closed under isomorphisms is $\Delta\left(\mathcal{L}^*\right)$-definable. [[Väänänen - Abstract logic and set theory. I. Definability]] [[Bagaria, Väänänen - On the symbiosis between model-theoretic and set-theoretic properties of large cardinals]] ![[Bagaria, Väänänen - On the symbiosis between model-theoretic and set-theoretic properties of large cardinals#Examples of symbiosis]]