interpolation property

A logic is said to have the interpolation property if whenever $\varphi (R,S)\to \psi (R,T)$ holds, then there is a sentence $\theta (R)$, so that both $\varphi (R,S)\to \theta (R)$ and $\theta (R)\to \psi (R,T)$ hold, where $R,S$ and $T$ are some sets of symbols that occur in the formulas, $R$ being the set of symbols common to both $\varphi$ and $\psi$.

The interpolation property holds for first order logic. The interpolation property is related to Beth definability property and Robinson’s consistency property. Also, a natural generalisation is the concept $\mathrm{\Delta }$-closed logic.