The Modelwise Interpolation Property of Semantic Logics

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi$ formulated in the intersectionof the vocabularies of \(\phi$ and \(\psi\), such that \(\mathfrak{M}\models \phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We discuss examples and show that while the \(n\)-variable fragment of first order logic and difference logic have no Craig interpolation, they both have the modelwise interpolation property. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. In particular, the class of finite dimensional cylindric set algebras enjoys this weak form of superamalgamation.