MAXIMALITY OF LOGIC WITHOUT IDENTITY

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( L − ωω ). In this note, we provide a fix: we show that L − ωω is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity

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 intersection of 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 compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. 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