Maximality of logic without identity

Abstract

Lindström theorem obviously fails as a characterization of ${\mathcal{L}}_{\omega \omega }^{-}$, first-order logic without identity. In this note we provide a fix: we show that ${\mathcal{L}}_{\omega \omega }^{-}$ is \emph{maximal} among abstract logics satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in \cite{Casa}), 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.