Maximality of Logic Without Identity

Abstract

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ 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.

First-Order Friendliness

“…One could look for other relations between models that make sense as specifications of the relation between A and A ; back-and-forth equivalence is mentioned in the Appendix as a possibility. In the context of equality-free logic, one might also consider variants using the relation of weak isomorphism from [3]. Finally, one might go beyond the context of classical logic to explore intuitionistic or modal contexts, in which further specifications of the relation between A and A are suggested by bisimulation and similar notions.…”