Generalized Keisler Theorems for First-order Logic and Protologics

The Keisler theorems dealing with the definability in first-order logic of classes of structures are generalized and adapted to non-classical logics. On the one hand, we generalize and prove by completely different means an analogue of the Keisler-Shelah isomorphism theorem for first-order logic and countable languages, where the notion of isomorphism is replaced in that theorem by a variant of partial isomorphism. On the other hand, we adapt the Keisler theorems for first-order logics to protologics, i.e. logics such that the truth conditions of their connectives are expressible by first-order formulas. Our results are based on similar theorems for atomic and molecular logics proved in a companion article [5]. We indeed show in this present article that first-order logic and protologics are as expressive as some atomic and molecular logics