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

Grades of Discrimination: Indiscernibility, Symmetry, and Relativity

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimi- nation have recently been the subject of much philosophical and technical dis- cussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formu- las. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of rela- tiveness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas, and draws con- nections with definability theory.This is the author accepted manuscript. It is currently under indefinite embargo pending publication by Duke University Press