Grades of discrimination: indiscernibility, symmetry, and relativity. (English) Zbl 1386.00057

Button, Tim

Notre Dame J. Formal Logic 58, No. 4, 527-553 (2017); erratum ibid. 59, No. 1, 135-138 (2018).

Summary: There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. 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 relativeness 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 connections with definability theory.

MSC:

00A30	Philosophy of mathematics
03C40	Interpolation, preservation, definability
03C99	Model theory

Keywords:

identity of indiscernibles; grades of indiscernibility; grades of symmetry; grades of relativity; equality-free model theory; identity-free model theory