Grades of Discrimination: Indiscernibility, symmetry, and relativity

TL;DR

This paper provides a comprehensive technical analysis of various grades of discrimination, including indiscernibility, symmetry, and relativity, exploring their interrelations and definability in logical structures.

Contribution

It introduces grades of relativity and examines their relationships with existing grades, completing the technical investigation of discrimination levels in structures.

Findings

Grades of indiscernibility are characterized by first-order formulas.

Grades of symmetry are defined via symmetries on structures.

Grades of relativity are introduced and related to existing grades.

Abstract

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.