## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 58, Number 4 (2017), 527-553.

### Grades of Discrimination: Indiscernibility, Symmetry, and Relativity

#### 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.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 58, Number 4 (2017), 527-553.

**Dates**

Received: 26 July 2013

Accepted: 20 March 2015

First available in Project Euclid: 25 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1493085740

**Digital Object Identifier**

doi:10.1215/00294527-2017-0007

**Mathematical Reviews number (MathSciNet)**

MR3707650

**Zentralblatt MATH identifier**

06803186

**Subjects**

Primary: 00A30: Philosophy of mathematics [See also 03A05]

Secondary: 03C40: Interpolation, preservation, definability 03C99: None of the above, but in this section

**Keywords**

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

#### Citation

Button, Tim. Grades of Discrimination: Indiscernibility, Symmetry, and Relativity. Notre Dame J. Formal Logic 58 (2017), no. 4, 527--553. doi:10.1215/00294527-2017-0007. https://projecteuclid.org/euclid.ndjfl/1493085740