## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 59, Number 1 (2018), 109-133.

### Invariance and Definability, with and without Equality

Denis Bonnay and Fredrik Engström

#### Abstract

The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in ${\mathcal{L}}_{\infty \infty}$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee’s theorem about quantifiers invariant under all permutations and definability in pure ${\mathcal{L}}_{\infty \infty}$ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 59, Number 1 (2018), 109-133.

**Dates**

Received: 7 August 2013

Accepted: 5 February 2016

First available in Project Euclid: 25 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1503626493

**Digital Object Identifier**

doi:10.1215/00294527-2017-0020

**Mathematical Reviews number (MathSciNet)**

MR3744354

**Zentralblatt MATH identifier**

06848194

**Subjects**

Primary: 03A99: None of the above, but in this section

Secondary: 03C75: Other infinitary logic 03C80: Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48]

**Keywords**

generalized quantifiers infinite languages invariance definability automorphism groups equality-free languages

#### Citation

Bonnay, Denis; Engström, Fredrik. Invariance and Definability, with and without Equality. Notre Dame J. Formal Logic 59 (2018), no. 1, 109--133. doi:10.1215/00294527-2017-0020. https://projecteuclid.org/euclid.ndjfl/1503626493