## Notre Dame Journal of Formal Logic

### Invariance and Definability, with and without Equality

#### 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 $\mathscr{L}_{\infty\infty}$ 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 $\mathscr{L}_{\infty\infty}$ 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
Accepted: 5 February 2016
First available in Project Euclid: 25 August 2017

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

Digital Object Identifier
doi:10.1215/00294527-2017-0020

Mathematical Reviews number (MathSciNet)
MR3744354

Zentralblatt MATH identifier
06848194

#### 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

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