Open Access
2018 Invariance and Definability, with and without Equality
Denis Bonnay, Fredrik Engström
Notre Dame J. Formal Logic 59(1): 109-133 (2018). DOI: 10.1215/00294527-2017-0020


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 L 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 L 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.


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Denis Bonnay. Fredrik Engström. "Invariance and Definability, with and without Equality." Notre Dame J. Formal Logic 59 (1) 109 - 133, 2018.


Received: 7 August 2013; Accepted: 5 February 2016; Published: 2018
First available in Project Euclid: 25 August 2017

zbMATH: 06848194
MathSciNet: MR3744354
Digital Object Identifier: 10.1215/00294527-2017-0020

Primary: 03A99
Secondary: 03C75 , 03C80

Keywords: automorphism groups , definability , equality-free languages , generalized quantifiers , infinite languages , Invariance

Rights: Copyright © 2018 University of Notre Dame

Vol.59 • No. 1 • 2018
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