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Invariance and Definability, with and without Equality

Journal article
Authors Dennis Bonnay
Fredrik Engström
Published in Notre Dame Journal of Formal Logic
Volume 59
Issue 1
Pages 109-133
ISSN 0029-4527
Publication year 2018
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 109-133
Language en
Keywords generalized quantifiers infinite languages invariance definability automorphism groups equality-free languages
Subject categories Mathematical logic, Logic


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∞∞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∞∞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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