This is a survey of work on set-theoretical invariance criteria for logicality.
It begins with a review of the Tarski-Sher thesis in terms, first, of
permutation invariance over a given domain and then of isomorphism invariance
across domains, both characterized by McGee in terms of definability in the
language $L_{\omega,\omega}$ . It continues with a review of critiques of the
Tarski-Sher thesis and a proposal in response to one of those critiques via
homomorphism invariance. That has quite divergent characterization results
depending on its formulation, one in terms of FOL, the other by Bonnay in terms
of $L_{\omega,\omega}$ , both without equality. From that we move on to a survey
of Bonnay's work on similarity relations between structures and his results that
single out invariance with respect to potential isomorphism among all such.
Turning to the critique that calls for sameness of meaning of a logical
operation across domains, the paper continues with a result showing that the
isomorphism invariant operations that are absolutely definable with respect to
KPU–Inf are exactly those definable in full FOL; this makes use of an
old theorem of Manders. The concluding section is devoted to a critical
discussion of the arguments for set-theoretical criteria for logicality.