Set-theoretical Invariance Criteria for Logicality

Abstract

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.