  It may seem that the only principled position is to demand invariance under all permutations. But even that position needs justification, especially when one sees that it is possible to formulate even stricter invariance conditions. Feferman (1999) defines a “similarity invariance” criterion that counts the truth-functional operators and first-order existential and universal quantifiers as logical constants, but not identity, the first-order cardinality quantifiers, or the second-order quantifiers. Feferman’s criterion draws the line between logic and mathematics much closer to the traditional boundary than the permutation invariance criterion does. Indeed, one of Feferman’s criticisms of the permutation invariance criterion is that it allows too many properly mathematical notions to be expressed in purely logical terms.  Bonnay (2008) argues for a different criterion, invariance under potential isomorphism, which counts finite cardinality quantifiers and the notion of finiteness as logical, while excluding the higher cardinality quantifiers—thus “[setting] the boundary between logic and mathematics somewhere between arithmetic and set theory” (37; see Feferman 2010, §6, for further discussion). Feferman (2010) suggests that instead of relying solely on invariance, we might combine invariance under permutations with a separate absoluteness requirement, which captures the insensitivity of logic to controversial set-theoretic theses like axioms of infinity.  He shows that the logical operations that are both permutation-invariant and absolutely definable with respect to Kripke–Platek set theory without an axiom of infinity are just those definable in first-order logic. 