  Which expressions get counted as logical constants, on this criterion? The monadic predicates “is a thing” (which applies to everything) and “is not anything” (which applies to nothing), the identity predicate, the truth-functional connectives, and the standard existential and universal quantifiers all pass the test. So do the standard first-order binary quantifiers like “most” and “the” (see  the entry on   descriptions).  Indeed, because cardinality is permutation-invariant, every cardinality quantifier is included, including “there are infinitely many”, “there are uncountably many”, and others that are not first-order definable. Moreover, the second-order quantifiers count as logical (at least on the standard semantics, in which they range over arbitrary subsets of the domain), as do all higher-order quantifiers. On the other hand, all proper names are excluded, as are the predicates “red”, “horse”, “is a successor of”, and “is a member of”, as well as the quantifiers “some dogs” and “exactly two natural numbers”. So the invariance criterion seems to accord at least partially with common intuitions about logicality or topic neutrality, and with our logical practice. Two technical results allow us to be a bit more precise about the extent of this accord: Lindenbaum and Tarski (1934–5) show that all of the relations definable in the language of Principia Mathematica are permutation-invariant. Moving in the other direction, McGee (1996) shows that every permutation-invariant operation can be defined in terms of operations with an intuitively logical character (identity, substitution of variables, finite or infinite disjunction, negation, and finite or infinite existential quantification). He also generalizes the Lindenbaum-Tarski result by showing that every operation so definable is permutation invariant.