  When the Tarski-Givant definition is applied to interpreted languages of mathematical theories with undefined mathematical primitives, it will generally yield the result that the notions denoted by these primitives are not Tarskian logical constants. Tarski’s example is set theory formalized in first-order with a single primitive predicate for membership as a relation among elements of the universe (see Tarski 1986a, p. 153). Obviously membership as a relation over a domain of individuals and sets is not invariant under all permutations of that domain, so it is not declared a logical notion of the language of set theory by Tarski’s proposed definition. Similarly, the class of all sets will be declared non-logical provided the class of individuals that are not sets is not empty. A predicate whose intended meaning is membership (among elements of the universe) is not a Tarskian logical constant, simply because there is a universe in which it denotes a non-logical object (with respect to that universe); similarly, a predicate “S” whose intended meaning is “is a set” (such as is used in some formalizations of set theory suitable for contemplating individuals other than sets in the universe) is not a Tarskian logical constant, for there is a universe in which “S” denotes a non-logical object (in such a universe, the class of objects that are not sets must be non-empty). These results are again in agreement with actual usage, for example in the model theory of first-order set theory, where the membership predicate is taken as a non-logical constant.