  In Tarski and Lindenbaum (1935) the authors had proved that given a basic universe U, all the notions in derivative universes of U which can be defined in the language of the simple theory of types are invariant under all permutations of U. In general the Tarski-Lindenbaum theorem guarantees that all mathematical notions definable in the logicist fashion in the simple theory of types are logical notions no matter what the universe of individuals is taken to be (Tarski says that “we may interpret [the universe of the intended interpretation of the language of the theory of types] as the universe of physical objects, although there is nothing in Principia mathematica which compels us to accept such an interpretation” (Tarski 1986a, p. 152)). Since the theorem applies to every universe U supplying an interpretation of the language of the theory of types, the definition of the concept of logical constant in Tarski and Givant (1987) implies that all primitive symbols denoting notions in that language (e.g. quantifiers of all orders) are Tarskian logical constants; also, if the definition were applicable to defined symbols, all these symbols would be Tarskian logical constants. Such results agree well with a “usage actually encountered in practice” (for example, the practice of the logicists, but of others as well) according to which the constants of the language of the theory of types are logical constants.