The Elimination of Variables by Regular Combinators 1

in a recent paper ² Quine has commented on the nature of variables and the possibility of eliminating them from logic. Such an elimination of variables was achieved by Schonfinkel ³ and has formed an essential feature of what has since been called combinatory logic. ⁴ But Quine finds the Schonfinkel theory unacceptable. He makes, in effect, two criticisms. In the first place he says that the Schonfinkel operators “operate on themselves and one another, whereas our [i.e., Quine’s] six operate only on the original predicates and the predicates thence derived by the operators.” In the second place he says that Schonfinkel operators “presuppose an abstract universe equivalent to that of higher set theory, whereas ours make no ontological demands…