 Another problem for a strategy of the type suggested by Boolos is that if the second-order quantifiers are interpreted so that they do not range over a separate domain of entities, then there is nothing appropriate to serve as the denotations of \(\lambda\)-expressions. Although Frege wouldn’t quite put it this way, our reconstruction suggests that Frege treats open formulas with free object variables as if they denoted concepts. Although Frege doesn’t use \(\lambda\)-notation, the use of such notation seems to be the most logically perspicuous way of reconstructing his work. The use of such notation faces the same epistemological puzzles that Frege’s Rule of Substitution faces.