Quine's Conjecture on Many-Sorted Logic

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it's true, at least according to one reasonable notion of theoretical equivalence. Our clarification of Quine's conjecture, however, exposes the shortcomings of his argument against many-sorted logic.