Frege’s Little Theorem and Frege’s Way Out

Shortly before the second volume of Basic Laws went to press, Frege appended an Afterword, analyzing the Russell paradox and suggesting an alternate formulation of basic law V. This attempted ‘way out’ has been characterized as “the wrong guess of a man in a hurry” and has been accused of leading to either paradoxes or absurd consequences. The former claim is wrong, and the latter, although correct, has until now been misunderstood. The bulk of the paper works though a new series of proofs—ones that pay closer attention to whether we are working in Frege’s original formalism or modern higher-order logic, and which do not consist merely of un-illuminating reductios—that highlight exactly where Frege’s attempted solution goes wrong. As a result, it becomes clear that Frege’s ‘way out’, although failing in the end to avoid Russellian problems, is far from merely a “guess” or the result of haste. On the contrary, the amended principle results from a natural application of theorems about, and insights into, abstraction provided by Frege himself in the Afterword. Ironically, however, natural generalizations of these theorems and insights (ones not noticed by Frege) point to precisely the problems that were later found to besiege the amended law V.