Permutation Arguments and Kunen’s Inconsistency Theorem

I offer a variant of Putnam’s “permutation argument,” originally an argument against metaphysical realism. This variant is called the “natural permutation argument.” I explain how the natural permutation argument generates a form of referential inscrutability which is not resolvable by consideration of “natural properties” in the sense of Lewis’s response to Putnam. However, unlike the classical permutation argument (which is applicable to nearly all interpretations of all first-order theories), the natural permutation argument only applies to interpretations which have some special symmetries. I give an analysis of the interpretations to which the natural permutation argument does apply, and I explain how, when it fails to apply, the referential inscrutability generated by permutation arguments is resolvable by a Lewisian strategy. In order to demonstrate how these problems of referential inscrutability play out in an a priori setting relevant to philosophy, I discuss the applicability of the natural permutation argument in set-theoretic reasoning. I use the well-known Kunen inconsistency theorem to show that, in Zermelo–Fraenkel set theory, the Axiom of Choice is sufficient to resolve referential inscrutability. I then explain how, as a result of a recent theorem of Daghighi–Golshani–Hamkins–Jeřábek, in certain non-well-founded set theories the natural permutation argument does yield an intractable inscrutability of reference.