What is PA + con(PA) about, and where?

Abstract

Justin Clarke-Doane offers what purports to be a stand-alone argument, relying on Gödel’s second incompleteness theorem, that if we hold that PA + Con(PA) and PA + ~ Con(PA) are equally true of their intended subjects, then there is no objective fact as to whether PA is consistent. It is shown that the argument is fallacious, although illuminating: The fallaciousness of the argument arises from a 20th-century shift in our understanding of interpreted languages from the view—derived from our experience of language—of sentences as intrinsically interpreted to one which sharply distinguishes syntax and semantics, and treats uninterpreted syntactic forms as endowed with interpretations by models. It is shown that this syntax/semantic distinction, because it is “unintuitive” induces fallacies such as the one that Clarke-Doane’s argument exemplifies.