A Benacerraf problem for higher-order metaphysics

Higher-order metaphysics is in full swing. Its proponents argue that higher-order logic should replace set theory at the foundations of mathematics and metaphysics. But amid the enthusiasm, surprisingly little attention has been paid to some serious epistemological challenges facing the program—foremost among them a variant of the Benacerraf challenge, developed by Field and Clarke-Doane. Roughly put, the challenge is to explain the reliability of our higher-order logical beliefs. A similar problem is familiar from the philosophy of set theory, where it has led to a pluralist reconception of the foundations of mathematics. In this paper, I argue that regardless of whether higher-order logic is preferable to set theory on abductive grounds, they stand or fall together when faced with this epistemological challenge. They are companions in guilt (or innocence). I conclude that, absent other solutions, a promising path forward is to adopt a pluralist approach to higher-order logic. The consequences of such a shift are difficult to overstate.