William McCarthy

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.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality.

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.