Newly Published: Logicism, Ontology, and the Epistemology of Second-Order Logic

In Ivette Fred and Jessica Leech, eds, Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (Oxford: Oxford University Press), pp. 140-69 (PDF here)

Abstract:

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms.

And from the acknowledgements:

It is the peculiar tradition of our tribe to express our respect for other members by highlighting our disagreements with them. So, in case it is not clear, let me just say explicitly how much I admire Bob Hale’s work. I learned a lot from him over the years—both in conversation and from his written work—and greatly enjoyed the time we were able to spend together. Bob’s enthusiastic support for me and my work, early in my career, was particularly important to me. So I am honored to be able to contribute to this volume and thank Ivette and Jessica for the invitation.

I'm particularly sad, for myself, that Bob passed before we had a chance to discuss these issues one more time....