Second-Order Relations and Nomic Regularities

Bird's Ultimate Argument sought to show that Armstrong's N relationships involving categorical universals can't entail nomic regularities. In N's place Bird offered a non-categorical SR relation. Two kinds of objection have been raised: either Bird's own alternative metaphysics fails in just the same way as Armstrong's or the target of Bird's argument may anyway have a way out of the problem. My aim is to reclaim the victory for Bird. I argue that the responses in defence of Armstong's N relationships fail to acknowledge that Bird was explicitly concerned with Armstrong's commitment to a categoricalist view of universals. Moreover, Bird's alternative account does not suffer the same problem since his metaphysics of properties is essentialist. Nevertheless, Bird's account does need elaborating on to explain why SR relationships entail their regularities. I offer Schaffer's Axiomatic Solution as a candidate for this purpose.