Revenge Russell‐Myhill and the Ramified Hierarchy

ABSTRACT The Russell‐Myhill paradox targeting the structuralist view of proposition in classical higher‐order logic is blocked by implementing restrictions associated with an ordinary account of predicativity. However, it has been shown that these restrictions are not safe: “revenge” versions of the paradox show the inconsistency of the structuralist view in what is ordinarily regarded as a predicative setting. This result has been interpreted as suggesting an expressive limitation of the structuralist view. In this paper, I argue otherwise. There is a metaphysical sense in which structured propositions presuppose properties appearing as their constituents. I use this idea to elaborate a more thoroughgoing understanding of predicativity. In this way, the structuralist has the resources to interpret revenge paradoxes as a byproduct of impredicativity and coherently block them by means of a more subtle regimentation on how properties are defined.