 This definition appeals to the notion of (type) influence; so it will not allow us to gives a reductive analysis of influence. But notice that it doesn’t appeal to influence between X and Y—and that is the relation which Woodward is attempting to characterize. So while the series of definitions are not reductive, they are not viciously circular. They tell us something about how influence between X and Y relates to influence between variables besides X and Y. (See the entry on  causation and manipulability  for more.)