  Say that a state of affairs is structural if it is atomic and  its constituent universal is structural or it is molecular and  includes a structural state of affairs; and say that it is  simple otherwise. The difference between structural and simple universals and states of affairs is particularly significant with  regard to the important concept of supervenience (Armstrong  1989, Ch   8).[45]  Entity or entities S supervene on entity or entities R  if and only if the existence of R necessitates that of  S (ibid., 103). (Necessitation here is, of course,  ultimately to be spelled out in terms of combinatorial possible  worlds.) Non-structural states of affairs supervene directly on their atomic   conjuncts.[46]  However, things are not in general quite so straightforward for  structural states of affairs. For, although structural states of  affairs are ultimately constituted entirely by simple states of  affairs, unlike non-structural states of affairs, structural states  of affairs typically supervene on more than the totality of their  constituents. For, in many cases, whether or not a structural fact  exists depends not only on the existence of its constituent facts but also on the absence of certain others (Armstrong 1997,  34ff). For example, as noted in our example above, our water molecule W comprises two further facts in which two hydrogen atoms  h1 and h2 both stand in the  covalent binding relation with an oxygen atom o. However, if  o were to bind with a further hydrogen atom  h3, then, despite the fact that the constituent  facts of W would still hold, W would not be water;  there would be no such fact as W's being   water.[47]  Rather, W would exist only as a complex part of a hydronium  ion; the new binding [B,o,h3] would,  so to say, “spoil” the instantiation of Water.  Thus, more generally, whether or not a structural state of affairs  S exists in a possible world typically requires something over and above its constituent states of affairs being “welded  together” in the right sort of way (Armstrong, 1997, 36); it  requires also that there be no relevant “spoilers” for   S.[48]  Armstrong draws directly on the initial passages of the   Tractatus[49]  for the necessary apparatus: a structural state of affairs S  in any possible world w, supervenes, not simply on its  constituent atomic states of affairs but on a certain  higher-order state of affairs Tw,  namely, the state of affairs that the (first-order) atomic states of  affairs of w are all the (first-order) atomic states of  affairs and, hence, that w includes no spoilers for  S. Armstrong (ibid., 35, 134–5, 196–201) calls  Tw the totality state of affairs for the  atomic states of affairs of   w.[50] 