 Schaffer (2005) additionally suggests that, if we say that causation is quaternary, we can defend a version of transitivity. Of course, transitivity is a property of a binary relation, but Schaffer proposes that the quaternary relation satisfies the following constraint: if c, rather than c*, is a cause of d, rather than d*, and d, rather than d* is a cause of e, rather than e*, then c, rather than c* is a cause of e, rather than e*. Think of it like this: corresponding to the 4-place causal relation is a two-place relation between event-contrast pairs; and Schaffer maintains that this two-place relation between pairs of events and their contrasts is a transitive relation. It’s not immediately obvious how this helps with switches like the one we saw in  §1.2.1  above. For it seems that Filipa’s flipping the switch to the right, rather than not flipping it, caused the right bulb to be on, rather than off. And it seems that the right bulb being on, rather than off, caused the room to be illuminated, rather than dark. But it doesn’t appear that Filipa’s flipping the switch to the right, rather than not flipping it, caused the room to be illuminated, rather than dark. The trick is that Schaffer has a view of events on which they are world-bound. So, in order to make the first causal claim true, the contrast “the right bulb’s being off” must be an event which only occurs in the world in which Filipa does not flip the switch to the right, and the left bulb is on. Therefore, the second causal claim will turn out to be false; the right bulb’s being on, rather than off in a world in which the left bulb is on, is not a cause of the room’s being illuminated, rather than dark.