Any adequate theory of causation must be able to account for the apparent fact that causes precede their effects and not the other way around. If event A causes event B, then it would seem that A occurs before B in time. Perhaps A is simultaneous with B and therefore causes B at the same moment of time, but we would still insist that A is causally prior to B if not temporally prior. To say that causes precede their effects but not visa-versa is to say that the relation of causation is “asymmetric” – i.e. the relation only goes in one direction.
As I understand him, David Lewis (1979) claims that the direction of causal relations can be explained by the conditions and laws of nature operating in a world. The conditions in a world refer to the events or states of affairs that hold in that world – at various times and places. The laws of nature are the truths about what must (or is likely to) happen when those conditions obtain. But how do laws and conditions tell us what the direction of causation is in a world?
To answer this question, Lewis distinguishes between necessary and sufficient conditions. Event A is a necessary condition of event B if A (together with the laws of nature) partly determines B but not fully. Thus, A is needed for B, but it does not guarantee B. By contrast, C is a sufficient condition of B if C (together with the laws of nature) fully determines B. C might not be needed for B, but if C occurs, then B’s occurrence is guaranteed by the laws of nature. To say that C is sufficient for B is just to say that C is a “determinant” of B.
Now, imagine a world W which is represented pictorially by events ordered on a time line. Suppose that event B is in the middle of the time line; that events of type A are mostly on the left side of B; and that events of type C are mostly on the right side of B. Given the above definition of what a “determinant” is, it follows that, in W, most of the determinants of B are on its right rather than on its left. In other words, B has more post-determinants than pre-determinants.
Can we learn anything about the direction of causation in worlds like W? Lewis thinks we can. Why? Because he analyses the direction of causation in terms of the direction of most determinants for an event. Since event B has more post-determinants than pre-, it follows (on Lewis’ theory) that the arrow of causation in W goes in the A-B-C direction, not the C-B-A direction.
To see why this is the case, consider the example of a world governed by indeterministic laws like those from quantum theory. In this world, the existence of an energy field (A) is nomically necessary for the occurrence of a tiny energy fluctuation (B), but the field itself is not nomically sufficient to make the fluctuation happen; A only increases the likelihood of B. Thus, event A is not a determinant of B. However, things are different in the reverse direction. Suppose that the energy fluctuation triggers the inflation of a baby universe (C). The inflation is sufficient to guarantee that a fluctuation has occurred since the laws of quantum theory require it. Thus, C is a determinant of B because it entails B.
What can we infer from this? We learn that when indeterministic laws are at work in worlds like one above, we generally find more post-determinants than pre-determinants of their events. If so, then Lewis has provided us with a basis for explaining the asymmetry of causation in those worlds. But does Lewis’ analysis similarly work for classically Newtonian worlds whose laws are deterministic? That will be the subject of a future post!
 I am assuming here that simultaneous causation is possible. Such an assumption would be false if the direction of time can be reduced to the direction of causation. For arguments in favour of this reduction, see Robin Le Poidevin’s Travels in Four-Dimensions: The Enigmas of Space and Time. Oxford University Press, 2003. pp. 202-233
 One might object that the prefixes of pre- and post- already assume a temporal direction and therefore cannot explain that direction. But this objection is misguided. The ordering of the events on the world-line is due to the laws of nature, whereas the direction of causation is due (for Lewis) to the location of most determinants on that ordering.
 This paragraph is not strictly speaking correct because the direction of most determinants fixes the direction of counterfactual dependence in W, which in turn fixes the direction of causation and time. But for the sake of simplicity, I have omitted mention of counterfactual dependence in the chain of Lewis’ reductive analysis.