Counterfactual relations are taken to supervene on the distribution of matter in our space-time and the space-times of other possible worlds. The distribution of matter in our space-time is asymmetric, an asymmetry that is supposed to yield the above-mentioned asymmetry in counterfactuals (Tooley 2002).
Lewis explains that counterfactual relations are what are behind causation and the counterfactual asymmetry implies a causal one: one can cause effects in the future but not in the past. A future pain that is certain is worse than a past pain that is certain in that normally the future depends on us to some extent, and while our impotence to remove a pain from the past is no evil at all but simply a consequence of the fact that we cannot affect the past, our impotence to prevent a future pain is particularly galling, because in general we can affect the future. Likewise, an evil followed by a good might be thought better than a good followed by an evil, because past events are sometimes causes of future ones and we might, perhaps illicitly, think this is so here: and certainly it is better that an evil should cause a good than that a good should cause an evil—since causing a good is itself a good over and beyond the caused good and causing an evil is an evil over and beyond the caused evil. Finally, punishment should be caused by the evil deed, and hence should follow it (Ginsberg 1986).
On Lewis’s account, closeness is analyzed in terms of overall similarity, a real relation of which we are supposed to have some intuitive grasp. A ; C is (a) vacuously true if there are no worlds where A is true (for example, if A is logically impossible); (b) non-vacuously true if, among the worlds where A is true, some worlds where C is true are closer than any world where C is not true; or (c) false otherwise.
Consider an example:
If I had eaten more at breakfast, I would not have been hungry at 11am.
On Lewis’s account, the truth of this statement consists in the fact that, among possible worlds where I ate more for breakfast, there is at least one world where I am not hungry at 11am and which is more similar to our world than any world where I ate more for breakfast but am still hungry at 11am.
Sometimes people condense this slightly difficult mode of expression and simply say that, at the closest world where A is the case, C is the case. So, at the closest world where I eat more breakfast, I don’t feel hungry at 11am. This mode of expression embodies the Limit Assumption, which is just the assumption, made of a given counterfactual, that there is a single closest world where the antecedent is true. Although it is commonly made, and can be useful for exposition, the Limit Assumption will often be strictly false. For example, consider whether there is a closest world where my coffee cup is to the left of its actual position. On the face of it, it seems not; for in principle, there might be an infinite series of worlds, each with my coffee cup a smaller fraction of an inch to the left of its actual position. (See Lewis 1973: 20.)
Ginsberg, M. L. (1986). “Counterfactuals”. Artificial Intelligence, 30: 35-79.
Tooley, M. (2002) Backward Causation and the Stalnaker-Lweis Approach to Counterfactuals University of Colorado at Boulder. University of Colorado University Press.