More on Causation

June 4, 2008

Say that an event, e1 is weirder than an event e2 just in case every world in which e2 doesn’t occur, there is a closer world where e1 doesn’t occur. e1’s failing to happen accords better with our laws and whatever else determines similarity, than e2’s failing to happen. Basically things would be more normal if e1 hadn’t happened, more normal than if e2 hadn’t happened.

Secondly suppose that for any two events, x and y, there is a fusion of those events, which occurs just in case x and y both occur.

• (1) $\Box(O(x + y) \leftrightarrow (O(x) \wedge O(y)))$

With this machinery in place, I think we can run a kind of triviality result for Lewis’s account of causation. Suppose that e2 is weirder than e1. Then we get the following counterfactual

• (2) $(\neg (O(e_1) \wedge O(e_2)) \Box\!\!\!\rightarrow \neg O(e_2))$

Why’s that? Suppose w is a closest world in which $(\neg O(e_1) \vee \neg O(e_2))$ holds. We want to show that $\neg O(e_2)$ holds at w. Suppose it doesn’t. Then $\neg O(e_1)$ must hold to make the disjunction true. Since e2 happening is weirder, there must be a closer world where e2 fails to happen. Thus there is a world v such that $(\neg O(e_1) \vee \neg O(e_2))$ holds, and v is closer than w. This contradicts the assumption that w was closest.

Now here is the problem for Lewis. By (1) we get the strict conditional $\Box (\neg O(e_1) \rightarrow \neg O(e_1 + e_2))$ which gives us the counterfactual

• (3) $(\neg O(e_1) \Box\!\!\!\rightarrow \neg O(e_1 + e_2))$

Substituting $O(e_1 + e_2)$ (by (1)) into the antecedent of (2) we get

• (4) $(\neg O(e_1 + e_2)) \Box\!\!\!\rightarrow \neg O(e_2))$

But (3) and (4) just constitutes a chain of causal dependence between e1 and e2.

But this is intolerable! An event causes every event weirder than it!

This was all a little abstract, so lets take a more down to earth example to fix ideas. Suppose we have two events, with one weirder than the other. For an uncontroversial example, the event that I sneeze, and the event that it rains down frogs (this occasionally happens when tornados suck them up out of the water.) The idea is that if I hadn’t sneezed, then the joint event of my sneezing and it raining frogs would not have occurred. But if at least one of the following obtained: (a) it hadn’t rained frogs or (b) I hadn’t sneezed, then it would have been the former that obtained: (a) it wouldn’t have rained frogs, because that is closer to normality. Thus, it seems as if my sneezing caused it to rain frogs.