In a previous post, we discovered that when indeterministic laws apply to a state of affairs, there tend to be more post-determinants of that state than pre-determinants. Thus, if the direction of causal relations is fixed by the direction of most determinants, then there is a basis for explaining the apparent asymmetry of causal relations. This is David Lewis’ view, which I summarized last week.
But does Lewis’ view give us the correct direction of causation when deterministic laws are at work, such as in classically Newtonian worlds?
THE PROBLEM OF NEWTONIAN WORLDS
To answer the above question, let’s revisit the distinction between necessary and sufficient conditions by applying it to an indeterministic world [W] which contains only three fully-described states: A, B, and C.
In W, state A is necessary for state B, and B is necessary for state C. We also know that state C is sufficient for B, and B is sufficient for state A. Since C is a determinant of B, but A is not a determinate of B, it follows by Lewis’ analysis that the arrow of causation runs in the A-B-C direction, not the other way around. So far so good.
But what happens when we apply the very same distinction to a simple deterministic world [W*] whose laws of physics are strictly Newtonian? In W*, state C is sufficient for B, and state B is sufficient for A, just like in W. But unlike W, state A is sufficient for B, and state B is sufficient for C. That’s because a full description of any state in W* is sufficient (along with the laws of physics) to determine all the states in that world. That’s part of what it means for a world like W* to be Newtonian.
But if Lewis analyzes the causal arrow in W* in terms of the direction of most determinants, then his theory is in trouble. Why? Because state B has just as many post-determinants as pre-determinants! Lewis would then be forced to conclude that the causal arrow goes in both or neither direction in W*, which is absurd. Thus, Lewis’ analysis does not give us the correct direction of causation in Newtonian worlds like the one above.
POSSIBLE WAYS OUT?
I can think of at least three alternative strategies which Lewis could use to derive the correct direction of causation in Newtonian worlds.
First, he could analyze the direction of causation in terms of time’s arrow. Unfortunately, this option is of little help to Lewis because his theory also reduces time’s arrow that of most determinants. Thus, the same problems with using determinants to explain causal asymmetry would apply with equal strength to his explanation of temporal asymmetry.
Second, Lewis could adopt a different brand of reductionism which analyzes the arrows of time and causation in terms of entropy increase – not in term of determinants. Other philosophers have tried this strategy, but they’ve met with serious difficulties. As Richard Swinburne (1997) points out, universes like ours might contain large regions of space which exhibit no overall increases in entropy (as when a universe ends in eternal expansion). This would mean that time eventually stops even as space continues to grow, which is rather bizarre if events are still occurring.
Third, Lewis could bite the bullet and defend a realist view of time. A realist would say that time-direction is a bedrock feature of reality which cannot therefore be reduced to anything more fundamental. This strategy would fix the direction of causation quite nicely, but it would also come at a cost. It would undercut the entire spirit of Lewis’ program, which is to reduce things like time, causation, and counterfactuals to contingent arrangements of matters of particular fact. Opting for a realist view of time would not be in keeping with this program.
 In Counterfactual Theories of Causation (2014), Peter Menzies says that “[Lewis] defines a determinant for an event as any set of conditions jointly sufficient, given the laws of nature, for the event’s occurrence.”
 David Lewis, Counterfactual Dependence and Time’s Arrow (1979), p.475. The asymmetry only holds in most cases, however, because Lewis still believes that backward causation is a logical possibility.
 There are, of course, several other reductionist strategies open to Lewis. I only mention one for the sake of brevity here.