Againt Armstrong on Causation

I’m working on a short paper arguing that Armstrong’s account of causation fails. The argument seems so simple that I’m worried I’ve missed something obvious. Any suggestions are much appreciated.

Armstrong identifies singular causal relations with instantiations of a law—with instances of the necessitation relation. Let N(P, Q) represent P’s necessitation of Q. In the case of determinism, N(P, Q) is something like P probabilifies Q to degree x.

I claim that Armstrong’s theory fails in indeterministic contexts. Whenever N(P, Q) holds, every instance of P is related by N to Q. After all, instances of universals are, according to Armstrong, nothing other than the universal itself. But then in cases where P occurs but does not cause Q, the instance of P is still related by N to Q. The law is instantiated, but causation does not occur. Hence, causation is not the instantiation of a law.

Here’s a slightly different way of putting the same problem: Assume indeterminism. Let it be an indeterministic law that N(P, Q). Assume there is an instance of P that is not followed by an instance of Q. (This is possible, given the assumption of indeterminism.) Either the instance of P is related by an instance of N to Q or it is not. Suppose it is. Then P caused Q, contrary to our assumption. On the other hand, suppose P is not related by N to Q. Then there is an instance of P not related by N to Q. But, then it can’t be a law that N(P, Q), contrary to our assumption.

It seems to me that Armstrong needs singular causal relations in addition to laws, so that in indeterministic contexts, the law can be instantiated without the singular causal relation holding.

Lewis and Vague Laws of Nature?

Sometimes people seem to assume that David Lewis took the notion of law of nature to be (somewhat) vague (which I take to mean that 'x is a law of nature at @' has borderline cases), but does Lewis say that explicitly anywhere? (On the face of it, it would seem to be in contrast with the best system thesis being expressed as a biconditional, but, on the other hand, Lewis seems to concede that strength and simplicity are somewhat vague criteria.) And would any other sophisticated regularity theorist be happy with that?

(Crossposted at It's Only A Theory)

Laws, Counterfactuals, and Essential Properties

I find it curious that nobody seems to be particularly bothered by the fact that the following three commonly-held and seemingly plausible theses seem to be somewhat at odds:

1. Unlike accidental generalizations, nomic generalizations support counterfactual conditionals. (So, for example, if it is a law that copper is a good conductor, then, if this piece of wood was made of copper, it would be a good conductor.)
2. Some properties are essential to their bearers (So, for example, it is metaphysically impossible for this piece of wood to be made of anything other than wood and a fortiori to be made of copper).
3. Counterfactuals whose antecedent is necessarily false are vacuously true.

The conflict seems to arise from the fact that, since laws of nature often involve essential properties, if (2) and (3) are true, (1) would not seem to be generally true--many accidental generalization would seem to support (vacuously true) counterfactuals just like nomic generalizations do.

Now, I'd be curious to hear which one(s) of the above theses (if any) the readers of this blog think should be amended/rejected in order to resolve the conflict and why. (I do have a main suspect, but, in order to avoid skewing my little survey, I'm not going to reveal its identity for the moment).