This kind of argument holds wherever causation of a supervenient property-instance is involved, provided the occurrence of that property-instance is insensitive to the precise manner of its realization. Interestingly, such cases don’t falsify (CX), but that’s because P fails to be a cause of M*. This is a case of M’s difference-making causal role with respect to M* excluding a similar difference-making role for P. What’s gone wrong with the exclusion argument, then? It’s not obvious what we should say. If we substitute ‘difference-making cause’ for ‘cause’ in (CC), then if we treat M* as a physical event (qua behavioural) then (CC) comes out false, because M* doesn’t have a physical difference-maker. If we substitute difference-making causation in (DC), it too comes out false. M isn’t a difference making cause of P*, since (M occurs  P* occurs). Perhaps we could then keep hold of (CC) by employing a more flexible notion of causation. There is, after all, a sense in which M* does have a sufficient physical cause. P is sufficient for P*, and the supervenience relation between P* and M* is synchronic and non-causal, so it would seem foolish to deny that some kind of causal relation holds between P and M*, and I see no reason why this causal relation shouldn’t satisfy proponents of (CC). Let’s say that difference-making causation involves a rejection of (DC): M causes M*, but not by causing P*, so there isn’t any over-determination, and M’s causal role is secure.
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