What I want to discuss are Barker's central claims connecting chance and cause, which he calls CC1 and CC2.

CC1: If c causes e, c contributes to the chance of e at tWe found both of these principles objectionable. In this post I'll discuss some of our worries about CC1; I'll discuss CC2 in a later post I hope._{c}, the time at which c occurs.

CC2: If at a time t, there is a non-zero chance of e and e obtains, then at least some of the conditions at t that determine the chance of e at t, caused e.

Discussing CC1, Barker says:

The general argument for CC1 might be summed up thus: causes explain their effects. If c causes e, then c explains e, and thus, at time t, c is a potential explanation of e. How then can c at t not contribute to fixing the chance of e at t?The obvious problem we saw with this argument came from cases like Hesslow's birth control pill example, where it could be that taking the pill causes thrombosis despite the fact that it makes no difference to the chances of an individual getting a thrombosis (because it exactly balances the risk, by inhibiting pregnancy, a potent promotor of thrombosis), and hence doesn't make a contribution to fixing them at their actual values—or, at least, no more of a contribution than non-causes do. Perhaps Barker is using 'fixing the chance' in some non-standard way, but he gives no indication of doing so

There are other problems too. If backwards causation is possible, as seems plausible in light of the possibility of time travel (and perhaps of some interpretations of quantum mechanics, such as those Huw Price has defended), then CC1 entails that some past events have non-trivial chances. But how can this be? If H is the history up until t, then no matter how or whether history fixes chances, it should be that the present chance of an event in a world w should be the same as the chance conditional on the history:

- Ch
_{wt}(A|H) = x and H is true iff Ch_{wt}(A) = x.

Barker mentions Lewis in this connection, as someone who accepts (1), and says

The spirit of CC1 is that there may be non-trivial backwards-directed chances. Lewis then must be wrong to have taken this line. Indeed, it is not clear why he takes it. Lewis accepts a chance-raising view about causation, and embraces the conceptual possibility of backwards causation.But Lewis does not accept a chance-raising view about backwards causation—in that case he explicitly thinks that (the ancestral of) regular non-backtracking counterfactual dependence is what enables prior effects to be caused (this is the case where a non-backtracking counterfactual just happens to have an antecedent made true after the time the consequent is made true, and doesn't have the evidential reading of backtracking counterfactuals). So I'm left no happier with CC1 despite these remarks about Lewis.

There are other worries about CC1 (e.g., Barker's invocation of infinitesimals despite the fact that it is no longer clear whether they can help with the problems of zero chance events, as Williamson recently argued). But I'll leave them, and invite comments on these problems here. Any defenders of CC1? I'm aware that the considerations I gave in favour of (1) aren't completely compelling, so anyone want to argue against it?