Suppose you have a series S of neural events in the actual world which lead to a physical movement P. Given Lowe's view that the mind does not directly cause any neural event there must be an external physical event E which causes the first event in the neural chain. So it'll look something like this:
- S: E > (N1 > N2 > ... > Nn) > P
Suppose Lowe is right and mental events do not directly cause any particular neural event, but instead explain why this whole series of neural events exists instead of another. For comparison, Lowe proposes that the mind acts in the same way that God could act as sustainer of the universe. Suppose the universe is an infinite series of physical events where God doesn't directly cause any of the events in the series:
- U: ... > P0 > P1 > P2 > ...
Now, every event in the series has a physical cause within the series, but there is still the question about why it is U which exists in the first place as opposed to another, distinct, series (say U*):
- U*: ... > P0* > P1* > P2* > ...
On this picture God does not directly cause any particular physical event in U, but rather explains why this whole series of physical events exists in the first place. Here God sustains the universe, as opposed to interacting with it.
If the mind's influence on the body is like this, then the mind would have to have counterfactual control over which series of events occurs given E. In other words, if the mind had decided differently, then had E happened a different chain of neural events would have existed than S.
Now presumably one chain of events is identical to another if all the events in the one chain are identical to the corresponding events in the other. Somewhat more precisely:
- (E1 > E2 > ... > Em) = (E1* > E2* > ... > Em*) if and only if (a) E1 = E1* and E2 = E2* and ... and Em = Em*; and (b) Ei > Ei+1 if and only if Ei* > Ei+1* for 1 < i < m
Obviously an identity criterion for infinite causal chains would be pretty easy to give too, but we're dealing with causal chains in the brain which lead to physical movement, so presumably this is unnecessary for our purposes. If you like you can think of this criterion as being restricted to causal chains in the brain.
Now then, suppose the mind does cause a different chain than the actual one. Call this chain S*. In order for a different causal chain to happen than the actual one it'd have to look something like this (for convenience I put the original chain above the new one):
- S: E > (N1 > N2 > ... > Nn) > P
- S*: E _ (N1* > N2* > ... > Nm*) _ P with Ni =/= Ni* for some i, 1≤ i ≤ n
The reason there must exist an i such that Ni =/= Ni* is because of our identity criterion above. If all the corresponding events in the chains are identical then the chains themselves are identical. But we supposed that some different chain was brought about, which again is possible given Lowe's view, since Lowe's view implies the mind has counterfactual control over which series of events occurs given E.
I leave blanks in between E and N1* and between Nm* and P since we don't want to assume too much; maybe the mind will cause a chain to exist which is uncaused or which does not lead to the same physical movement P. Now, there are a few possibilities here.
(i) If E does not cause N1* then a law of nature is violated, since (in the actual world) it is a law that E will cause N1. (Actually, if I were being more precise a bit more detail and argument would be necessary here, but this is right.)
(ii) If E does cause N1* and N1 =/= N1* then a law of nature is violated, since it is a law that E will cause N1. Moreover, the laws of nature are changed, since E causes something else than it normally would.
(iii) If E does cause N1* and N1 = N1* then no law of nature is violated yet, and we must look for the smallest i in the chain such that Ni =/= Ni*.
(iv) Again, such an i exists because of our identity criterion and the assumption of a different series S*. So Ni =/= Ni*. Then it will follow that Ni-1 = Ni-1*. So Ni-1 does not cause Ni. So a law of nature is violated, since it is a law that Ni-l will cause Ni.
Moreover, Ni-1 must cause Ni*, since this is a chain of neural events leading to a physical movement. By a metaphysically motivated syntactic rule for these representations of causal chains, there must be a '>' between every node in the chain. (If there were causal 'gaps' how could you really call it a causal chain; the only real causal chains would be the ones before and after the 'gap'.) So that means a law of nature is changed, since Ni-1 causes something different than it normally would.
So, no matter what, if the mind ever exercises the power it has to bring about another neural causal series, a law of nature must be violated. Moreover, if the first event in this series is caused then the laws of nature must be changed by the mind at some point. This is bad.
However, there's more to say than just that it's bad. Lowe could reply that, in the actual world, there are psychophysical laws which hold and which tell us the mind never actually uses this power of counterfactual control over laws. Thus, the normal patterns of events we observe still obey the laws of nature. The laws of nature continue to hold in this world.
But that leaves two options: (1) Does that mean minds never actually do anything? Are they causally effete and just let the world stay its course? Or else (2) Do minds still do something, and thereby contribute some causal influence? If they do, does that mean they cause the actual laws of nature to hold in the case of neural causal chains?
Both possibilities seem weird and unpalatable. If the first view is true that means mental causation never actually occurs in this world, though it could. On the second view there are two possibilities: Either (A) mental causation is superfluous, or else (B) laws of nature are really really weird.
On the one hand, it could be that the laws of nature by themselves are enough to explain why each event in the series causes the next, in which case the mental decision is superfluous and just 'backs up' the law of nature by its influence. The mind gives the laws of nature more 'oomph', though they are completely sufficient even without this 'oomph' at all. This would be case (A).
On the other hand, it could be that, once we move from E to N1, all of a sudden the mind is needed to hold up the laws of nature from then on. This is option (B). And that seems weird. That would mean when the brain comes into play, all of a sudden the laws of nature by themselves aren't sufficient for producing later events and the influence of the mind is needed to 'keep them going'. When it comes to the brain, the laws of nature need crutches; the mind is a metaphysical crutch. This is very metaphorical of course, but the point could certainly be made more precisely.
To be fair, this theory can't be disproven by the science. Neither option (1) nor (2) above is even possibly ruled out by the actual scientific evidence. So maybe this is one way to reject causal completeness while holding a theory that is empirically equivalent. I'm not sure it's a plausible one though.