On p. 162 of Causation: A User's Guide, Paul and Hall present a neuron diagram which they call a 'black-box case' and use it to argue that it refutes Lewis's Influence account and also teaches important lessons about causation. (The lessons: Don't jump to conclusions about causation until you know there are no further facts about internal causal structure. Don't assume that what is presented as a case of trumping is what it claims to be, rather than a case of overdetermination.)
I don't dispute their second claim; I think they are right to challenge the assumption that the "trumped" cause is not a cause. But they are wrong about the first. Their black box case does not refute the Influence account. If we think it does, it is because we have been misled by a neuron diagram.
Let's take a look.
The "black box" is represented by the neuron diagram they call 'Figure 30'.
Hall and Paul provide a brief description of the neurons C, D, and E and stipulate some facts about the firings of these neurons on a particular occasion, and also stipulate some counterfactuals about what would have happened if one or both of the neurons had not fired, or had fired on its own, or in a different color. They then say that these stipulated actual and counterfactual "outside" facts are consistent with at least three different internal causal structures and three different internal causal scenarios. They provide three neuron diagrams to depict these structures and scenarios.
They tell us how to read the diagrams: Black and grey circles represent the strong and weak firing of neurons; empty circles represent non-firings. If a line ending with an arrow connects two circles, then the first neuron's firing (strong or weak) has the power to cause the second neuron's firing. If a line ending with a dot connects two circles, then the first neuron's firing has the power to prevent ("inhibit") or weaken the second neuron's firing. The checkerboard and striped neurons A and H of Figures 32 and 33 represent shunt neurons which have the job of "deciding" what to do when both neurons fire.
In Figures 31 and 32, the process initiated by C's firing goes through to completion, causing E's firing. But the process initiated by D's firing is cut off before it reaches E. In Figure 31, this happens because A's firing prevents G's firing. In Figure 32, this happens because A deflects the signal from neuron D to B, and sends only the signal from neuron C to E. In both cases, it seems intuitively clear that the causes of E's firing trace back to C's firing, and not to D's firing.
In Figure 33, by contrast, it is the process initiated by C's firing that doesn't go through to completion. In this diagram there are two pathways to E's firing, one along the upper channel, initiated by C's firing; and one along the lower channel initiated by D's firing. There are two causal gaps (empty circles) in the upper pathway: the failure of neurons A and F to fire. The lower pathway has no causal gaps. It seems intuitively clear that the causes of E's firing trace back to D's firing, and not to C's firing.
Hall and Paul's claim, then, is that the causal scenario depicted by Figure 30 underdetermines the causal facts. In two scenarios -- those depicted by Figs. 31 and 32 -- C's firing pre-empts D's firing as cause of E's firing. But in Figure 33, it is the other way around: D's firing pre-empts C's as the cause of E's firing. This refutes Lewis's Influence account because the Influence account must say that C's firing caused E's firing in all four scenarios. It must say this because the account says that an event E "is caused by every event with which it counterfactually covaries..to a sufficient degree." And in Figure 30, as well as Figs. 31, 32, and 33, E counterfactually covaries with C in exactly the way that Lewis deems sufficient for influence. Or so say Hall and Paul.
Are they right? Let's take a closer look.
Figure 30 is under-described, not just in terms of inner structure, but also in terms of "outside" laws and counterfactuals.
What actually happens is that C fires in black, D fires in grey (a shade of black), and E fires in black. We are told that if neither C nor D fire, E does not fire. We are told that if D fires, with or without C also firing, then E fires. But we aren't told whether E fires whenever C fires (with or without D). We are told that if C had fired in another color while D still fired grey, E would have fired in C's color, but that if D had fired in another color, while C still fired black, E would still have fired in black. But we aren't told whether it is always the case, when C and D fire in the same color, that E fires in the same color as C. From this limited information, we can draw only limited causal conclusions. We can conclude that E's firing is at least sometimes caused by D's firing, but we can't rule out the possiblity that it is also sometimes caused by C's firing. We can conclude that the color of E's firing is at least sometimes caused by the color of C's firing, but we can't rule out the possibility that it is sometimes (e.g. when both fire in the same color) caused by the color of D's firing. And because we can't rule out these causal possibilities, we cannot conclude that in the actual scenario (when C and D fired in different shades of the same color), C's firing caused either E's firing or the color of E's firing.
If Lewis is committed to false claims about this case (the Figure 30 case), on the basis of such incomplete information about counterfactuals, this would be very bad. However, I don't think he is committed to such claims. Let's take a closer look at his account.
The central idea behind the Influence account is that an event E causes (or is one of the causes of) another event F if the first event makes enough of a difference to the second event. (Since Lewis defines causation as the ancestral of this making-enough-of-a-difference relation, this condition is sufficient, but not necessary, for causation.) There are different ways in which an event might make enough of a difference to another event. One way -- the most common way -- is by counterfactual dependence: if it's true that if E hadn't happened, then F would also not have happened, then E made enough of a difference to F to count as a cause of F. But cases of causal preemption teach us that this isn't the only way in which one event might make enough of a difference to another event. In the classic late preemption case in which Suzy and Billy both throw rocks at a window, but Suzy's rock strikes first, shattering the window before Billy's rock arrives, the (non-fragile) breaking of the window doesn't depend on Suzy's throw. But the timing of the window-breaking depends on her throw; if her throw had not happened, the window would have been broken slightly later, by Billy's rock. And the manner of the window-breaking depends on her throw; if her throw had happened in any of a number of different ways (different spin, different aim point, different velocity, with a rock of greater mass), the window-breaking would have happened in one of a number of different ways.
At this point you might object on the grounds that Billy's rock throw also makes a difference to the timing and manner of the window-breaking. If Billy's rock throw had happened early enough -- earlier than Suzy's throw -- the window would have been broken earlier, by Billy's rock. And if Billy's rock throw had happened in a suitably different manner (if he had thrown harder, or faster), his rock would have reached the window before Suzy's rock, and the breaking would have happened differently. Lewis grants this, but, he insists, if Suzy's rock throw preempted Billy's as cause of the window-breaking, then it must be the case that her throw made more of a difference to the breaking than his did. The window-breaking is more counterfactually sensitive to Suzy's throw than it is to Billy's. It takes only small differences in the timing or manner of Suzy's throw to make a difference to the window-breaking; it takes greater differences in the timing or manner of Billy's throw to make similar differences to the breaking. He contrasts this with a case in which both rocks arrive at exactly the same time, and strike the window in a way such that neither makes more of a difference to the manner of the breaking. In this case, the two are equals with respect to Influence, and we should count each of the rock throws as a cause of the window-breaking. Intuitively this seems right; how exactly this can be cashed out in every case of preemption and overdetermination is more dubious. But set aside these worries and notice only this central and crucial feature of Lewis's account: to conclude that one event caused another (in cases where we don't have information about whether there is a causal process connecting the two events), we must know not only that the first event made a difference to the second event; we must also know that the first event made more of a difference than any other event that was a potential cause of the second event. In other words, the Influence account says that Influence is both a "graded" or comparative relation (enough of a difference, where this is a matter of balancing and comparing different difference-making respects) and also a contextually dependent relation (since a very slight difference may still count as enough of a difference in cases where there are two potential causes of a single effect).
With this in mind, let's see what Lewis would say about the Black Box case of Figure 30.
The first thing to note is that the stipulated counterfactuals about Figure 30 are incomplete, so far as the Influence relation is concerned. They tell us that C's firing makes a how-how difference to E's firing (if C had fired in a different color, E would have fired in a different color) and they tell us that D's firing makes a when-when difference to E's firing (if D had fired earlier or later, E would have fired earlier or later. And they tell us that C's firing doesn't make a whether-whether difference to E's firing (If C had not fired, E would still have fired, thanks to D.) But we aren't told about other ways of making a difference. For instance, we are not told whether it is also true that if C had fired earlier or later, then E would have fired earlier or later. So we don't know whether C's firing makes a when--when difference to E's firing. And we aren't told what would have happened if D had not fired. So we don't know if D's firing makes a whether--whether difference to E's firing.
Given that all the relevant counterfactual facts are not known, Lewis is not committed to any causal judgments about Figure 30. He should say that, so far as we know, it might be that C's firing makes more of a difference than D's firing, or that D's firing makes more of a difference, or that the two events are equals with respect to the difference they make. Until we are in possession of further counterfactual facts about the case, we should reserve judgment about causation.
The stipulated description of Figure 33 provides us with more details.
We are now told that that if C fires, with or without D also firing, then E fires. This means that C's firing makes a when-when difference to E's firing and D's firing makes no whether-whether difference to E's firing. So we now know that C's firing and D's firing are equals with respect to when-when influence (both have it) and whether-whether influence (neither has it) with respect to E's firing. Since C's firing makes a how-how difference to E's firing, and D's firing does not, it follows that C's firing makes more of a difference to E's firing than does D's firing. The case now looks like a neuron version of Schaffer's Major-Sergeant trumping case.
And it looks like Hall and Paul are right to conclude that Lewis must say what he says about that case: that E's firing is caused by the event with greater Influence -- C's firing (the Major's command) -- and not by the event with lesser Influence -- D's firing (the Sergeant's command).
Hall and Paul are partly right. Lewis would say that E's firing is caused by C's firing. But they are partly wrong. Lewis would not deny that E's firing is (also) caused by D's firing. Recall that Lewis defines causation as the ancestral of Influence. Influence is not transitive. It follows that one event (e.g. D's firing) may cause another event (e.g. E's firing) even though the first event doesn't stand in the Influence relation to the second, provided that the first event is connected to the second event by a sequence of events, each of which stands in the Influence relation to the subsequent one. And this is true of D's firing and E's firing. D's firing Influenced (and thus caused) B's firing, B's firing Influenced (and thus caused) G's firing, G's firing Influenced (and thus caused) E's firing.
Is this consistent with what Lewis says about trumping? Well, Lewis does say, about the Major/Sargeant case, that we are not entitled to assume that the case works by 'cutting', and so we are not entitled to assume that there is a causal chain of events between the Major's order and the soldiers' advance. That's why he accepted the trumping case as a counterexample to his earlier, simpler analysis of causation, and this is why his new Influence account allows us to say, as the old account did not, that the Major's command directly caused the soldiers' advance. The Major's command directly caused the soldiers' advance because his command had greater Influence on their advance than did the Sergeant's command. But since influence is only sufficient and not necessary for causation saying that the Major's order caused the advance is consistent with saying that the advance was also caused (by way of a causal chain of intermediate events) by the Sergeant's command. In other words, Lewis can say that what looks, at first glance, like a case of trumping might, on further investigation, turn out to be a case where both potential causes are actual causes of the effect.
At this point you might object: This is all well and good, but it doesn't rescue Lewis, Even if he can say that D's firing is a cause, by transitivity, of E's firing, his account commits him to the claim that C's firing is also a cause of E's firing. And that's wrong. In Figure 33, the causal pathway to E's firing goes by way of D's firing, and not by way of C's firing. This shows that Influence, even when understood in the more complex way that I've explained, does not suffice for causation. Therefore Hall and Paul are right after all. The Black Box case refutes the Influence account.
But I respond thus: Yes, Lewis is committed to the claim that in the Figure 33 scenario, C's firing is one of the causes of E's firing. But a closer look at the stipulated facts of the case shows that this is the right result. The shunt neuron, H, "decides" whether to direct neuron signals along the upper or lower pathway. It "decides" this on the basis of incoming information from neurons D and C. If neither C nor D fire, H "does nothing". If C and D fire in different colors, H fires one way (let's call it "down"), inhibiting D's signal and allowing C's signal to go through along the upper pathway. If C and D fire in the same color, H fires another way (call it "up"), inhibiting C's signal and letting D's signal go through along the lower pathway. This is the actual case, and since the neuron diagram depicts all the neurons in the lower pathway as firing, and depicts two neurons in the upper pathway not firing, we conclude that E's firing is caused only by D's firing and not by C's firing. But this is a mistake that comes from confusing what we say about object causation with event causation.
Compare: Sylvie's throw was aimed away from the window, and would have missed, while Bruno's throw was aimed directly at the window. But the two rocks collided in mid-air, and the result of the collision was that Sylvie's rock struck and broke the window, while Bruno's missed altogether. In this case, it was only Sylvie's rock, and not Bruno's rock, that caused the window-breaking. (We might say that his rock caused her rock to break the window, and we might even say that his rock caused the window-breaking, by causing her rock to break it. But we wouldn't say that his rock broke the window, or that his rock caused the window breaking, full stop.) But Bruno's throw - the event - was among the causes of the breaking. It (jointly with Sylvie's throw) caused the collision, which caused the breaking.
The analogy isn't perfect, but if we compare neuron signals to rocks, we can compare Sylvie's rock to the signal from neuron D and Bruno's rock to the signal from neuron C. The firing of the shunt neuron, H, is like the collision of the rocks. The upshot of H's firing, in the Figure 33 case, is that the signal from neuron C is stopped while the signal from neuron D makes it all the way to neuron E. So if we think in terms of object causation, we conclude that E's firing was caused by D (the neuron, by way of the signal sent by the neuron), and not by C. But if we stay focused on event causation, we'll see that C's firing -- the event -- was among the causes of E's firing. It (jointly with D's firing) caused H's firing, which caused E's firing.
Therefore Hall and Paul are wrong. Lewis's Influence account delivers the correct verdict about this complex case. Contrary to what I said earlier, our initial classification of Figure 33 as a case of causal pre-emption is mistaken. C's firing and D's firing are joint causes of an intermediate event, H's firing, and this intermediate event causes E's firing. If some alleged cases of trumping turn out to have this kind of causal structure, then it is a mistake to classify them as trumping (since trumping is a species of preemption). This is an important point. But Hall and Paul are mistaken, for the reasons I have given, to think that this point counts against the Influence account.
Why did Hall and Paul think otherwise? Why did they think that Figure 33 is a kind of causal pre-emption scenario, as opposed to being a case where C's firing and D's firing are both causes of E's firing?
Well, one of the reasons is the one I've already explained -- the tendency to slide from talk of what objects (rocks, bullets, neuron signals) cause to talk of what events cause. But there is another reason, having to do with the limitations of neuron diagrams.
The way that neuron diagrams are presented encourages us to think that they are highly simplified but nevertheless unambiguous representations of the causal facts. But this isn't the case because the diagrams are set up to represent two different kinds of causal information: information about causal laws and information about causation in a particular case.
The circles (filled and blank) do double duty; they represent objects (neurons) and also occurrences and non-occurrences of event-types (neuron firings). The lines with arrows and dots connecting the circles also do double duty; they represent nomologically possible sequences ("pathways") of occurrences (filled circles) or non-occurrences (empty circles) of events and they also represent actual sequences of occurrences and non-occurrences of events.
Our instructions for how to read the diagrams are incomplete, but we are not warned of this. For instance, we are not told how to read the following simple neuron diagram: two filled circles, A and B, are connected by lines with arrows to a third filled circle, C.
Is this a case where A and B are joint causes of C? Or is it a case where A and B are overdetermining causes of C? Or might it even be a case where one of the two is the actual cause and the other is the merely potential but pre-empted cause? The instructions for reading the diagrams do not tell us.
It follows, from this simple example, that neuron diagrams, by their very nature, cannot be relied upon to depict the causal facts of a particular case.
You might think that they can at least be relied upon to tell us what the causal laws are. But this isn't true either. In order to know whether A-like events are joint causes, with B-like events, of C-like events, or whether A-like events are capable of causing C-like events without the help of B-like events, we need to know more than what the diagram can depict.
It follows that we should not rely on a neuron diagram for verdicts or intuitions about what either the particular or general causal facts are. Yet this is precisely what we did when we first looked at Figures 31, 32, and 33.
You might protest, at this point, that neuron diagrams can still be useful. Even if they don't provide an unambiguous depiction of all the causal facts, they depict some of them, and so long as we are careful to supplement our interpretation of the diagrams with the stipulated description of a particular case, we will not be led astray.
But we were led astray, when interpreting Figure 33. And I speak from bitter personal experience here. It took me a long time to figure out the true causal facts about Figure 33.
The problem, as I already noted, is that it is easy to speak the language of object causation, and then switch from this to the language of event causation. Sometimes this is harmless, but not always.
Recall that in describing Figures 31 and 32 we followed Hall and Paul in speaking of a signal "travelling" and the neuron A "sending a signal along a path". When we speak this way, we are thinking of neuron signals as persisting objects, like rocks or baseballs. And this encourages us to think that if the signal sent by one neuron (e.g. D, in Figure 32; e.g. C in Figure 33) doesn't make it all the way down its "pathway" to another neuron (e.g. E, in Figure 32 or 33), then that neuron's firing did not cause the firing of the second neuron. (Compare: Billy's rock never made it to the window because Suzy's rock got there first, so his rock throw didn't cause the breaking of the window.)
Thinking this way doesn't get us into trouble with Figure 32. In Figure 32, the causes of E's firing trace back, via A's firing, to C's firing and not to D's firing. But in Figure 33, thinking in terms of causation by a persisting object does get us into trouble. We can agree that it was the signal sent by D, and not the signal sent by C, that caused E's firing. But in this case, unlike the case when only one neuron fires, both neuron firings were causes of H's firing, and thus, by transitivity, both were causes of E's firing.