Narrative · The single case
Suzie and Billy
Both throw a stone, Suzie's hits first. Both throws are causally relevant to the shattering, but only one brought it about in this case.
Chapter 1
Two throws, one bottle
Suzie and Billy both throw a stone at a bottle. Suzie's stone arrives first, the bottle shatters, and Billy's stone flies through empty air. Who broke the bottle? Intuition is unambiguous: Suzie. The example is a standard case in the literature on causation in the single case.
S Suzie throws · B Billy throws · ST Suzie's stone hits · BT Billy's stone hits · E the bottle shatters.
The regularities about these kinds of events are simple. If Suzie throws, her stone hits. If Billy throws, his stone hits the bottle only if Suzie's stone has not already hit it. If either of the two stones hits, the bottle shatters.
Chapter 2
At the level of regularities
At the level of regularities both throws are causally relevant to the shattering: Suzie's throw via her hit, Billy's throw via his. The theory must name both, for it speaks about kinds of events, and on other days, when Suzie does not throw, Billy's stone breaks the bottle.
All arrowheads are hollow. A hollow arrowhead stands for a causal regularity over kinds of events, not yet for what happened on a particular day.
Chapter 3
The single case
On this day both throw. The shaded nodes are present in the case: Suzie's throw, Billy's throw, Suzie's hit, the shattering. Billy's hit is not present; its node is dashed.
A cluster is fired in a case when all its conjuncts are present. Suzie's throw fires her hit, her hit fires the shattering. The heads of these arrows are filled: they show the active cluster, the instantiated regularity of this case.
Billy's cluster requires two things: that Billy throws and that Suzie's stone has not already hit. The second conjunct is missing, for Suzie's hit is present, so the cluster is not fired, and Billy's hit does not occur. Billy's throw is causally relevant to the shattering, but in this case it lies on no active cluster. He did not break the bottle.
Chapter 4
Without a counterfactual assumption
The counterfactual analysis runs into difficulties with this case: had Suzie not thrown, the bottle would still have shattered, through Billy's stone. The shattering therefore does not depend counterfactually on Suzie's throw, and yet her throw brought it about.
Volume III treats the case as late preemption: Billy's process is cut off by the effect itself, by the shattering that Suzie's stone has already brought about (Baumgartner 2013). The regularity theory does not need the counterfactual assumption. It asks which clusters are fired in this case and follows the active cluster from the cause to the effect. The diagram shows the distinction between what holds of kinds of events and what happens in a case by the fill of the arrowheads (Vol. III, §4.2).