Narrative · No difference to the outcome
The switch
The switch decides which track the train takes, but not whether it arrives. The minimal theory of arrival does not contain it.
Chapter 1
Two tracks, one destination
A train approaches a switch. If the switch is set to the right, the train continues on the right track, otherwise on the left. Both tracks lead to the same station.
A the train approaches · B the train is at the switch · F the switch is set to the right · C the train runs on the right track · D on the left · E the train arrives.
The switch is causally relevant to which track the train takes: it is part of the cluster for the right track, its absence part of the cluster for the left. In the literature on causation in the single case the example is a touchstone (Baumgartner 2013) (Vol. III, §6).
Chapter 2
The theory of arrival
Does the train arrive? If arrival is expressed through the factors before the switch, two things are sufficient: the train at the switch with the switch set to the right, or the train at the switch with the switch not set to the right.
Neither of the two conditions is minimally sufficient. The train at the switch is sufficient on its own, for the switch is either set to the right or not, and both tracks lead to the destination. In both conditions the switch conjunct is redundant; minimal sufficiency strikes it out.
What remains is B ⇒ E. The switch does not occur in the minimal theory of arrival. It is not causally relevant to the train's arriving.
Chapter 3
This journey
Today the switch is set to the right. The active cluster leads from the approaching train via the switch and the right track to the arrival; the filled arrowheads show it. The left track stays empty.
The switch lies on this active cluster, and yet it made no difference to the arrival: had it been set to the left, the train would have arrived just the same. The regularity theory bears out this intuition without justifying it counterfactually, because its verdict on the single case remains bound to the minimal theory: whatever does not occur in the theory of arrival is not a cause of the arrival in this case either.