1
Preemption
A cause brings about the effect and in doing so stops a backup from doing so. Flip the cause and the backup takes over, so the effect is unchanged.
Lesson · Science, Induction and Causation · Causation
Preemption, Overdetermination and Redundant Causes
Suzy throws a rock and shatters the bottle; Billy, a moment behind her, would have shattered it had hers missed. Suzy's throw is not a but-for cause, yet it plainly caused the shattering. Holding Billy's rock at what it actually did, flipping Suzy's throw does flip the shattering. When two rocks arrive together, each fails alone and the pair passes.
backup present: flipping the cause is idle
hold the backup at its actual value, flip
two at once: neither alone, both together
no but-for does not mean no cause
When the effect was coming anyway
Core ideas
What this lesson establishes
The previous lesson ended on a failure of the but-for test. The failures have two shapes, and each has a repair that is computed rather than described.
2
Overdetermination
Two causes act at once, each sufficient. Flip either and the other still produces the effect, so neither is a but-for cause.
3
Holding something fixed
A variable is a cause, in the lab's sense, if for some set of other variables held at their actual values, flipping it flips the effect. The held set cancels the backup.
The material
Two rocks and a bottle
The preemption model first, the failure of the test, the repair, and then the case of two rocks that arrive together.
Suzy and Billy each throw a rock at a bottle. Suzy throws first and her rock hits. Billy's rock is a moment behind, and would have hit if hers had not. Write ST and BT for the throws, SH and BH for the hits, and SHAT for the shattering.
ST = 1
BT = 1
SH = ST
BH = BT ∧ ¬SH
SHAT = SH ∨ BH
The equation for BH says that Billy's rock hits only if he threw and Suzy's rock has not already hit. Solving: SH = 1, so BH = 1 ∧ 0 = 0, and SHAT = 1. Suzy's rock did the work.
Now apply the test of the previous lesson. Flip ST to 0. Then SH = 0, and BH = 1 ∧ 1 = 1, and SHAT = 1. The bottle shatters anyway, so Suzy's throw is not a but-for cause. The lab reports No for the but-for test, and a counterfactual account of causation that stopped there would say Suzy did not break the bottle.
Definition · Preemption
Preemption is the case in which a cause brings about an effect and thereby prevents a backup that was ready to bring it about. The effect would have occurred without the cause, so the cause is not a but-for cause.
The repair uses what actually happened to the backup. In the actual course of events Billy's rock did not hit: BH = 0. Hold BH at 0 and flip ST. Then SH = 0, BH stays at 0, and SHAT = 0 ∨ 0 = 0. With the backup held at its actual value, flipping Suzy's throw flips the shattering.
Theorem · A cause with a held set
A variable C is a cause of E when, for some set W of other variables held at their actual values, flipping C changes E. If W can be empty, C is a but-for cause. The lab reports the smallest W that works.
The held values must be the actual ones, and that restriction matters. It is what keeps Billy's hit from qualifying: select BH as the cause and the lab reports not a cause, because in the actual course Suzy's rock has already hit and nothing done to BH changes the shattering. Billy's throw BT is reported only as a joint cause with Suzy's, since flipping both throws leaves the bottle whole. That is true, and it is not what we mean by Billy having broken it. The joint rule asks only whether a pair changes the effect, and it cannot tell equal partners from a cause and a bystander.
The second preset removes the ordering. Both throw, both rocks reach the bottle at once, and BH = BT. Each hit is enough on its own. Flip ST and BH still shatters the bottle, and holding BH at its actual value 1 does not help, since it is already doing the work. Neither throw passes alone. Flip both, and the bottle survives, so the pair passes together; the lab calls Suzy's throw a joint cause with Billy's.
The third preset is Schaffer's case of trumping, in which a major's order and a sergeant's order agree and the soldiers do what the major says. Here it is written with the sergeant's order obeyed only if the major gave none, and the equations are those of preemption. That is a limit of the model: a Boolean model cannot represent a difference between preemption by timing and by rank, unless the equations already contain it.
Whether holding a variable fixed is a faithful account of causation is disputed. The lab implements a simplified version of the Halpern–Pearl definition, and the full definition adds conditions that exclude some cases the simple version accepts. The reader who rejects the whole approach owes a different account of why Suzy broke the bottle.
Interactive
Causes in a structural model
But-for, held fixed, and joint
Method
Testing a cause when a backup is present
Five steps, in this order, when the but-for test says No for something that looks like a cause.
1
Run the but-for test
Flip the candidate and recompute. If the effect changes you are done: it is a but-for cause.
2
Look for a backup
If the effect did not change, find the variable that took over, and note its actual value in the case. That variable is the candidate to hold fixed.
3
Hold it at its actual value and flip again
Fix the backup at what it actually did, flip the candidate, and recompute. If the effect now changes, the candidate is a cause once the backup is held.
4
Try the pair if nothing works alone
If no held set works, flip the candidate together with another variable. If the effect changes, the two are a joint cause of an overdetermined effect.
5
Check that the held value was the actual one
A set held at a value that did not occur proves too much, since almost anything is a cause then. Say what each held variable actually did.
Backups and pairs
Score 0 / 0Choose the answer you can justify.
Suzy's throw and Billy's backup
Inputs ST = 1, BT = 1; the equations as in the lab.
actual: SH = 1, BH = 1 ∧ ¬1 = 0, SHAT = 1 ∨ 0 = 1
flip ST to 0: SH = 0, BH = 1 ∧ ¬0 = 1, SHAT = 1
but-for: no, the bottle shatters either way
hold BH at actual 0, flip ST: SH = 0, SHAT = 0 ∨ 0 = 0
cause, holding BH fixed
Suzy's throw is a cause once Billy's rock is held at what it actually did, which was nothing. Billy's hit, by the same recipe, is not a cause: whatever is held, Suzy's rock hits and the bottle breaks.
Common mistakes
1
Thinking that if the effect would have happened anyway, nothing caused itThe bottle would have shattered without Suzy's throw, and the lab still finds the throw a cause once Billy's rock is held at what it actually did. A missing but-for dependence shows that a backup was present and not that the actual cause was idle.
2
Holding the backup at whatever value is convenientOnly actual values may be held. If any value were allowed, Billy could be made a cause by setting Suzy's rock to a miss, and the test would name causes that did nothing.
3
Expecting each of two simultaneous causes to pass aloneIn the overdetermination model neither throw is a but-for cause and neither passes with the other's hit held at its actual value. The two pass together, and the lab names the partner.
Check your understanding
✓
Finish when you can build a preemption model, show the but-for test fail and recover the cause.
Given a case with a backup, you should be able to write its equations, show that the candidate is not a but-for cause, find the variable to hold at its actual value, show the cause pass, and show two simultaneous causes pass only jointly.
Note: The held-fixed test is a simplified version of the Halpern–Pearl definition and is contested, as the lesson says. The trumping preset is a Boolean rendering with the same equations as preemption, and it is offered as a limit of the model, not as an analysis of trumping.