# How to Distinguish Hasteners from Delayers

Counterfactual theories of causation have a hard time distinguishing hasteners from delayers. Or so I used to believe.

This past semester, while teaching a seminar on Causation, I learned better from a footnote (number 15) hidden at the end of this article by Jonathan Schaffer. There, I found a solution to the problem of hasteners and delayers, due to Chris Hitchcock, which is, to my mind, completely satisfying. The point of this post is to expand upon that solution.

First, the problem:

On Monday, Dr. Costello gives Abbott a palliative dose of steroids. Without the drug, Abbott would die on Monday; but, with the drug, Abbott dies on Tuesday instead. The steroids did not cause Abbott to die.

On Monday, Dr. Hardy gives Laurel a lethal dose of morphine to ease his pain. Without the morphine, Laurel would have died on Wednesday; but, with the morphine, Laurel dies on Tuesday instead. The morphine caused Laurel to die.

Lewis’s counterfactual theory has trouble getting both of these cases right. If he says the steroids didn’t cause Abbott to die, consistency pushes him to say that the morphine didn’t cause Laurel to die. And if he says that the morphine caused Laurel to die, he’s pushed to say that the steroids caused Abbott to die.

That’s because, on Lewis’s theory, causation is the transitive closure of counterfactual dependence between distinct events. The ‘transitive closure’ bit of the account is only relevant for cases of preemptive overdetermination; since there are no such worries in either of these cases, we can fudge things slightly and say that, according to Lewis’s theory, $c$ caused $e$ if and only if $e$’s occurrence counterfactually depends upon $c$’s occurrence.

An Event-based Counterfactual Theory (ECT)
The actually occurring event $c$ caused the actually occurring event $e$ iff, had $c$ not occurred, $e$ would not have occurred either. \begin{align} c \text{ caused } e \iff \neg O( c ) > \neg O(e) \end{align}

where ‘$>$’ is the counterfactual conditional, and ‘$O(e)$’ is true at a world iff the event $e$ occurs at that world.

So, to see whether the morphine caused Laurel to die, we ask: if the morphine hadn’t been administered, would Laurel’s death have occurred? To my mind, the natural answer is ‘yes, it still would have occurred, just a day later on Wednesday’. But then, ECT tells us that the administration of the morphine didn’t cause Laurel to die. Bad verdict. Perhaps the natural answer to our question was the wrong one. Perhaps the death Laurel would have died on Wednesday without the morphine isn’t the death he actually died. If, instead of the death being delayed a day, it fails to occur at all and is replaced by a numerically distinct death, then ECT can get this case right and say that the morphine caused Laurel to die.

To get the Laurel and Hardy case right, ECT had to stipulate that death may not be delayed a day. But if we say this, then surely the death Abbott actually died (on Tuesday) would not have occurred were it not for the steroids. Rather, a numerically distinct death would have occurred on Monday. For, if Abbott’s actual Tuesday death were identical with the merely possible Monday death, then death could be delayed a day. So, if the steroids had not be administered, then Abbott’s actual death would not have occurred. Rather, a different death a day earlier would take its place. So ECT says that the steroids caused Abbott to die. Bad verdict.

So, if we try to get Laurel and Hardy right, we get Abbott and Costello wrong. If we try to get Abbott and Costello right, we get Laurel and Hardy wrong. That’s the problem.

A natural reaction to the question I posed earlier—would a Wednesday death be the same death as the Tuesday death?—is that it is a pseudo-question, and we would be within our linguistic rights to answer either ‘yes’ or ‘no’. This is what Lewis (2000) eventually decided, and it led him to abandon his ECT-like account. But accounts like ECT are not the only kinds of counterfactual accounts of causation. Counterfactual accounts making use of variables have become increasingly popular. On the simplest possible version of that kind of view, a variable value $C=c$ caused a distinct variable value $E=e$ iff, had $C$ taken on some value other than $c$, $E$ would have taken on some value other than $e$.

A Variable-based Counterfactual Theory (VCT)
The variable $C$'s taking on the value $c$ caused the variable $E$ to take on the value $e$ iff there’s some value of $C$, $c^*$, such that, had $C$ taken on the value $c^*$, $E$ wouldn’t have taken on the value $e$. \begin{aligned} C=c \text{ caused } E=e \iff (\exists c^*) ( C=c^* > E \neq e) \end{aligned}

You could look at VCT and decide to think of the variables as standing for the occurrence or non-occurrence of events. Then, you could say that the variable $C$ takes on the value $1$ iff the event $c$ occurs and it takes on the value $0$ otherwise. If you said that, then VCT would be precisely the same as ECT; and you might be inclined to think that the variables have added nothing metaphysically interesting.

That reaction to the use of variables is not uncommon in the metaphysics literature. For instance, Ned Hall says, about the recent interest in causal modeling (which eschews an ECT-like account for a VCT-like account) that “anything that can be accomplished by means of causal models can be accomplished just as straightforwardly without them”. In making this claim, I think that Hall is implicitly relying upon an understanding of the variables on which they are representing whether an event occurred or not. Variables may be used this way (and, in fairness to Hall, they basically are used this way by almost all of the philosophical work that’s been done with them), but they are more profitably used to represent properties of the world at various times. One advantage of this shift in perspective is that it absolves us of the responsibility to provide any theory of event individuation. If we’re using a variable to represent whether Abbott’s death occurred, then we had better say something about how long a death may be delayed. If we’re using a variable to represent whether Abbott is alive or dead at $t$, then we don’t have to futz with any of that messy business (and just as well). We merely have to say what it takes for Abbott to be alive or dead at a given time. And here, matters are far less murky. The causal relata are still events, but they are the events of certain variables taking on or retaining certain values at certain times. So, if $A_{t}$ is a variable which takes on the value $1$ if Abbott is alive at $t$ and takes on the value $0$ if Abbott is dead on Tuesday, then the English expression “Abbott’s death” refers to the event of $A_t$ taking on the value $0$ at $t=$ Tuesday. Perhaps this event could have occurred even if $A$ had taken on the value $0$ earlier, or later. Perhaps not. That fortunately won’t matter at all for what VCT has to say about the causes of Abbott’s death. And if $S_t$ is a variable which takes on the value $1$ if Abbott is given steroids at $t$ and takes on the value $0$ if Abbott is not given steroids at $t$, then the English expression “the administration of steroids” refers to the event of $S_t$ taking on the value $1$ at $t=$ Monday. Could this event have been delayed a day? Perhaps, perhaps not. But that’s no concern of a theory of causation, once we’ve opted for VCT. Did the administration of steroids cause Abbott’s death? To answer that question, we should ask whether, if $S_t$ had not taken on the value $1$ on Monday (if it had instead remained at the value $0$ on Monday), the variable $A_t$ would have taken on the value 0 on Tuesday. And the answer to that question is “yes”. Without the steroids, $A_t$ would have first taken on the value 0 on Monday, but it would still have taken on the value $0$ at Tuesday as well.

$S_{\text{Monday}} = 0 ~~>~~ A_{\text{Tuesday}} = 0$

So, VCT tells us that the steroids did not cause Abbott’s death. We can similarly let $L_t$ be a variable which takes on the value $1$ if Laurel is living at $t$ and takes on the value $0$ if Laurel is not living at $t$. And we can let $M_t$ be a variable which takes on the value $1$ if Laurel is given morphine at $t$ and takes on the value $0$ if Laurel is not given morphine at $t$. Again, we should take the English expression “the administration of morphine” to refer to the event of $M$ taking on the value $1$ at $t=$ Monday; and we should take the English expression “Laurel’s death” to refer to the event of $L$ taking on the value $0$ at $t=$ Tuesday. Did the administration of morphine cause Laurel’s death? VCT asks us to check whether, had $M$ remained at the value $0$ on Monday, $L$ would have taken on the value $0$ on Tuesday. Ex hypothesi, it would not have. Rather, it would have still had the value $1$ on Tuesday. (It wouldn’t have taken on the value $0$ until Wednesday, without the morphine.)

$M_{\text{Monday}} = 0 ~~>~~ L_{\text{Tuesday}} = 1$

So, VCT tells us that the morphine caused Laurel’s death. So, properly understood, accounts like VCT are able to cleanly distinguish between hasteners and delayers. Of course, not all delayers are non-causes. Some palliative care both delays and causes death. (I’m inclined to think that all hasteners are causes, but others may disagree.) A full theory of causation needs to deal with such cases. I think that they can be adequately dealt with, but that’s another story for another day. Today’s story is just this: the bit of philosophical lore that counterfactual theories have a hard time distinguishing hasteners from delayers is wrong. What’s right is that event-based counterfactual theories have a hard time making this distinction. Variable-based counterfactual theories, properly understood, do not.