Relevant texts:
Carroll, J. & Markosian N. An Introduction to Metaphysics focusing on pp. 20–44.
Fenton-Glynn, L. Causation focsuing on pp. 1–38.
Hitchcock, C. ‘The Intransitivity of Causation Revealed in Equations and Graphs’.
Glynn, L. ‘A Probabilistic Analysis of Causation’.
where to find: libgen.is, or search up papers on google or reddit
Causal Relations and Events
• The lighting of the fuse caused the explosion
• The lighting of the fuse caused the bomb to explode
• The lighting of the fuse was a cause of the bomb’s exploding
• The lighting of the fuse was the cause of the bomb’s exploding
• The bomb exploded because the fuse was lit
We will follow most recent authors in concentrating on the relation c is a cause of e, where c and e are events (like the lighting of the fuse, or the explosion of the bomb). This relation is nowadays typically called ‘actual causation’. Note the use ‘a’ rather than ‘the’. Uncontroversially, most events have many causes. The question what we are up to when we single out one of them as ‘the’ cause is interesting, but plausibly introduces an extra layer of context-sensitivity. Some notes on events:
To any event there corresponds a proposition—the proposition that that event occurs. But perhaps not every proposition is such that for some event, the proposition is equivalent to the proposition that that event occurs. (Example: the proposition that either Obama is president of the US in 2010 or Gaddafi is president of Libya in 2008.) Do not take it for granted that when we refer to an event using a name of the form ‘the φ- ing of x’, e.g. ‘the death of Socrates’, the proposition that that event occurs is equivalent to the proposition that x φs at some time or other. Perhaps the particular event that was the actual death of Socrates would not have occurred if Socrates had died peacefully in his sleep five years later.
Events sometimes stand in acausal ‘determinative’ relations to one another. For example: the Battle of Britain partly constituted World War II; Xanthippe’s widowing depended on Socrates’ death; I wore a red shirt to philosophy class in virtue of wearing a scarlet shirt; Martha Stewart broke the law because she insider traded. These kinds of intimate connection seem different from mere causation. When two events are not related by acausal determination, nor do they share a common acausal determiner, we can say that they are ‘metaphysically independent’.
Nomic Sufficiency Accounts of Causation
Say that a is a nomically sufficient condition for b iff it is nomically necessary that if a
occurs, b occurs iff b occurs in every world with the same laws as ours in which a occurs.
Simple nomic sufficiency analysis of causation:
c is a cause of e iff c and e are metaphysically independent and c is a nomically sufficient condition for e.
Five problems for this proposal:
(i) The problem of effects: In some cases, a is nomically sufficient for b although intuitively b is a cause of a and not vice-versa. For example: the presence at t of an α- particle shooting out from a nucleus might be a nomically sufficient condition for its decay at t–∆.
(ii) The problem of epiphenomena: In some cases, a is nomically sufficient for b although intuitively, neither is a cause of the other, but rather they are effects of a common cause. For example: the presence at t of an α-particle with such and such energy shooting out from a nucleus might be a nomically sufficient condition for the presence at t, or at t+∆, of a β-particle with such and such energy shooting out on the opposite side.
(iii) The problem of ‘completeness’: Under realistic physical assumptions, the only propositions about a time t that nomically necessitate the occurrence of some ordinary event e at a later time t′ are very rich propositions that characterise the state of some very wide spatial region at t—e.g. under relativity, the intersection of the entire past light cone of E with t.
(iv) The problem of preemption: Two trails of gunpowder are laid out on the ground, both leading to a pile of TNT. Wile E Coyote lights the long trail; Road Runner lights the short one. The bomb explodes when the fire reaches it along the short trail. The lighting of the long trail (in conjunction with lots of background facts) is nomically sufficient for the explosion, but is not (even taken together with all those background facts) a cause of the explosion.
(v) The problem of chancy causation: Suppose that any event always has some chance of occurring, so long as it is nomically possible. Then no proposition entirely about events occurring at an earlier time ever nomically necessitates a proposition entirely about events occurring at a later time, unless that proposition is nomically necessary all by itself. Given the simple nomic sufficiency analysis, it follows that no earlier event ever causes a later one!
Refining the Nomic Sufficiency Approach
c is a cause of e iff c and e are metaphysically independent and for some other events d1…dn, metaphysically independent from c and e, the occurrence of c together with d1…dn is nomically sufficient for the occurrence of e. i.e.: c is part of a nomically sufficient condition for e. This is no good: if there exist any d1…dn, metaphysically independent from e, which together nomically necessitate e, then every event e that is metaphysically independent from e and from all of d1…dn, will count as a cause of e! c is a cause of e iff c and e are metaphysically independent and for some other events d1…dn, metaphysically independent from c and e, the occurrence of c together with d1…dn is nomically sufficient for the occurrence of e. but the occurrence of d1…dn is not nomically sufficient for the occurrence of e. i.e.: c is part of a minimal nomically sufficient condition for e. Mackie calls this an ‘INUS’ condition. Not clear that this solves any of the above five problems though.
The Simple Counterfactual Analysis
Define: a counterfactually depends on b =df if b hadn’t occurred, a wouldn’t have occurred.
Analysis: c is a cause of e iff c and e are metaphysically independent and e counterfactually depends on c.
How does this deal with our five problems?
(i) Problem of effects
(ii) Problem of epiphenomena
So long as we can understand the relevant counterfactuals in such a way that there is not widespread counterfactual dependence of later events on earlier ones, the counterfactual analysis does much better with these two problems.
(iii) Problem of completeness
Gone.
(iv) Problem of preemption
Still arises in a different form. Now the problem is not that Wile E Coyote’s lighting of the fuse does count as a cause of the explosion, but that Road Runner’s lighting doesn’t. Can we get around this by saying that if the bomb had exploded later it would have been a numerically different explosion? This doesn’t seem promising.
(v) Problem of chancy causation
What we say about this depends on what we say about counterfactuals in a chancy universe. If you think ‘The bomb wouldn’t have exploded if the fuse hadn’t been lit’ can be true even in a world where bombs always have some small background chance of going off spontaneously at any time, you’re OK.
Refining the Counterfactual Approach
The main goal has been to solve the problem of preemption.
Lewis’s analysis: c is a cause of e iff c is metaphysically independent from e and there exist d1…dn such that d1 counterfactually depends on c, and d2 counterfactually depends on d1, and… and e counterfactually depends on dn.
How is this supposed to help with the problem of pre-emption? Lewis was focused on cases of what has come to be called ‘early’ pre-emption, like the following:
Two Assassins: Boss independently orders Assassin 1 and Assassin 2 to see to it that Vic- tim dies. Both go to Victim’s house. Assassin 1 shoots and kills Victim. Assassin 2, waiting in the wings, sees Assassin 1 shooting, concludes that Victim must be dead, and goes home. If Assassin 1 hadn’t shot at Victim, Assassin 2 would have done so, and Victim would still have died. Although Victim’s death does not counterfactually depend on Assassin 1’s shooting, we still have a chain of counterfactual dependence: the death counterfactually depends on Victim’s being hit by Assassin 1’s bullet, which counterfactually depends on the shooting.
However:
• This strategy doesn’t seem to generalise to other cases of pre-emption, e.g. our example with Wile E Coyote.
• It also has the controversial consequence that causation is transitive. Many reject this on the basis of examples like the following:
Boulder: A boulder is dislodged and rolls down the mountain towards me; I see it coming, duck in time and live to see another day. The fall of the boulder causes my ducking. My ducking causes my survival. But does the fall of the boulder cause my survival?
Previously we started off with, the simple counterfactual analysis: c is a cause of e iff c and e are metaphysically independent and e counterfactually depends on c.
The problem of pre-emption: cases where e is intuitively caused by e without counter-
factually depending on c, because some other event d, which is not actually a cause of e,
would have caused e if c hadn’t occurred.
Examples:
• Road Runner
• Backup (Hitchcock, p. 276): an assassin-in-training is on his first mission. Trainee is an excellent shot: if he shoots his gun, the bullet will fell Victim. Supervisor is also present, in case Trainee has a last minute loss of nerve (a common affliction among student assassins) and fails to pull the trigger. If Trainee does not shoot, Supervisor will shoot Victim herself. In fact, Trainee performs admirably, firing his gun and killing Victim.
• Billy and Suzy (Hitchcock, p. 278): Billy and Suzy both throw rocks at a bottle. Suzy’s throw gets there first, shattering the bottle. Billy’s throw arrives at the scene a split second later, encountering nothing but air where the bottle used to be.
Lewis’ Response
The problem of pre-emption: cases where e is intuitively caused by e without counter- factually depending on c, because some other event d, which is not actually a cause of e, would have caused e if c hadn’t occurred. Lewis 1973: c is a cause of e iff c is distinct (or, perhaps better, metaphysically independent) from e and there exist d1…dn such that d1 counterfactually depends on c, and d2 counterfactually depends on d1, and… and e counterfactually depends on dn.
How this supposed to help: idea is that in Backup, the death counterfactually depends on Victim being struck by Trainee’s bullet, which counterfactually depends on Trainee’s pulling the trigger. (Alternatively: the presence of a bullet en route from T to V). But this doesn’t seem to help with examples like Billy and Suzy and Road Runner. E.g.: the bottle’s breaking does not counterfactually depend on its being hit by Suzy’s rock, since if it hadn’t been hit by her rock it would still have been hit by Billy’s. Moreover, Lewis’ response entails that causation is transitive. Many reject this on the basis of examples like the following:
Boulder (Hitchcock, p. 276): a boulder is dislodged, and begins rolling ominously toward Hiker. Before it reaches him, Hiker sees the boulder and ducks. The boulder sails harmlessly over his head with nary a centimetre to spare. Hiker survives his ordeal.
Intuitively, the fall of the boulder does not cause Hiker to survive, and this despite the causal chain connecting the two. But, on reflection, how convincing is this? Notice that we can sort of convince ourselves that transitivity holds even here by attending to the following considerations. Ask yourself: did the boulder do anything the make the hiker survive? The answer, it seems, is ‘yes’. What it did was make the hiker duck. And it did this by falling, which seems to imply that the boulder made the hiker survive by falling. This, in turn, seems to imply that the boulder’s falling caused the hiker to survive. A thought about Billy and Suzy from Lewis (2000): It seems Suzy’s throw gets to count as a cause of the bottle shattering because if it hadn’t taken place, the bottle would have shattered at a different time. Problem: if so, then all kinds of things that just make a slight difference to the timing of an effect count among its causes.
It seems plausible that Trainee’s pulling the trigger counts a cause of Victim’s death because the following counterfactual is true: if Trainee hadn’t pulled the trigger but Supervisor still hadn’t shot, Victim wouldn’t have died. (Holding fixed the fact that Supervisor didn’t shoot, V wouldn’t have died if T hadn’t pulled the trigger.) Suzy’s throw counts as a cause of the bottle breaking because: if Suzy hadn’t thrown her rock, but Billy’s rock still didn’t strike the bottle, the bottle wouldn’t have broken. (Hitchcock calls these ‘explicitly nonforetracking’ , or ‘ENF’ counterfactuals). How to work this thought up into an analysis of causation? That’s been done in various ways; Hitchcock’s paper describes one of the most fruitful approaches, which involves a level of mathematical abstraction that takes some getting used to.
Causal Models
Variables: We have been talking about events, each of which can either occur or fail to occur. Hitchcock works with a more flexible framework of ‘variables’ which may be allowed to take more than two values. (For example: a variable that takes value 0 if I don’t push a certain button, 1 if I push it with my left hand, and 2 if I push it with my right hand.) We can apply all mathematical operations that make sense for numbers to variables. For example: given two variables A and B, there are variables A + B, A × B, max(A,B), etc. Call expressions like ‘0’, ’A’, ‘A + 1’, ’A + B’, ‘max(A, B)’ and so forth expressions. Say that an expression is minimal if it does not contain any algebraically redundant variables: e.g. ‘(A + B) − B’ is not minimal.
A structural equation is something of the form ‘A := T’, where A is a variable and T is a minimal expression.
A system of structural equations is an ordered pair 〈V, E〉, where V is a set of variables, and E is a set of structural equations such every variable in V occurs on the left-hand-side of exactly one member of E, and all the right-hand-sides of members of E are expressions in which all variables belong to V. When S is a system of structural equations and A and B are variables, A directly depends on B in S if some equation of the form A := T, where T contains B, is a member of E where S = 〈V, E〉. (A system of structural equations thus generates a directed graph among its variables, with relations of direct dependence represented by arrows.)
A is exogenous in S iff it does not immediately depend on any other variable in S. That is: its structural equation is of the form A := [constant].
A causal model is a system of structural equations S that is acyclic: there are no variables V1…Vn such that V1 immediately depends on V2 in S, and… and Vn directly depends on V1 in S.
The value of a variable in a causal model is the unique number you get by successively replacing variables with constants, starting with the exogenous variables in the obvious way. A counterfactual of the form ‘(V1 = x1 ∧ … ∧ Vn = xn) □→ Vn + 1 = xn + 1’ is true in a causal model S iff all the variables V1…Vn + 1 occur in S, and the value of Vn + 1 is xn + 1 in the mutilated causal model S′ which is arrived at by replacing all the equations for V1…Vn in S with exogenous equations V1 := x1 … Vn := xn.
A causal model S is apt (H, p. 287) only if
(i) The actual value of every variable that occurs in S equals its value in S.
(ii) Every counterfactual of the form ‘V1 = x1 ∧ … ∧ Vn = xn □→ Vn + 1 = xn + 1’ that is true in S is true in fact.
(iii) The model does not represent relations of dependence between variables that are not ‘distinct’. (Perhaps the ‘metaphysical independence’ constraint is better).
(iv) No conjunction of assignments of values to variables in the model should be true only at possibilities that are ‘too remote’.
Hitchcock’s Theory of Causation
Let c and e be distinct occurrent events, and let C and E be variables such that C = c represents that c occurs and E = e represent that e occurs. Then c is a cause of e if and only if there is an active causal route from C to E in an apt causal model 〈V, E〉, where
Act: The route (C, D1…Dn, E,) is active in the causal model 〈V, E〉 if and only if it it is true that C = c* □→ E = e* (for some c* and e*, where c* ≠ c and e* ≠ e) in the model that results from replacing E with the new system of equations E′ constructed from E as follows: for all X ∈ V; if X is intermediate between C and E but does not belong to the route (C, D1…Dn, E,), then replace the equation for X with a new equation that sets X equal to its actual value in E.
Backup: T := 1, S := ¬T, V := T ∨ S
Boulder: F := 1; D := F; S := ¬F ∨ D
Billy and Suzy: S := 1; B := 1; SH := S; BH := ¬SH ∧ B; B := SH ∨ BH
Problems that might arise are cases like these
Commander: Commander orders the two shooters in a firing squad to fire, which they do simultaneously, so that both their bullets hit Prisoner’s heart at precisely the same time. Prisoner dies. Either shot on its own would have been enough to kill Prisoner. It is natural to think that the following causal model would be apt in this case: O := 1; S1 := O; S2 := O; P := S1 ∨ S2
Bet: You and I bet about whether the atmospheric pressure will drop: I bet that it will, you bet that it won’t. In fact, the pressure does drop, I win our bet, and a storm results from the drop in pressure.
Another Worry: do ENF counterfactuals work in the way that causal models assume? Dominoes: Four dominoes A, B, C, D are arranged in a row in the usual way. Each can either remain upright (value 0) or fall (value 1). In fact, none of them fall. It is natural to think that the following causal model would be apt in this case: A := 0; B:=A; C:=B; D:=C. But note that the following counterfactual is true according to this model: A:=1 ∧ C:=0 □→B:=1. Is this really true? If not, the use of causal models will have to be seriously rethought. And so will the more general idea of analysing causation using ENF counterfactuals.