In reading the SEP entry on counterfactual theories of causation, I had the following question occur, and I haven’t been able to satisfactorily resolve it for myself.

An event *e* is said to causally depend on an event *c* if and only if *e* would occur if *c* were to occur and *e* would not occur if *c* were not to occur.

The article makes a point of articulating that causal dependence entails causation (if *e* causally depends on *c*, *c* is a cause of *e*) but not vice versa. It then defines a causal chain as a fine sequence of events *c*, *d*, *e*,… where *d* causally depends on *c*, *e* on *d*, and so on, before defining *c* to be a cause of *e* if and only if there exists a causal chain leading from *c* to *e*.

What I’m having trouble with is understanding how *c* can cause *e* according to the given definition without *e* causally depending on *c*. If there’s a causal chain from *c* to *d* to *e*, then *d* causally depends on *c*, and *e* causally depends on *d*, so if *c* were to not occur, *d* would not occur, and if *d* were to not occur, *e* would not occur. But doesn’t this directly entail that if *c* were to not occur, then *e* would not occur and therefore that *e* causally depends on *c*?

So how can *c* cause *e* according to the definition without *e* causally depending on *c*??

On Lewis’s account of counterfactuals, this isn’t true, i.e. causal dependence is non-transitive. Hence, he defines causation as the transitive closure of causal dependence.

Lewis’ semanticsLet W be a set of worlds. A proposition is characterised by the subset A⊆W of worlds in which the proposition is true.

Moreover, assume each world w∈W induces an ordering ≤w over worlds, where w1≤ww2 means that world w1 is closer to w than w2. Informally, if the actual world is w, then w1 is a smaller deviation than w2. We assume w′≤ww⟹w′=w, i.e. no world is closer to the actual world than the actual world.

For each w∈W, a “neighbourhood” around w is a downwards-closed set of the preorder (W,≤w). That is, a neighbourhood around w is some set N such that w∈N and for all w′∈N and w′′∈W, if w′′≤ww′ then w′′∈N. Intuitively, if a neighbourhood around w contains some world w′ then it contains all worlds closer to wthan w′. Let Nw denote the neighbourhoods of w∈W.

NegationLet Ac denote the proposition ”A is not true”. This is defined by the complement subset W∖A.

CounterfactualsWe can define counterfactuals as follows. Given two propositions A and B, let A?B denote the proposition “were A to happen then B would’ve happened”. If we consider A,B⊆W as subsets, then we define A?B as the subset {w∈W∣A=∅, or for some N∈Nw,∅≠A∩N⊆B∩N}. That’s a mouthful, but basically, A?B is true at some world w if

(1) ”A is possible” is globally false, i.e. A=∅

(2) or ”A is possible and A→B is necessary” is locally true, i.e. true in some neighbourhood N∈Nω.

Intuitively, to check whether the proposition “were A to occur then B would’ve occurred” is true at w, we must search successively larger neighbourhoods around w until we find a neighbourhood containing an A-world, and then check that all A-worlds are B-worlds in that neighbourhood. If we don’t find any A-worlds, then we also count that as success.

Causal dependenceLet A⇝B denote the proposition ”B causally depends on A”. This is defined as the subset (A?B)∩(Ac?Bc)

Nontransitivity of causal dependenceWe can see that (−?−) is not a transitive relation. Imagine W={0,1,2,3} with the ordering ≤0 given by 1≤02≤03. Then {3}⇝{2,3} and {2,3}⇝{2} but not {3}⇝{2}.

Informal counterexampleImagine I’m in a casino, I have million-to-one odds of winning small and billion-to-one odds of winning big.

Winning something causally depends on winning big:

Were I to win big, then I would’ve won something. (Trivial.)

Were I to not win big, then I would’ve not won something. (Because winning nothing is more likely than winning small.)

Winning small causally depends on winning something:

Were I win something, then I would’ve won small. (Because winning small is more likely than winning big.)

Were I to not win something, then I would’ve not won small. (Trivial.)

Winning small doesn’t causally depend on winning big:

Were I to win big, then I would’ve won small.

(WRONG.)Were I to not win big, then I would’ve not won small. (Because winning nothing is more likely than winning small.)

Thanks so much for this—it was just the answer I was looking for!

I was able to follow the logic you presented, and in particular, I understand that {3}⇝{2,3} and {2,3}⇝{2} but not {3}⇝{2} in the example given.

So, I was correct in my original example of c->d->e that

(1) if c were to not happen, d would not happen

(2) if d were to not happen, e would not happen

BUT it was incorrect to then derive that if c were to not happen, e would not happen? Have I understood you correctly?

I’m still a bit fuzzy on the informal counterexample you presented, possibly because of the introduction of probability. For example, I don’t understand how not winning big entails not winning something because winning nothing is more likely than winning small. If you did not win big, you might still have won small (even if it’s unlikely); I don’t understand how the likelihood comes into account.

I wish I had an intuitive example of a

candewherecis a cause ofewithoutecausally depending onc, but I’m struggling to imagine one.Suppose Alice and Bob throw a rock at a fragile window, Alice’s rock hits the window first, smashing it.

Then the following seems reasonable:

Alice throwing the rock caused the window to smash.

True.Were Alice ot throw the rock, then the window would’ve smashed.

True.Were Alice not to throw the rock, then the window would’ve not smashed.

False.By (3), the window smashing does not causally depend on Alice throwing the rock.

If I understand the example and the commentary from SEP correctly, doesn’t this example illustrate a problem with Lewis’ definition of causation? I agree that commonsense dictates that Alice throwing the rock caused the window to smash, but I think the problem is that you cannot construct a sequence of stepwise dependences from cause to effect:

Is the example of the two hitmen given in the SEP article (where B does not fire if A does) an instance of causation without causal dependence?

tbh, Lewis’s account of counterfactual is a bit defective, compared with (e.g.) Pearl’s

I think you want to define A?B to be true if A is true when we restrict to some neighbourhood N∈Nw

. Otherwise your later example doesn’t make sense.such that A∩N is nonemptyEdit: Wait, I see what you mean. Fixed definition.

~~For Lewis,~~∅?B=W~~for all~~B~~. In other words, the counterfactual proposition “were~~ϕ~~to occur then~~ψ~~would’ve occurred” is necessarily true if~~ϕ~~is necessarily false. For example, Lewis thinks “were 1+1=3, then Elizabeth I would’ve married” is true. This means that~~A∩N~~may be empty for all neighbourhoods~~N∈Nω~~, yet~~A?B~~is nonetheless true at~~w~~.~~~~Source: David Lewis (1973), Counterfactuals. Link:~~https://perso.uclouvain.be/peter.verdee/counterfactuals/lewis.pdf~~Elaborate?~~