Question about Lewis' counterfactual theory of causation

Jun 08, 2024*

170

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

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' semantics

Let be a set of worlds. A proposition is characterised by the subset $A \subseteq W$ of worlds in which the proposition is true.

Moreover, assume each world $w \in W$ induces an ordering $\leq_{w}$ over worlds, where $w_{1} \leq_{w} w_{2}$ means that world $w_{1}$ is closer to $w$ than $w_{2}$ . Informally, if the actual world is $w$ , then $w_{1}$ is a smaller deviation than $w_{2}$ . We assume $w^{'} \leq_{w} w ⟹ w^{'} = w$ , i.e. no world is closer to the actual world than the actual world.

For each $w \in W$ , a "neighbourhood" around $w$ is a downwards-closed set of the preorder $(W, \leq_{w})$ . That is, a neighbourhood around $w$ is some set $N$ such that $w \in N$ and for all $w^{'} \in N$ and $w^{''} \in W$ , if $w^{''} \leq_{w} w^{'}$ then $w^{''} \in N$ . Intuitively, if a neighbourhood around $w$ contains some world $w^{'}$ then it contains all worlds closer to $w$ than $w^{'}$ . Let $N_{w}$ denote the neighbourhoods of $w \in W$ .

Negation

Let $A^{c}$ denote the proposition " $A$ is not true". This is defined by the complement subset $W ∖ A$ .

Counterfactuals

We 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 $A, B \subseteq W$ as subsets, then we define $A ? B$ as the subset ${w \in W ∣ A = \emptyset, or for some N \in N_{w}, \emptyset \neq A \cap N \subseteq B \cap N}$ . That's a mouthful, but basically, $A ? B$ is true at some world $w$ if

(1) " $A$ is possible" is globally false, i.e. $A = \emptyset$

(2) or " $A$ is possible and $A \to B$ is necessary" is locally true, i.e. true in some neighbourhood $N \in N_{ω}$ .

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 dependence

Let $A ⇝ B$ denote the proposition " $B$ causally depends on $A$ ". This is defined as the subset $(A ? B) \cap (A^{c} ? B^{c})$

Nontransitivity of causal dependence

We can see that $(- ? -)$ is not a transitive relation. Imagine $W = {0, 1, 2, 3}$ with the ordering $\leq_{0}$ given by $1 \leq_{0} 2 \leq_{0} 3$ . Then ${3} ⇝ {2, 3}$ and ${2, 3} ⇝ {2}$ but not ${3} ⇝ {2}$ .

Informal counterexample

Imagine 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:
1. Were I to win big, then I would've won something. (Trivial.)
2. 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:
1. Were I win something, then I would've won small. (Because winning small is more likely than winning big.)
2. Were I to not win something, then I would've not won small. (Trivial.)
Winning small doesn't causally depend on winning big:
1. Were I to win big, then I would've won small. (WRONG.)
2. Were I to not win big, then I would've not won small. (Because winning nothing is more likely than winning small.)

[-]jbkjr1y21

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 counterexam... (read more)

4Cleo Nardo1y

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: 1. Alice throwing the rock caused the window to smash. True. 2. Were Alice ot throw the rock, then the window would've smashed. True. 3. Were Alice not to throw the rock, then the window would've not smashed. False. 4. By (3), the window smashing does not causally depend on Alice throwing the rock.

1jbkjr1y

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?

2Cleo Nardo1y

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

[-]Dacyn1y10

I think you want to define $A ? B$ to be true if $A$ is true when we restrict to some neighbourhood $N \in N_{w}$ such that $A \cap N$ is nonempty. Otherwise your later example doesn't make sense.

2Cleo Nardo1y

Edit: 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?

LESSWRONG
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LESSWRONG
LW

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Question about Lewis' counterfactual theory of causation

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Jun 08, 2024*

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Question about Lewis' counterfactual theory of causation

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Jun 08, 2024*

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