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 $W$ 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 ${\le }_w$ over worlds, where $w_1{\le }_w{w}_2$ 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 $w^{\prime }{\le }_{w}w⟹w^{\prime }=w$, 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 $(W,{\le }_w)$. That is, a neighbourhood around $w$ is some set $N$ such that $w\in N$ and for all $w^{\prime }\in N$ and $w^{\prime \prime }\in W$, if $w^{\prime \prime }{\le }_w{w}^{\prime }$ then $w^{\prime \prime }\in N$. Intuitively, if a neighbourhood around $w$ contains some world $w^{\prime }$ then it contains all worlds closer to $w$than $w^{\prime }$. 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 $W\setminus A$. 

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 $\{w\in W\mid A=\varnothing \text{, or for some}N\in N_w,\varnothing \ne A\cap N\subseteq B\cap N\}$. That's a mouthful, but basically, $A?B$ is true at some world $w$ if

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

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

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 ${\le }_0$ given by $1{\le }_{0}2{\le }_{0}3$. Then $\left\{3\right\}⇝\{2,3\}$ and $\{2,3\}⇝\left\{2\right\}$ but not $\left\{3\right\}⇝\left\{2\right\}$.

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.

1. 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.)
2. 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.)
3. 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.)