David Lewis, answered from the texts and cited to the page.
The basic case: C causes E when E counterfactually depends on C — that is, when had C not occurred, E would not have occurred either. Counterfactual dependence between distinct events is sufficient for causation between them in the simple case; the analysis extends to more complex cases by allowing chains of counterfactual dependence, so that C causes E whenever there is a sequence C, D₁, D₂, ..., E with each step a case of counterfactual dependence on its predecessor.12
The counterfactuals are evaluated by similarity-of-worlds in the system Counterfactuals (1973) provides, with the four-condition similarity metric of "Counterfactual Dependence and Time's Arrow" (1979) keeping the asymmetry between past and future right — we hold the past mostly fixed, accept a small miracle at the time of antecedent realization, and minimize subsequent miracles.3
The 1973 chain-of-dependence form handles early pre-emption (the back-up assassin) cleanly; it struggles with late pre-emption (the wood-splitting case, the truck and the fragile vase) where the back-up process is cut off by the effect itself rather than before. In "Causation as Influence" (2000) I revised: C influences E if there is a pattern of counterfactual dependence relating substantial ranges of variations in how/when/whether C occurs to ranges of variations in how/when/whether E occurs, and causation is a chain of such influences.
The influence formulation handles late pre-emption while preserving the counterfactual-dependence intuition that underwrote the 1973 analysis.45
As before, we must take an ancestral to ensure that causation will come out transitive; thereby providing not only for chains of stepwise dependence, but also for chains of stepwise quasi-dependence, or mixed chains. To this end we could redefine a causal chain as a sequence of two or more events, with either dependence or quasi- dependence at each step.Part 6: Causation, pp. 206–207
Perhaps there are special reasons in special cases— but then these may be cases of backward causal dependence. PREEMPTION Suppose that c1 occurs and causes e; and that c2 also occurs and does not cause e, but would have caused e if c↑ had been absent. Thus c2 is a potential alternate cause of e, but is preempted by the actual cause c1.Part 6: Causation, pp. 171–172
See also Fred I. Dretske, "Laws of Next, counterfactuals. I take them to be governed by similarity of worlds, according to the analysis given in "Counterfactuals and Com parative Possibility," in this volume.Philosophical Papers Vol II, pp. xii–xiii
Therefore I shall not follow the popular strategy of approaching events by way of nominalisations. Events made in the image of nominalisations are right for some purposes, but not for mine. When I introduce nominalisations to denote events, as I shall, it will not be analysis of natural language but mere stipulative definition.Part 6: Causation, p. 242
Let us see where this one leads us. A process in a region may exhibit a pattern of counterfactual depen dence that makes it causal, according to my original analysis.Part 6: Causation, p. 206
The most influential analysis of causation in late-twentieth- century analytic philosophy, organizing literature still generating new arguments today. The principal rivals are regularity theories (the contemporary descendants of Mackie's INUS-conditions analysis), process theories (Salmon, Dowe — causation as the transfer of a conserved quantity), interventionist theories (Woodward, Pearl — causation as what would happen under hypothetical interventions), and probabilistic theories. The pre- emption case literature (Hall, Schaffer, McDermott, Hitchcock, Hall-Paul) tests every analysis against the same set of standard examples — late pre-emption, trumping, symmetric overdetermination, causation by omission and double prevention. The 2000 "Influence" revision is the official late position; some of the literature treats the 1973 chain-of-dependence view as the canonical Lewis line and the influence view as a refinement under pressure.