We might be tempted to say that causes are necessary and sufficient conditions for their effects. As J. L. Mackie points out in "Causes and Conditions," however, this will not do. Consider the sentence "The short circuit caused the fire" (and suppose it true). The occurrence of a short circuit was in no way necessary for the occurrence of a fire, for a fire could have been caused by Mrs. O'Leary's cow knocking over a lamp. And the short circuit was not sufficient for the fire, for without the presence of oxygen and flammable material, the short circuit could have occurred without the ensuing fire.
Mackie proposes, however, that this naive judgment is not completely off the mark. This is what is true of the short circuit, according to Mackie: it is an indispensable (i.e. necessary) member of a set of conditions that are jointly sufficient for the fire, although these conditions are not themselves necessary for the occurrence of the fire. Mackie refers to such a condition as an INUS condition: an Insufficient but Necessary part of a condition which is itself Unnecessary but Sufficient for the result.
Formally, Mackie's definition of an INUS condition is as follows: A is an INUS condition of P iff for some X and some Y, (AX or Y) is necessary and sufficient for P, although neither A nor X is sufficient for P. [AX is the conjunction of A and X.]
Mackie's official analysis of causation actually makes use of the notion of A's being at least an INUS condition for P. A is at least an INUS condition just in case there is a necessary and sufficient condition for P of one of the following forms: (AX or Y), (A or Y), AX, A. According to Mackie, we may then analyze "A caused P" as follows:
(i) A is at least an INUS condition for P,
(ii) A was present on the occasion in question,
(iii) X [if there is an X in the NS conditions] was present
on the occasion in question, and
(iv) Every disjunct in Y not containing A was absent.
In "Causation," David Lewis notes that Hume gave two different definitions of "cause": first, Hume said that "we may define a cause to be an object followed by another, and where all objects similar to the first are followed by objects similar to the second." We may call analyses of causation that follow this general strategy "regularity analyses."
Following Lewis, we may give a general schema for a regularity analysis. Let C be the proposition that event c occurs, and let E be the proposition that event e occurs. Then, according to a regularity analysis, c causes e iff (1) C and E are true, (2) for some non-empty set L of true law-propositions and some set F of true propositions of particular fact, L and F jointly imply C->E, although L and F alone do not jointly imply E and F alone does not imply C->E.
Lewis thinks that the prospects for a workable regularity analysis of causation are not good. In particular, it does not seem that a regularity analysis will be able to distinguish between genuine causation and other causal relations. Rather than c causing e, e might be a cause of c, one which, given the laws and the circumstances, have occurred otherwise than by being caused by e. Or c might be an epiphenomenon of some genuine cause of e, an event caused by some genuine cause of e. Or c might be a preempted potential cause of e, an event that did not cause e, but which would have, in the absence of whatever actually did cause e.
Hume's second definition (which he apparently thought synonymous with the first, was this: "an object followed by another, . . . where, if the first had not existed, the second had never existed." We may call analyses of causation that follow this general strategy "counterfactual analyses." This is the strategy that Lewis proposes to follow. The classic objection to counterfactual analyses, that counterfactuals are ill-understood, no longer holds: thanks to the work of Stalnaker and Lewis, we now have an adequate semantics for counterfactual discourse. We may now put this understanding to use in an analysis of causation.
First, we need to define causal dependence among actual events. Let O(c) be the proposition that event c occurred, and let O(e) be the proposition that event e occurred. (Note that O(c) is not the proposition that some event occurs satisfying some description that c actually satisfies, but the proposition that c itself occurs. If Socrates had fled Athens and died of old age, there still would have been an event satisfying the description "the death of Socrates," but this would not have been the same event that was actually the death of Socrates.) Then we may define "e depends counterfactually on c" as follows: ~O(c) []-> ~O(e), i.e. if c had not occurred, e would not have occurred.
Counterfactual dependence implies causation, according to Lewis, but causation does not imply counterfactual dependence, for causation must be transitive, although counterfactual dependence may not be. (Counterfactual conditionals, unlike material conditionals, do not obey the law of hypothetical syllogism.) Perhaps c is a cause of e, even though e would have occurred even in c's absence: for if c had not occurred, something else would have caused e. Instead, we may define causation as the ancestral of counterfactual dependence. An event c causes an event e just in case there is some finite sequence of events d1...dn, such that e depends counterfactually on dn, d1 depends counterfactually on c, and each member (after the first) of the sequence d1...dn depends counterfactually on the preceding member.
Lewis, David. "Causation", Journal of Philosophy 70 (1973), 556-67. Reprinted in Philosophical Papers, Vol. II. Oxford, 1986.
Mackie, J. L. "Causes and Conditionals," American Philosophical Quarterly 2 (1965), 245-65.
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