Causation and counterfactuals

Let us now consider the way Mackie characterizes sufficient and necessary conditions. Standard definitions of these notions are as follows:
A is a sufficient condition of B iff, if A occurs, B occurs
A is a necessary condition of B iff, if B occurs, A occurs (or, equivalently, if A doesn’t occur, B doesn’t occur).

But these definitions are correct only when A and B are general types of events, and not names of individual objects (in that case the right-hand sides of the equivalences have to be interpreted as general statements: “For all x, if x is A and x occurs, then there is a y such that y is B and y occurs”). If A and B are singular names, the aforementioned definitions wrongly imply that all actual events are sufficient and necessary conditions of each other. Mackie attempts to give a better analysis applicable to singular claims, not general ones. His analysis is presented with the help of following equivalences:

x is a sufficient condition of y iff since x occurred, y occurred
x is a necessary condition of y iff if x had not occurred, y would not have occurred.

Mackie stresses that the conditionals used in each explanans can’t be interpreted as material conditionals. But this creates an immediate problem for the regularity approach, as one of its main assumptions is that causation should be explicated without resorting to modal notions, such as necessity or possibility. Mackie suggests the following interpretations of the non-material conditionals used above. He treats them as “telescoped arguments” in which some premises are omitted. For instance, the statement “If the short circuit had not occurred, there would have been no fire” can be expanded into the statement that there are some true universal propositions which together with true statements about the conditions of the house and together with the supposition that the short circuit did not occur logically imply that there was no fire. A similar analysis can be given for the statement “Since x occurred, y occurred”.


One of the most serious challenges for any account of causation is presented by the so-called redundant causation. There are two main types of redundant causation: overdetermination and pre-emption. Overdetermination occurs when there is more than one acting cause, each of which is sufficient for the effect to occur. An example can be an execution by a firing squad, in which each bullet causes a lethal injury. Is each individual shot a cause of the death of the condemned man? In Mackie’s approach the answer is negative, because one of his characteristics of causation is that no alternative sufficient conditions are present (a cause is necessary post factum for the effect). Only the disjunction of all shots is necessary in this sense.

The case of pre-emption can be described using the following example. Two children, Billy and Suzie, are throwing stones at a bottle. Seeing that Suzie has thrown her stone and shattered the bottle, Billy does not hurl his stone, but if Suzie had not thrown, Billy would have thrown his stone. We call this pre-emption, because Suzie’s throw pre-empts Billy’s action which would otherwise have taken place. Any reasonable theory of causation should imply that Suzie’s throw was the actual cause of the shattering. But wasn’t her throw unnecessary, given that Billy was present as a backup? Are the conditions imposed by Mackie satisfied? It turns out that they are, in spite of our initial worries. According to Mackie’s definition, there has to be a set of conditions X such that together with Suzie’s throw (let’s call it A) they would constitute a sufficient condition for the shattering, and moreover no alternative sets of sufficient conditions can be present. We can reasonably believe that without A Bill’s throw together with its conditions would create a different sufficient condition for the shattering of the bottle, but this set is not present at the time of Suzie’s throw. Suzie’s throw is still a necessary part of its own set of conditions – without it this set would not be sufficient, although a different one would be. So Mackie’s definition gives the right answer in the case of pre-emption.

David Lewis has noted that all regularity theories of causation, including Mackie’s, have problems with distinguishing causes from effects, and direct causal links from correlations arising due to a common cause. If only events of type A can cause B, we can say that a given event B is a sufficient condition (or part of a sufficient condition) of A, and hence B is wrongly classified as a cause of A. Even if we artificially exclude this possibility by stipulating that a cause has to be earlier than its effect, still the problem remains. Suppose that an event A causes B and C in succession, and that B can be created only by events of type A. In such a case B is a sufficient condition (or, as in Mackie’s conception, a necessary part of a sufficient condition) of A, and A is in turn a sufficient condition of C, hence B comes out to be a cause of C.

Lewis suggests an alternative account of causation: a counterfactual analysis. The simplest version of such an analysis (the so-called naive counterfactual analysis) is as follows: x is a cause of y iff if x had not occurred, y would not have occurred. But this definition is in need of serious corrections. For instance, suppose that I shut the door by slamming it. If I hadn’t shut the door, I wouldn’t have slammed it (we assume that each slamming shuts the door), but the shutting of the door is not a cause of its slamming. Similarly, if I hadn’t written the letter “L”, I would not have written the word “Lewis”, but the first did not cause the second. To eliminate these counterexamples, we have to assume that x and y are distinct events (i.e. they are not identical, nor one is part of the other). But in order to further advance the counterfactual analysis, we have to understand better the meaning of counterfactual conditionals.

Counterfactual conditionals can be generally presented as statements of the form “If it were (had been) the case that p, then it would be (have been) the case that q”. It is well known that sentences of that form are not truth-functional, i.e. the truth value of the entire complex statement is not determined by the truth values of its components. To see this, it suffices to compare the following two statements: “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would float” and “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would fly”. In both cases the antecedent and the consequent are false, and yet the first conditional is true, while the second false. The counterfactual connective is considered to be a modal one, and it receives an interpretation in terms of possible worlds, similar to that of necessity and possibility. The most commonly accepted analysis stipulates that the conditional “If it were p, then it would be q” is true if and only if q is true in all possible worlds in which p is true and which are closest to the actual world of all p-worlds. Thus the statement “If I threw a stone at a window, it would shatter” is true if in the possible worlds in which I throw the stone and which otherwise are as similar to the actual world as the truth of the antecedent allows, the window shatters. And this is what we expect to get, because in such worlds the laws of nature and the properties of materials such as glass and rock will be the same as in our world. On the other hand, if we considered a far away world in which glass is tougher than rock, the consequent would not be true. This shows that for a counterfactual to be true, the consequent does not have to be true in all worlds in which the antecedent is true (this truth condition defines the so-called strict conditional).

Counterfactual conditionals follow a slightly different logic than material conditionals or strict conditionals. Let us consider the three following logical laws: the law of the strengthening of the antecedent, the law of transposition and the law of transitivity. The first one states that if p implies q, then the conjunction p and r also implies q. But consider the following example: If someone shot a gun pointed at my chest, I would be dead, but if someone shot at me and I was wearing a bulletproof vest, I would survive. Given that in the actual world I am not wearing a bulletproof vest now, both statements seem to be true, which shows that the law of the strengthening of the antecedent is violated for counterfactual conditionals. The reason behind this is that we choose different possible worlds to evaluate both statements: the first one is the world where I am being fired at, but I am not wearing any protection, and the second one is the one in which I am protected by a bulletproof vest.

The law of transposition states that if p implies q, not-q implies not-p. A counterexample to this law is as follows: it is true that if I didn’t come to my lecture, the building in which I am lecturing would still stand. But from this it does not follow that if the building collapsed (for instance because of an earthquake), I would still come to the lecture. Finally, the law of transitivity prescribes that if p implies q, and q implies r, then p implies r. We already presented a case which violates this law when we discussed the problem of the transitivity of the causal relation. In this case p = the bomb was not planted, q = the bomb was not defused, r = the politician was assassinated. We can also note that the violation of transitivity follows directly from the violation of the strengthening of the antecedent, given that it is a logical truth that if it were p and r, then it would be p. Now we can choose p, q and r such that it is true that if it were p, it would be q, but it’s false that if it were p and r, then it would be q, and the transitivity is violated.

Reading:

E.J. Lowe, Chapter 8 "Counterfactual conditionals", pp. 137-154, A Survey of Metaphysics.