Notes on David Lewis

NOTES ON DAVID LEWIS

(What follows are two separate handouts with some overlapping comments.)

I. Definitions and theorems from David Lewis's "Causation"

D1. Let the A's be a family A₁, A₂, ... of possible propositions, no two of which are compossible, and let the C's be another such family C₁, C₂, ... . Then

The C's depend counterfactually on the A's iff all the counterfactuals A₁ ---> C₁, A₂ ---> C₂, ... between corresponding propositions in the two families are true.

D2. Actual particular event e depends causally on actual particular event c iff the family O(e), ~O(e) ... depends counterfactually on the family O(c), ~O(c), ..., i.e. iff c ---> e and ~c ---> ~e.

D2a. (To accommodate "chancy causation"): Actual particular event e depends causally on actual particular event c iff c ---> (x)e and ~c ---> (1-y)~e, where x > 0 and x>> y.

(See below for an objection to (D2a) according to which (D2) is right, or at least closer to the truth, for probabilistic as well as deterministic causation.)

D3. CC is a causal chain iff there is a finite sequence c, d, e ... of actual particular events such that d depends causally on c, e depends causally on d, ...

D4. c is a cause of e iff there is a causal chain CC such that CC leads from c to e.

Question: How is this account related to a regularity account, the fundamental idea of which is that causation is the instantiation of regularities? Lewis's answer: The regularity account captures something akin to (but not quite identical with) nomic dependence (see below), where nomic dependence implies causal dependence as explicated above, but not vice versa. The problem is that there seem to be imaginable instances of causation in which causal dependence is present without nomic dependence and hence is not "explicable" by appeal to nomic dependence.

D5. B is counterfactually independent of the family A₁, A₂, ... of alternatives iff B would be true no matter which of the A's were true.

D6. Let the A's be a family A₁, A₂, ... of possible propositions, no two of which are compossible, and let the C's be another such family C₁, C₂, ... . Then

The C's depend nomically on the A's iff there is a set L of true law-propositions and a set F of true propositions of particular fact such that

(a) L and F jointly imply all the material conditionals A₁ ---> C₁, A₂ ---> C₂, ..., and
(b) F does not by itself imply all these material conditionals.

In such a case we can say that the C's depend nomically on the A's in virtue of L and F.

T1. If (i) the C's depend nomically on the A's in virtue of the premise sets L and F and (ii) all members of L and F are counterfactually independent of the A's, then the C's depend counterfactually on the A's (though not necessarily vice versa).

Kim questions how the notion of law figures in this derivation; if this cannot be shown clearly, he claims, then it is not clear how Lewis's account relates to the classical regularity theory. (In passing we should briefly consider the five contexts for causal talk noted by Kim on p. 194; an account of causality, he claims, should be judged by its account of such talk. The five contexts are:

a. explanation of the occurrence of particular events;
b. prediction of future events;
c. control of events;
d. attributions of moral responsibility, legal liability, etc.;
e. special use of notion of cause in physical theory.)

II. Putative counterexamples

Lewis next claims his account is not subject to the three sorts of counterexamples that plague even sophisticated regularity accounts:

1. Problem of effects: Suppose c is a cause of subsequent event e and that e does not also cause c; suppose further that given the relevant laws and circumstances, c could not have failed to cause e. In such a case it seems to follow that c is counterfactually, and thus causally, dependent on e--contrary to our assumption.

Reply: It is not the case that c is counterfactually dependent on e. That is, it does not follow that if e had not been actual, c would not have been actual. In this sort of case similarity with respect to actual present and past fact (c's being actual) outweighs similarity with respect to laws.

2. Problem of epiphenomena: Suppose c is a cause first of e and then of f, but e is not a cause of f. Suppose further that given the relevant laws and circumstances, c could not have failed to cause e and f could not have been caused otherwise than by c. It seems to follow that f is counterfactually, and thus causally, dependent on e--contrary to our assumption.

Reply: It is not the case that f is counterfactually dependent on e. That is, it does not follow that if e had not been actual, then f would not have been actual. Once again, similarity with respect to past and present fact (f's being actual) outweighs similarity with respect to laws.

3. Problem of preemption: Suppose c occurs and causes e and that d also occurs and does not cause e, but would have caused e if c had not occurred. So e is not counterfactually dependent on either c or d. How, then, on the present account can it be that c, but not d, is a cause of e?

Reply: This can happen only if there is some actual event f such that there is a casual chain from c to e via f. But even then it seems that e is not causally dependent on f, since if f had not been actual, then c would not have been actual and d would have caused e. Not so, says Lewis; it's just not the case that if f had not been actual, then c would not have been actual. (Question: is this satisfactory? Answer: No, but Lewis has a lot more to say about preemption in the Postscript to this paper that appears in volume 2 of his collected papers (see below).) In particular, he distinguishes redundant causation from preemption and distinguishes two sorts of preemption, viz., early and late).

III. Some reflections

Kim claims that D2 is too broad because there are other sorts of counterfactual dependence among events besides causal dependence, viz., (i) logical or analytical dependence, (ii) constitutive dependency, (iii) act-generation dependencies, (iv) non-causal determination (see his examples). Also, the problem of preemption, he thinks, shows that counterfactual dependence is too narrow to cover (even deterministic) causation.

Some comments on chancy causation:

a. Lewis emphasizes that (D2a) has to do with singular cases and with counterfactuals about probabilities rather than with conditional probabilities. This helps him to avoid common cause problems and also to allow for the conceptual possibility of a wholly deterministic world. (The reason for the latter is that the conditional probability P(~e/~c) = P(~e~c)/P(~c) and hence is undefined if the world is completely determined and thus P(c) = 1.)

b. On this view c might be a cause of e even if it makes e less probable than it would otherwise have been (since c might be part of a causal chain that is a less reliable producer of e than another causal chain that would have been in place if c had not occurred. (This is definitely an advantage, since analyses of causality in terms of probability differences run into serious problems with cases of this sort).

c. Lewis then considers an objection to (D2a) that is very interesting from an Aristotelian point of view. The objection goes as follows: "The original analysis of causal dependence, (D2), is better. For if there would have been some residual chance of e even without c, then the raising of probability only makes it probable that in this case c is a cause of e. Suppose, for instance, that the actual chance of e, with c, was 88%, but that without c, there would still have been a 3% probability of e. Then most likely (probability 97%) this is a case in which e would not have happened without c; then c is indeed a cause of e. But this just might be (probability 3%) a case in which e would have happened anyway; then c is not a cause of e. We can't tell for sure which kind of case this is." Notice that on an Aristotelian view this sort of case is perfectly possible; it depends on which substances or powers happen to have acted or operated in the case in question. Lewis, however, casts aspersions on appeal to any hidden factors here and simply denies that in such a situation it is either definitely true that e would have occurred without c or definitely true that e would not have occurred without c. Interestingly, in a footnote (p. 182) Lewis notes than an appeal to "hidden factors" might be supported by either theological reasons (having to do with divine providence and put forth by Molina and Suarez in support of the doctrine of middle knowledge) or physical reasons (having to do with the attempt to derive Bell's theorem by taking certain counterfactuals as basic and without recourse to hidden variables). But he rejects both sets of reasons and accuses those who put them forth as the victims of an ambiguity in the notion of chance.

This raises an interesting question about the relation between what we might call simple counterfactuals, i.e., counterfactuals of the form A ---> B, and counterfactuals about probabilities, i.e., counterfactuals of the form A ---> (x)B or A --->x B. My own suspicion is that the standard possible-world semantics for counterfactuals works only for the latter and not for the former. But this would take us too far afield for the present. Still, it does raise the question of just what Lewis's account is lacking from an Aristotelian perspective. Is it the case that despite its much higher level of sophistication, it is still vulnerable to the Kim-like objections that undermine Biel's analysis as well? Perhaps a lot depends on how one analyzes the notion of an event. (See Lewis, Philosophical Papers, volume II, pp. 241-69).

CAUSALITY, PROBABILITIES, AND SUBJUNCTIVE CONDITIONALS: MORE REFLECTIONS

The following are some thoughts that I hope will clarify Lewis's reply to the objection that definition 2 is preferable to definition 2A as an account of probabilistic causality. For convenience I will reprint the two definitions plus the objection itself.

D2. Actual particular event e depends causally on actual particular event c iff the family O(e), ~O(e) depends counterfactually on the family O(c), ~O(c), i.e. iff c ---> e and ~c ---> ~e.

D2a. (To accommodate "chancy causation"): Actual particular event edepends causally on actual particular even c iff c ---> (x)e and ~c --->(1- y)~e, where x >0 and x >>y.

The objection: "The original analysis of causal dependence, (D2), is better. For if there would have been some residual chance of e even without c, then the raising of probability only makes it probable that in this case c is a cause of e. Suppose, for instance, that the actual chance of e, with c, was 88%, but that without c, there would still have been a 3% probability of e. Then most likely (probability 97%) this is a case in which e would not have happened without c; then c is indeed a cause of e. But this just might be (probability 3%) a case in which e would have happened anyway; then c is not a cause of e. We can't tell for sure which kind of case this is."

Let me try to restate the objection by constructing two cases:

CASE 1: Suppose that (i) c and e both occur, and (ii) c ---> (.88)e, and (iii) ~c ---> (1)~e. Then even though c's occurrence made e only 88% probable, it is 100% probable that ~c ---> ~e is true in this case, and it is 0% probable that ~c ---> e is true in this case. So in a case like this definition 2A gives the right answer, but 2 would have served just as well.

CASE 2: Suppose that (i) c and e both occur, and (ii) c ---> (.88)e, and (iii) ~c ---> (.97)~e. Then even though c's occurrence made e only 88% probable, it is 97% probable that ~c ---> ~e is true in this case, and it is 3% probable that ~c ---> e is true in this case. Now suppose further that, as seems conceivable, this is one of the cases in which ~c ---> e is true. It follows that definition 2A gives the wrong result, since according to it c is a cause of e, whereas definition 2 gives the correct result, since according to it c is not a cause of e.

LEWIS'S REPLY TO CASE 2: Lewis denies that in case 2 ~c ---> e is true, though he also denies that in case 2 ~c ---> ~e is true, since he is taking ~c ---> ~e to mean that e would definitely not have occurred without c:

"It is granted, ex hypothesi, that it would have been a matter of chance whether e occurred. Even so, the objection presupposes that the case must be of one kind or the other: either e definitely would have occurred without c, or it definitely would not have occurred. If that were so, then indeed it would be sensible to say that we have causation only in case e definitely would not have occurred without c. My original analysis [definition 2 above] would serve, the amendment suggested in this postscript [definition 2A] would be unwise, and instead of having a plain case of probabilistic causation we would have a probable case of plain causation. But I reject the presupposition that there are two different ways the world could be, giving us one definite counterfactual or the other. That presupposition is a metaphysical burden quite out of proportion to its intuitive appeal; what is more, the intuition can be explained away. The presupposition is that there is some hidden feature which may or may not be present in our actual world, and which if present would make true the counterfactual that e would have occurred anyway without c. If this counterfactual works as others do, then the only way this hidden feature could make the counterfactual true is by carrying over to the counterfactual situation and there being part of a set of conditions jointly sufficient for e. What sort of set of conditions? We think at once that the set might consist in part of laws of nature, and in part of matters of historical fact prior to the time t, which would together predetermine e. But e cannot be predetermined in the counterfactual situation. For it is supposed to be a matter of chance, in the counterfactual situation as in actuality, whether e occurs ... So the hidden feature must be something else. But what else can it be? Not the historical facts prior to t, not the chances, not the laws of nature or the history-to-chance conditionals that say how those chances depend on prior historical facts. For all those are already taken account of, and they suffice only for a chance and not a certainty of e" (pp. 180-181 of volume II).

MY REPLY TO LEWIS'S REPLY TO CASE 2:

First of all, the "hidden factor" here is action or operation. (Later we will see that Cartwright introduces an operation functor to designate just this factor.)

Second and more important, Lewis has mischaracterized the objection, or at least he has not presented the most sympathetic version of it. However, the most sympathetic rendition of the objection raises some serious questions, as I see it, about Lewis's semantics for counterfactuals and how it is to be applied. For the ultimate claim will be that certain counterfactuals, which I will call simple counterfactuals, are indeed not like other counterfactuals.

Suppose that N is a two-sided coin which has a 88% bias in favor of heads. Consider the following four subjunctive conditionals:

(1) If N were flipped, the objective probability of its coming up heads would be .88 [N ---> (.88)H]

(2) If N were flipped, the objective probability of its not coming up heads would be .12 [N ---> (.12)not-H]

(3) If N were flipped, the objective probability of its not coming up heads would be 1 [N ---> (1)not-H]

(4) If N were flipped, it would not come up heads [N ---> not-H]

Notice that according to the standard semantics (1)-(2) present no serious problem once we solve the problem of how to incorporate probabilistic consequents into the theory. (There are various ways to do this, I assume.) The more interesting question has to do with the relation between (3) and (4). I take (3) to be equivalent to

(3*) If N were flipped, it would definitely not come up heads [N ---> (1)not-H]

This seems unproblematically false in the case in question--even to this Aristotelian diehard. And it would be counted as false by the Lewis semantics. Likewise, take

(5) If N were flipped, it would definitely come up heads [N ---> (1)H]

(5) would also be unproblematically false in the case in question and false according to the Lewis semantics--and once again this diehard Aristotelian would agree. Yet I also endorse the objection posed above to definition 2A. My conclusion is that Lewis has misinterpreted the objection by claiming that one who proposes it is insisting that in this case either H would definitely occur if N occurred or else H would definitely not occur if N occurred.

As I see it, the real problem is whether (3) and (4) express the same thing, and my own view is that they do not. To see this, suppose that before you flip the coin I bet that on this occasion the coin will not come up heads. That is, or so I say, I assert (4), even though I agree that (1) and (2) are true and that (3) is false. If the coin does not come up heads, you have to pay up. You cannot get out of your obligation by claiming that when I said "If you were to flip the coin, it would not come up heads," I was asserting a falsehood, viz., (3), and that therefore what I said was false. My reply will be: "Look, I predicted that it would not come up heads even though I knew that it was not a sure thing, as (3) asserts." There are other examples as well (see my introduction to Luis de Molina, On Divine Foreknowledge: Part IV of the "Concordia", section 5.6).

My radical conclusion is that (4) simply cannot be handled via an appeal to similarity relations among worlds as long as that semantics allows only the laws of nature and the history of the world to count as the basis for similarity among worlds. Such similarity relations are helpful only for propositions like (1)-(3), which I will call counterfactuals about probabilities. On an Aristotelian view such counterfactuals make claims about the way the world is tending given the truth of the antecedent, with the limiting case being one in which the world is tending deterministically (or definitely) in a certain direction. But this is different from making a claim, as simple counterfactuals do, about what would in fact occur given the truth of the antecedent.

Take another example, this one predicated on the claim that free choice is indeterministic, even though the behavior of free beings is heavily influenced by all sorts of causal factors. Suppose that Jones is a heavy smoker who intends to quit. Taking all the relevant factors into consideration, the following might well be true, where C is a situation in which Jones has to decide whether or not to smoke this cigarette:

(1) If Jones were in C, the objective probability of his deciding to smoke the cigarette would be .88 [C ---> (.88)S]

(2) If Jones were in C, the objective probability of his deciding not to smoke the cigarette would be .12 [C ---> (.12)not-S]

(3) If Jones were in C, the objective probability of his deciding not to smoke the cigarette would be 1 [C ---> (1)not-S]. That is, if Jones were in C, he would definitely decide not to smoke.

(4) If Jones were in C, he would decide not to smoke the cigarette [C ---> not-S]

Once again, (1) and (2) are true, let us suppose; once again (3) is false, since it is not true that Jones would definitely decide not to smoke if C obtained. Nonetheless, it might very well be that (4) is true; and if it is, then you would win if you bet beforehand that Jones would not smoke in C. But also, if libertarianism is true, then no similarity relation based solely on laws of nature and past history is able to yield the result that (4) is true, since the presupposition of libertarianism is that any such similarity relation, though perhaps relevant to the question of what the objective probabilities are, is irrelevant to the question of what Jones would in fact do in C. (4) is, in short, a simple and irreducible truth about our world. Only if such irreducible subjunctive truths themselves are partially determinative of similarity relations among worlds can the Lewis-type semantics succeed.

How does this affect Lewis's account of causation? It seems to me that 2 is better than 2A as long as we take the counterfactuals asserted in 2 to be simple counterfactuals, even if our warrant for asserting them in many cases will be judgments concerning counterfactuals about probabilities. However, 2A is superior if we take the relevant counterfactuals to be themselves counterfactuals about probabilities. But if we do so take them, then, as the objection shows, it is not clear what they have to say about causation itself, since they have to do with tendencies rather than with actions.