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 A_{1}, A_{2},
... of possible propositions, no two of which are compossible, and
let the C's be another such family C_{1}, C_{2},
... . Then

The C's depend counterfactually on the A's iff all
the counterfactuals
A_{1} > C_{1}, A_{2} > C_{2}, ... 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 > (1y)~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
A_{1}, A_{2}, ... of alternatives iff
B would be true no matter
which of the A's were true.

D6. Let the A's be a family A_{1}, A_{2}, ... of possible
propositions, no two of which are compossible, and let the C's be
another such family C_{1}, C_{2}, ... . Then

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

(a) L and F jointly imply all the material conditionals A_{1}
> C_{1}, A_{2} > C_{2}, ..., 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 econtrary
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 econtrary 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).

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)
actgeneration dependencies, (iv)
noncausal 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 possibleworld 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 Kimlike 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. 24169).

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 historytochance
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. 180181 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 twosided 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)notH]

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

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

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)notH]

This seems unproblematically false in the case in questioneven 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 semanticsand 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)notS]

(3) If Jones were in C, the objective probability of his deciding
not to smoke the cigarette would be 1 [C > (1)notS]. 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 > notS]

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 Lewistype 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.
