Lewis Counterfactual Theory of Causation Summary An account only of deterministic, token causation. Causation = counterfactual dependence + right semantics for counterfactuals + transitivity E depends counterfactually on C iff the counterfactuals C E, ~C ~E are both true. Whether or not E occurs depends on whether or not C occurs. [Counterfactual Influence Theory: whether, when or how E happens depends counterfactually on whether, when or how C happens.] 2. Counterfactual semantics. A B is true at w if (roughly) at all closest A-worlds, B is true. Closest A-worlds to w (assuming A is false in w) are the worlds w with exact match of particular fact up to the very recent past, a small miracle (= violation of the laws in w) in w to bring about A, and agreement with the laws of w from that point on. 3. Transitivity. A is a cause of B if either i) B counterfactually depends on A, or ii) there is a causal chain of counterfactual dependence linking A and B.
I. Causation 1. Regularity theories Core idea: Causation = regular succession Neo-Humean accounts: - nomic regularity (causal laws rather than accidental regularities) - causal factors (rather than total causes) [+ causal chains (direct and indirect causes)] Lewis version: c causes e if and only if: (1) c and e both occur (2) Laws and particular facts together (but not separately) entail that if c occurs then e occurs, but do not entail that c occurs. [Leaves out temporal priority] Three objections: 1) Fails to distinguish causes from effects (regular connection goes both ways) [Temporal priority? Objects to building it in.] 2) Fails to distinguish causes from epiphenomena (two effects of a common cause might satisfy the definition) [Horwich s causal chains may give an answer] 3) Fails to exclude preempted causes (if c then e is still entailed because the particular facts include the presence of the preempting cause c ) [In the end, this causes more trouble for Lewis than for regularity theories]
2. Preliminary ideas Core idea: Hume s second definition. If c had not occurred, e would not have occurred. Initial objection: no clear truth-conditions for counterfactuals, so don t use them to get clear on causation (or anything else)! Reply: The truth-conditions are somewhat vague, but: i) The standards for clarity are too high (causation itself is vague); ii) There are clear semantics for counterfactuals (Lewis-Stalnaker); iii) The overall similarity relation can be interpreted with sensitivity to deal with standard objections to the counterfactual account; and iv) The counterfactual theory is able to handle the three main deficiencies identified for regularity theories.
3. Counterfactual Semantics, part 1: Similarity and Possible Worlds Primitives: A set of possible worlds, W A relation of comparative overall similarity (or inter-world distance): d(w 1, w) < d(w 2, w) if w 1 is closer to w than is w 2. The relation d must satisfy two constraints: a) Weak ordering. Any two worlds are comparable in closeness to w, but they may be equally close. b) Centring. Any w is closest to itself: d(w, w) < d(w, w) if w w. In particular, the actual world w 0 is the unique closest world to itself. Remarks on overall similarity: i) The term overall similarity is misleading, since the choice of d is contextual. Different choices of d are appropriate in different contexts. ii) Assessment of similarity requires a balancing of similarities (or exact matches) of laws and similarities (or exact matches) of particular fact. iii) Similarity of law is not automatically top priority. <We often consider counterfactuals that involve some violation of law, especially if we are assuming determinism and counterfactually alter the actual course of recent events.> iv) Comprehensive and exact similarity of particular fact through large regions of space and time has great weight. [Question: Does Lewis have independent arguments for iii) and iv)?]
4. Counterfactual semantics, part 2: truth-conditions for counterfactuals i) Semantics of modal logic. Truth at worlds: Val(A, w) = T iff A is true at possible world w. Parlour game analogy. More complex clauses for the usual operators. An A-world is a possible world in which A is true. Proposition = a set of possible worlds (intuitively: those where the proposition holds) Clauses for the usual modal operators and - the accessibility relation R. - necessity ( ) and possibility ( ) To these basic modal notions, Lewis adds the similarity relation d. ii) Counterfactuals For A C, read If A were true, then C would be true. A C is true at w iff there are no possible A-worlds, or some A-world where C is true is closer to w than any A-world where C is false. Special cases: i) If there are closest A-worlds, then condition is that C is true at all closest A-worlds. ii) If w is an A-world, then condition is that C is true at w. <So the counterfactual implies the material conditional A C.> iii) Counterfactual dependence. (1) General case. Given mutually exclusive propositions A 1, A 2, and C 1, C 2,, we say the C s depend counterfactually on the A s if each of the counterfactuals A 1 C 1, A 2 C 2, is true. (2) Special case. E depends counterfactually on C if each of C E, ~C ~E is true. <Examples>
5. Causal dependence i) Initial thesis: For token causation, causal dependence of events just is counterfactual dependence among the corresponding propositions. Given event e the corresponding proposition O(e) is just that e occurs O(e) = set of possible worlds where e occurs ii) Causal dependence. (1) General case. Given exclusive events c 1, c 2, and e 1, e 2,, we say the e s depend causally on the c s if each of the counterfactuals O(c 1 ) O(e 1 ), O(c 2 ) O(e 2 ), is true. (2) Special case. e depends counterfactually on c if each of O(c) O(e), ~O(c) ~O(e) is true. If c and e are actual events, then the special case reduces to the second condition. If c and e do not actually occur, it reduces to the first condition. [That one of the counterfactuals becomes trivial is one difference from Mackie.] iii) Causation. c is a cause of e iff: (a) e depends causally on c <direct cause>; or (b) there is a causal chain of particular events, each a <direct cause> of the next, linking c to e.
Critique of Lewis definition 1) Too broad (Kim). Ex: If yesterday had not been Tuesday, today would not be Wednesday. <Analytical dependence, not causal dependence.> Ex: If I had not written r twice in succession, I would not have written Larry. <Dependence as constituent part, not causal dependence.> Ex: If I had not turned the knob, I would not have opened the window. <Turning is part of the action of opening the window, not a cause of that action.> Ex: If my sister had not given birth at t, I would not have become an uncle at t. <E is determined by C, but not caused by it.> 2) Too narrow (Kim, Horwich). Overdetermination cases. (e.g., firing squad) Lewis account yields a definite negative, but in fact either the shots are causes or it should be unclear whether they are causes.
6. Comparison between counterfactual and nomic dependence Nomic dependence (which plays a role in most updated regularity theories). Comparison: General case. Given exclusive propositions A 1, A 2, and C 1, C 2,, we say the C s depend nomically on the A s if each of the material conditionals A 1 C 1, A 2 C 2, is implied jointly (but not separately) by some nonempty set L of true law statements and F of true singular propositions. (1) Counterfactual dependence of the C s on the A s is implied in the case where the C s depend nomically on the A s in virtue of L and F, and the members of L and F are counterfactually independent of the A s. [Proof: In the nearest A 1 -worlds, all members of L and F hold by independence; so A 1 C 1 and hence C 1 in these worlds.] But without this independence, the implication fails. [Kim: what role is played by laws in this argument?] [Puzzling: what about epiphenomena? Let A falling barometer, C coming storm. Laws plus particular facts imply A and C and hence A C. The laws and particular facts (about air pressure at t) are surely counterfactually independent of A and C. So we seem to get counterfactual dependence of C on A! Absent backtracking counterfactuals, how could particular facts about the air pressure at t depend counterfactually on A?]
(2) Nomic dependence is reversible: If C s depend nomically on the A s in virtue of L and F, then A 1, A 2, depend nomically on AC 1, AC 2, in virtue of L and F, where A = A 1 A 2. [Proof: L, F, and (A 1 A 2 ) C 1 must imply C 1 A 1. For one of the A i s holds, and if i 1 then some C i will hold for i 1, contradicting the exclusiveness of C 1, C 2, ] But counterfactual dependence is irreversible in general (example, 168). Example: If the barometer reading had been higher, we don t say the pressure would have been higher, but that the barometer would have been malfunctioning. [This is explained better in the other paper: fixity of the past.] (3) Nomic dependence is almost (but not quite) sufficient for causal dependence (169).
7. The counterfactual analysis solves the three problems with regularity accounts A. Effects and epiphenomena Problem of effects: suppose c causes e and, given laws and particular conditions, c could not have failed to cause e. Then (assuming e does not also cause c) it appears that if e had not occurred, c would not have occurred, and the reverse counterfactual dependence (hence reverse causal dependence) is established. Problem of epiphenomena: suppose c is a common cause of e and f (where e temporally precedes f but is not in any way a cause of f). And given the laws and particular facts, c could not have failed to cause e, and f could not have been caused except by c. Then it appears that if e had not occurred, c would not have occurred and thus f would not have occurred, and spurious counterfactual dependence (hence causal dependence) between e and f is established. Common Solution: The relevant counterfactuals, ~O(e) ~O(c) and ~O(e) ~O(f), are in fact false. If e had not occurred, c would still have occurred but just failed to cause e. Fixity of the past: less of a departure to give up some of the laws and circumstances that made it so c could not have failed to cause e than to keep the laws and circumstances and change the past. [2 nd article helps here] To get rid of e, best not to diverge from actual course of events until just before e occurs. Prolong perfect match : in a deterministic universe, any divergence will require some violation of laws. No reason to diverge any earlier. [Qu: What pre-theoretical rationale, as Horwich asks (211)?]
B. Preemption c 1 causes e (the preempting cause), but c 2 occurs and would have caused e if c 1 were absent. So e does not depend causally (or counterfactually) on c 1 or c 2. We have a problem! Solution. Thanks to c 1, there is no causal chain between c 2 and e. But there is a causal chain c 1 - d - e (simplest case). Then e depends causally on d and d depends causally on c 1. Objection: If d had not happened then c 1 would not have happened and hence c 2 would have kicked in and brought about e. So e is not causally dependent on d. Questions: Solution: Says Lewis: the true counterfactual is that if d had not happened then c 1 would still have happened, yet failed to bring about d. But the causal chain from c 2 would still have been interrupted and so e would not have happened; hence, e does depend causally on d. Example: The desert traveler. e = death; c 1 = hole; c 2 = poison. d = the flask after the water is gone but a short time before the traveler picks it up for a drink. Here, d depends causally on c 1 and e depends causally on d. In the closest world where not d, the past right up to d is kept the same (so the hole was made and the water leaked out) we have to posit some small miracle just before d, like hydrolysis that fills the canteen with fresh water. Why would anyone think this is a closer world than one in which (e.g.) the hole was gummed up and the poisoned water did not drain (so not identical for a stretch of time, granted )? How can anybody settle such intuitions? And in fact, are not preemption cases better handled by the Humean account (e.g., by Mackie or Horwich) than by Lewis?
Some counterexamples: 1. Fertilization. Sperm s fertilizes an egg, which instantaneously seals off the egg and prevents other sperm from penetrating. Suppose that if sperm s had not fertilized the egg, some other sperm t would have done so. Here, we have last-minute or late preemption of the causal chain from sperm t. The effect, fertilization, would have been produced by t if not by s. There is no intermediate event d that occurs after the chain from t is interrupted, but before the effect e: it is the effect itself that ruptures the alternate causal chain. 2. Picking up baby. Baby cries; c is my response; e is the event of the baby s being picked up. But if I had not picked him up, my wife would have done so. Here again, it is only the effect that stops the alternate causal chain. But further: my action is itself caused by the baby s cry. So there is a second causal process (crying) that brings about my action, and would bring about my wife s action if I did not respond, and it is only stopped by the effect e. Here too, it is impossible to find any intermediate d such that e is counterfactually dependent on d. Horwich has a third type of counterexample where C causes E directly, with no intermediary. 3. Rolling balls. Lewis discusses trumping, a fourth type of preemption counterexample, in 2000 paper. 4. The sergeant and the major. Major s order trumps, but both shout Advance! at the same time.
III. Lewis: Counterfactual dependence and Time s Arrow Main objectives: A defense of sorts for the way in which similarity of worlds is evaluated An account of causal asymmetry The paper does not help with late preemption cases, but does handle some objections. 1. Asymmetry of counterfactual dependence - temporal priority of causes is not a constitutive part of causation - consequence of his counterfactual account plus contingent facts about the world - theory counts as a substantive account (on Horwich s classification). Counterfactual Assymetry Thesis: counterfactually on later events. Earlier events do not (ordinarily) depend Linguistic practices argument. When reasoning from counterfactual suppositions, we freely make use of knowledge about the past, but not always the present and future. This is to assume counterfactual independence of past from present. [What does this really show?] Back-tracking counterfactuals. Standard counterfactuals: Maintain counterfactual independence of prior events. Back-tracking counterfactuals: Allow counterfactual dependence of prior events. Past causes would be modified because of a counterfactual supposition about the present. - true under non-standard resolution of vagueness, but false under standard resolution - marked by complex syntax: it would have to have been that Ex: If Jim asked Jack for help, there would have to have been no quarrel yesterday. Ex: If Kennedy had not been assassinated, Oswald would have to have aimed poorly or not fired at all. For standard resolution: if time of C precedes time of A, A C is true iff C is true. [Qu: Is standard resolution defined so as to make this a trivial claim? cf. Analysis 1]
2. Other asymmetries a) Asymmetry of causation Causes precede their effects, and not the reverse. On Lewis account, explained by asymmetry of counterfactual dependence. Reverse causation remains possible, but requires back-tracking counterfactuals. [Note: simultaneous causation is not ruled out.] b) Asymmetry of openness The future is open; the past is fixed. Not merely epistemic Not a case of multiple actuality of possible futures: also have multiple pasts. Not accounted for by indeterminism (multiple potential futures): - asymmetry between future and past is more basic and certain than indeterminism - indeterminism goes in both directions (past and future) Determinism: Two worlds with the same (deterministic) laws will either be exactly alike throughout history, or never exactly alike through any stretch of time. Not about our ability to change the future but not the past. - the future is as immutable as the past Correct explanation: counterfactual dependence. The future depends counterfactually on the present; the past does not. Different possible futures are brought about on different counterfactual assumptions, but the past remains the same.
3. Counterfactuals: source of the asymmetry Analysis 1: fixity of the past guaranteed by brute force. A C true in the actual world, where A occurs in stretch of time t A, if C is true at all closest A-worlds that: i) are exactly like the actual world until shortly before A occurs (before t A ) ii) conform to the actual laws of nature except possibly for the temporal blip t A around A - guarantees temporal asymmetry of counterfactual dependence, except for the immediate past (the temporal blip). Objections: 1) Generality: Unable to handle counterfactuals whose assumptions are not about a particular time. [Weak objection: he s after a theory of singular causation.] 2) Rigidity. Makes it impossible a priori to have an exception to the asymmetry. But (e.g.) time travel is not impossible a priori. Analysis 2: Similarity of worlds. (Standard counterfactual semantics) A C is true in the actual world iff C is true in some A-world that is closer than any A-world where C is false. Remarks: 1) Skeletal account: supplement with an account of the appropriate similarity relation, which varies with context. 2) We want a similarity relation that prescribes standard resolution in most contexts, ruling out backtracking counterfactuals. 3) Overall similarity is a matter of balance, but not just every similarity has positive weight. (Ex: grueness) 4) Maxim: don t fix similarity relation in advance. Use knowledge of counterfactuals to try to find out the appropriate similarity relation. [Qu: Given 2) and 4), how is this different from the brute force approach of Analysis 1? Only in that our knowledge of which counterfactuals are true is contingent.]
4. Future similarity objection Putative counterexample: N: If Nixon had pressed the button at t, there would have been a nuclear holocaust shortly thereafter. N is true, but must be false on the Lewis semantics. Closest button-worlds are those most like our world, and hence (assuming we have no nuclear holocaust) ones in which there is no nuclear holocaust. Response: presence of a holocaust counts heavily towards *some* similarity relation, but not towards the relevant one for assessing the counterfactual claim. N is true on Lewis semantics in actual world w 0 or close approximation, but only in virtue of a careful survey and assessment of similarity of types of worlds to w 0. Type w 1 : Small miracle. Exactly like w 0 until a small miracle moments before t leads Nixon to press the button; thereafter, vastly different. Type w 2 : No miracle. - small miracle = violation of the laws of w 0 - w 1 is Lewis candidate for most similar button-world - Same laws of nature, but (by determinism) vastly different sequence of events. - w 2 is not the most similar button-world to w 0 : makes counterfactuals useless, since it permits unlimited back-tracking. Type w 3 : Second small miracle plus resemblance. - begins like w 1, but second tiny miracle prevents holocaust and restores close approximation (but not exact match) of w 0. - Lewis claims that w 3 is not closer to w 0 than is w 1 : close similarity will not last, and not worth much (cf. exact matching). - In any case, this is the similarity relation he needs. Type w 4 : Big miracle plus exact match. - begins like w 3, except all traces of button-pressing are erased and w 4 reconverges with w 0 - big miracle = many small violations of the laws of w 0 - Again, not the most similar button-world to w 0
Upshot: w 1 must count as closer than any of the others considered. So in deciding on a similarity relation to get the correct semantics for counterfactuals: 1) First, avoid widespread violation of law (even at cost of divergence of particular fact). 2) Second, maximize the period of exact match of particular fact. 3) Third priority, avoid small violations of laws. 4) Low priority: achieving approximate similarity of particular fact. Objection (Horwich): What pre-theoretical rationale? Why not use a different notion of similarity?
5. Origin of the asymmetry of counterfactual dependence The asymmetry: if Nixon had acted differently, the future would have been different, but the past exactly the same. Not built into Analysis 2. Explanation: asymmetry in the range of candidates. Consider w 5 : vastly different past, small miracle, convergence to w 0. - mirror image of w 1, so would be equally similar - but no such possible world, for convergence requires a big miracle Note: this fails in very simple worlds. [Qu: w 5 would be a world with a vastly different past prior to, say, 1972, but exactly matching ours after 1972. So Lewis talk about erasing traces, etc. is a bit misleading. It is very hard to imagine such a world, but is it impossible? Why could not things be set up just so the button yields convergence? Perhaps the reason is that so many other causal processes are going in the other direction.] 6. Asymmetry of overdetermination Past events are extremely overdetermined by later ones (lots of traces left), but overdetermination of future events by past ones is rare. Miracles break the links between any determinant and what is determined. - for w 1, only need to break the link to past conditions that determine that Nixon not press the button - for w 5, have to break links to all future conditions that overdetermine that he not press it. The existence of this type of asymmetry is a contingent feature of our world. Objection (Horwich): Does not Lewis theory implausibly commit us to backward causation in a case where we deliberately create overdetermination from past to future e.g., several particles simultaneously reaching one spot?
III. Causation as Influence Preemption: i) Early cutting. Desert traveler. Handled by causal chains. ii) Late cutting. Throwing rocks at a bottle; fertilization. Handled by quasidependence. iii) Trumping. Sergeant and the major. Handled by causation as influence. Causation as influence is a version of the fragile events approach. Definition: C causes E directly if whether, how or when E occurs depends counterfactually on whether, how or when C occurs. Sergeant-major case: if we vary the sergeant s order, there is no change in E. If we vary the major s order, there is. Objection: Then just about every event in the past light-cone of E counts as a cause (e.g., because of the universal law of gravitation). Reply: A pragmatic affair. In most cases, we ignore negligible causes; in some contexts, it s legitimate to include them as causes. [Qu: Far from satisfactory we have a qualitative, not a quantitative, theory of causes.] Generalization: C causes E if C causes E directly, or there is a chain of direct causes and effects linking C and E.