Monday, February 11, 2013

From "Counterfactual Dependence and Time's Arrow"

By David Lewis (text here).

[...] Let W0 also be a world that fits our worst fantasies about the button: there is such a button, it is connected to a fully automatic command and control system, the wired-in war plan consists of one big salvo, everything is in faultless working order, there is no way for anyone to stop the attack, and so on. Then I agree that Fine's counterfactual is true at W0: if Nixon had pressed the button, there would have been a nuclear holocaust.

There are all sorts of worlds where Nixon (or rather, a counterpart of Nixon) presses the button at t. We must consider which of these differ least, under the appropriate similarity relation, from woo Some are non-starters. Those where the payload of the rockets consists entirely of confetti depart gratuitously from W0 by any reasonable standards. The more serious candidates fall into several classes.

One class is typified by the world Wl. Until shortly before t, Wl is exactly like W0. The two match perfectly in every detail of particular fact, however minute. Shortly before t, however, the spatio-temporal region of perfect match comes to an end as Wl and W0 begin to diverge. The deterministic laws of W0 are violated at Wl in some simple, localized, inconspicuous way. A tiny miracle takes place. Perhaps a few extra neurons fire in some corner of Nixon's brain. As a result of this, Nixon presses the button. With no further miracles events take their lawful course and the two worlds Wl and W0 go their separate ways. The holocaust takes place. From that point on, at least so far as the surface of this planet is concerned, the two worlds are not even approximately similar in matters of particular fact. In short, the worlds typified by Wl are the worlds that meet the conditions listed in Analysis 1, our asymmetry-by-fiat analysis. What is the case throughout these worlds is just what we think would have been the case if Nixon had pressed the button (assuming that we are at W0, and operating under the standard resolution of vagueness). Therefore the worlds typified by Wl should turn out to be more similar to W0, under the similarity relation we seek, than any of the other worlds where Nixon pressed the button.

(When I say that a miracle takes place at Wl, I mean that there is a violation of the laws of nature. But note that the violated laws are not laws of the same world where they are violated. That is impossible; whatever else a law may be, it is at least an exceptionless regularity. I am using "miracle" to express a relation between different worlds. A miracle at W1, relative to W0, is a violation at WI of the laws of W0, which are at best the almost-laws of WI. The laws of W1 itself, if such there be, do not enter into it.)

A second class of candidates is typified by W2. This is a world completely free of miracles: the deterministic laws of W0 are obeyed perfectly. However, W2 differs from Wo in that Nixon pressed the button. By definition of determinism, W2 and W0 are alike always or alike never, and they are not alike always. Therefore they are not exactly alike through any stretch of time. They differ even in the remote past. What is worse, there is no guarantee whatever that W2 can be chosen so that the differences diminish and eventually become negligible in the more and more remote past. Indeed, it is hard to imagine how two deterministic worlds anything like ours could possibly remain just a little bit different for very long. There are altogether too many opportunities for little differences to give rise to bigger differences. Certainly such worlds as W2 should not turn out to be the most similar worlds to W0 where Nixon pressed the button. That would lead to back-tracking unlimited. (And as Bennett observes in [2], it would make counterfactuals useless; we know far too little to figure out which of them are true under a resolution of vagueness that validates very much backtracking.) The lesson we learn by comparing WI and W2 is that under the similarity relation we seek, a lot of perfect match of particular fact is worth a little miracle.

A third class of candidates is typified by W3. This world begins like WI. Until shortly before t, W3 is exactly like W0. Then a tiny miracle takes place, permitting divergence. Nixon presses the button at t. But there is no holocaust, because soon after t a second tiny miracle takes place, just as simple and localized and inconspicuous as the first. The fatal signal vanishes on its way from the button to the rockets. Thereafter events at W3 take their lawful course. At least for a while, worlds W0 and W3 remain very closely similar in matters of particular fact. But they are no longer exacly alike. The holocaust has been prevented, but Nixon's deed has left its mark on the world W3. There are his fingerprints on the button. Nixon is still trembling, wondering what went wrong - or right. His gin bottle is depleted. The click of the button has been preserved on tape. Light waves that flew out the window, bearing the image of Nixon's finger on the button, are still on their way into outer space. The wire is ever so slightly warmed where the signal current passed through it. And so on, and on, and on. The differences between W3 and W0 are many and varied, although no one of them amounts to much.

I should think that the close similarity between W3 and W0 could not last. Some of the little differences would give rise to bigger differences sooner or later. Maybe Nixon's memoirs are more sanctimonious at W3 than at W0. Consequently they have a different impact on the character of a few hundred out of the millions who read them. A few of these few hundred make different decisions at crucial moments of their lives, and we're off! But if you are not convinced that the differences need increase, no matter. My case will not depend on that.

If Analysis 2 is to succeed, such worlds as W3 must not turn out to be the most similar worlds to W0 where Nixon pressed the button. The lesson we learn by comparing W1 and W3 is that under the similarity relation we seek, close but approximate match of particular fact (especially if it is temporary) is not worth even a little miracle. Taking that and the previous lesson of W2 together, we learn that perfect match of particular fact counts for much more than imperfect match, even if the imperfect match is good enough to give us similarity in respects that matter very much to us. I do not claim that this pre-eminence of perfect match is intuitively obvious. I do not claim that it is a feature of the similarity relations most likely to guide our explicit judgments. It is not; else the objection we are considering never would have been put forward. (See also the opinion survey reported by Bennett in [2].) But the preeminence of perfect match is a feature of some relations of overall similarity, and it must be a feature of any similarity relation that will meet our present needs.

A fourth class of candidates is typified by W4. This world begins like W1 and W3. There is perfect match with W0 until shortly before t, there is a tiny divergence miracle, the button is pressed. But there is a widespread and complicated and diverse second miracle after t. It not only prevents the holocaust but also removes all traces of Nixon's button-pressing.

The cover-up job is miraculously perfect. Of course the fatal signal vanishes, just as at W3, but there is much more. The fingerprint vanishes, and the sweat returns to Nixon's fingertip. Nixon's nerves are soothed, his memories are falsified, and so he feels no need of the extra martini. The click on the tape is replaced by innocent noises. The receding light waves cease to bear their incriminating images. The wire cools down, and not by heating its surroundings in the ordinary way. And so on, and on, and on. Not only are there no traces that any human detective could read; in every detail of particular fact, however minute, it is just as if the button-pressing had never been. The worlds W4 and W0 reconverge. They are exactly alike again soon after t, and exactly alike forevermore. All it takes is enough of a reconvergence miracle: one involving enough different sorts of violations of the laws of W0, in enough different places. Because there are many different sorts of traces to be removed, and because the traces spread out rapidly, the cover-up job divides into very many parts. Each part requires a miracle at least on a par with the small miracle required to prevent the holocaust, or the one required to get the button pressed in the first place. Different sorts of unlawful processes are needed to remove different sorts of traces: the miraculous vanishing of a pulse of current in a wire is not like the miraculous rearrangement of magnetized grains on a recording tape.

The big miracle required for perfect reconvergence consists of a multitude of little miracles, spread out and diverse. Such worlds as W4 had better not turn out to be the most similar worlds to Wo where Nixon pressed the button. The lesson we learn by comparing W1 and W4 is that under the similarity relation we seek, perfect match of particular fact even through the entire future is not worth a big, widespread, diverse miracle. Taking that and the lesson of W2 together, we learn that avoidance of big miracles counts for much more than avoidance of little miracles. Miracles are not all equal. The all-or-nothing distinction between worlds that do and that do not ever violate the laws of W0 is not sensitive enough to meet our needs.

This completes our survey of the leading candidates. There are other candidates, but they teach us nothing new. There are some worlds where approximate reconvergence to W0 is secured by a second small miracle before t, rather than afterward as at W3: Haig has seen fit to disconnect the button. Likewise there are worlds where a diverse and widespread miracle to permit perfect reconvergence takes place mostly before and during t: Nixon's fingers leave no prints, the tape recorder malfunctions, and so on.

Under the similarity relation we seek W1 must count as closer to W0 than any of W2, W3, and W4. That means that a similarity relation that combines with Analysis 2 to give the correct truth conditions for counterfactuals such as the one we have considered, taken under the standard resolution of vagueness, must be governed by the following system of weights or priorities.

(1) It is of the first importance to avoid big, widespread, diverse violations of law.
(2) It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails.
(3) It is of the third importance to avoid even small, localized, simple violations of law.
(4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.

(It is a good question whether approximate similarities of particular fact should have little weight or none. Different cases come out differently, and I would like to know why. Tichy ([20]) and Jackson ([7]) give cases which appear to come out right under Analysis 2 only if approximate similarities count for nothing; but Morgenbesser has given a case, reported in Slote ([ 19]), which appears to go the other way. This problem was first brought my attention by Ernest Loevinsohn.)

Plenty of unresolved vagueness remains, of course, even after we have distinguished the four sorts of respect of comparison and ranked them in decreasing order of importance. But enough has been said to answer Fine's objection; and I think other versions of the future similarity objection may be answered in the same way.


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