Lewis on Counterfactuals, Similarity, and Morgenbesser's Coin
There is a mistake on page 49 of Lewis's "Counterfactual dependence and time's arrow" (1979). Since the mistake seems to be repeated all the time, it might be worth pointing it out.
Page 49 is where Lewis lists similarity standards for his analysis of counterfactuals. The analysis, recall, says that "if A were the case, then C" is true iff the closest A-worlds are C-worlds (or, more precisely, iff either there are no A-worlds or some A&C-worlds are closer to the actual world than any A&~C world). Closeness is a matter of similarity, and Lewis indicates what the relevant respects of similarity might be for certain ordinary counterfactuals in section 3.3 of his 1973 book, and again in the 1979 article on counterfactual dependence. Roughly, the closest A-worlds are those that perfectly match the actual world across as much of spacetime as possible without diverse and widespread violations of the actual laws. This won't do for indeterministic worlds, where generally no laws need to be violated at all in order to ensure perfect match of futures even after earlier divergence. So Lewis restricts his standards to deterministic worlds, returning to the indeterministic case in the 1986 postscript to the 1979 paper.
Now consider the following situation, due to Sydney Morgenbesser.
You bet that a certain fair coin toss will result in tails; but the coin lands heads. If you had bet on heads, you would have won.
This is widely supposed to be a problem for Lewis's 1973/1979 account: by this account, the closest antecedent worlds are worlds where a small miracle makes you bet on heads instead of tails. Without a big and widespread miracle, all such worlds differ from the actual world throughout the future light cone of the betting event. Introducing further small miracles would therefore needlessly detract from similarity. Hence all those worlds evolve by the actual laws of nature. Since the coin is indeterministic, laws and history leave it open whether the coin lands heads or tails. So it is not true that at all the closest worlds where you bet on heads, you win. The intuitively true counterfactuals comes out false.
But this is all wrong. For one, the 1973/1979 account is silent on indeterministic cases. And even if we ignore the restriction to determinacy and apply Lewis's standards, we can't assume that the coin toss is the only indeterministic event in the history of the world. If Morgenbesser's situation takes place in a thoroughly indeterministic universe, the closest antecedent worlds (by the 1979 standards) are miracle-free worlds where an entirely lawful, indeterministic event makes you bet on heads, and where other indeterministic events make the future perfectly match the actual future. In all such worlds, the coin lands heads. Morgenbesser's counterfactual comes out true.
One of those who mistakenly took Morgenbesser's case to be relevant to Lewis's 1973/1979 account was Lewis himself. On page 49 of the 1979 paper, he adds a further condition to the similarity standards given above. The added condition (numbered "4") says that "approximate overall similarity" is of "little or no importance" when comparing worlds. Lewis notes that "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 and Jackson give cases which appear to come out right [...] only if approximate similarities count for nothing; but Morgenbesser has given a case [...] which appears to go the other way."
Apparently Lewis thought that if approximate similarities get some weight in Morgenbesser's example, then worlds where the coin lands heads come out closer to the actual world than worlds where it lands tails. But the patch is not only unnecessary, it also doesn't work. There are many respects in which the worlds with heads are less similar to the original world, e.g. with respect to your losing the bet. Depending on the story (think: high stakes), this dissimality can easily outweigh any approximate similarity gained by keeping fixed the outcome of the coin toss.
The upshot is twofold.
First, Lewis never had any good reason for assigning positive weight to approximate overall similarity in his standards for deterministic worlds. The only reason he gives is the irrelevant Morgenbesser case. As he points out, the addition yields wrong results in various other cases, which, unlike Morgenbesser's, do not involve indeterminacy. We should therefore just ignore the mistakenly added condition 4. Sadly, many philosophers still think that approximate similarity is somehow very important for Lewis's analysis, and that examples of the Tichy-Jackson type pose a big problem for his account.
Second, we should stop repeating Lewis's mistake to think that Morgenbesser-type cases are relevant to his 1973/1979 account. Such cases are interesting, but they should be discussed at the right place: when we're dealing with similarity standards for indeterministic worlds. In this context, the 1973/1979 standards are not meant to apply.
I'd thought the Morgenbesser intuitions could be got going without the presumption that the laws were indeterministic. Presumably there's some content to "fair coin" even without fundamentally chanciness (if we wanted to get fancy, maybe we could explicate this in terms of stat mechanical probabilities).
Suppose I'm offered the chance to bet before the coin lands (or is even tossed, if you like), and the coin actually lands heads. The intuition is just as good by my lights as in the fundamentally chancy case: If you'd've bet heads, you'd've won.
Now consider the nearest bet-worlds. Typically for Lewis these'll have perfectly match up to a short time before pre-bet (but before the coin lands). Then you get a small violation of the laws, leading to my betting. If we weight approximate similarity zero, is there any reason for thinking that all the most similar such worlds would be ones where the coin lands heads? Why can't the differences mean that the state of the world (post-miracle) leads (deterministically) to it landing tails? I think it's plausible enough that without weighting approximate similarity to some extent, we'll have some nearest bet-world where the coin lands tails, falsifying "if were bet, then heads", for Lewis). Am I missing something here?
Even given all this, I'm not terribly sympathetic to clause 4. I'm inclined to agree I can get myself into a state where the Morgenbesser counterfactual seems at least not clearly true. And I'd prefer to handle these cases by appealing to contextual restrictions on the whole space of worlds, rather than trying to fine-grain similarity.
(In fairness to them, the people who do want to fine-grain it, that I know of, don't tend to want to add a blanket clause "approximate similarity counts for a little", but rather, some selective version of it: "such-and-such kind of approx similarity counts for a little, but such-and-such kind counts for nothing". The difference between relevant and irrelevant approx similarity is typically cashed out by appeal to something like preservation of causal chains. So they hope to get a single account that fits with Tichy et al as well as Morganbesser. I don't know whether you intended to suggest otherwise, but in any case, that seems to me the position that competes hereabouts.)