One upshot of taking the line on the scattered match case I discussed below is the following: if @ is deterministic, then legal worlds (aside from @) are really far away, on grounds of utterly flunking the “pefect match” criterion utterly. If perfect match, as I suggested, means “perfect match over a temporal segment of the world”, then legal worlds just never score on this grounds at all.
Here’s one implication of this. Take a probability distribution compatible with determinism—like the chances of statistical mechanics. I’m thinking of this as a measure over some kind of configuration space—the space of nomically poossible worlds. So subsets of this space correspond to propositions that (if we choose them right) have high probability, given the macro-state of the world at the present time. And we can equally consider the conditional probability of those on x pushing the nuclear button. For many choices of P which have high probability conditionally on button-pressing, “button-pressing>~P” will be true. The closest worlds where the button-pressing happens are going to be law-breaking worlds, not legal worlds. So any proposition only true at legal worlds will not obtain, given the counterfactual. But sets of such worlds can of course get high conditional probability.
There’s an analogue of this result that connects to recent work on safety by Hawthorne and Lasonen-Aarnio. First, presume that the safety set at w,t (roughly set of worlds such we musn’t believe falsely that p, if we are to have knowledge that p) is a similarity sphere in Lewis’s sense. That is: any world counterfactually as close as a world in the set must be in the set. If any legal world is in the set, all worlds with at least some perfect match will also be in that set, by the conditions for closeness previously mentioned. But that would be crazy—e.g. there are worlds where I falsely believe that I’m sitting in front of my computer, on the same base as I do now, which have *some* perfect match with actuality in the far distant past (we can set up mad scientists etc to achieve this with only a small departure from actuality a few hundred years ago). So if the safety set is a similarity sphere, and the perfect match constraint is taken as I urged, then there better not be any legal worlds in the safety set.
What this means is that a fairly plausible principle has to go: that if, at w and t, P is high probability, then there must be at least one P-world in the safety set at w and t. For as noted earlier, law-entailing propositions can be high-probability. But massive scepticisim results if they’re included in the safety set. (I should note that Hawthorne and Lasonen don’t endorse this principle, but only the analogous one where the “probabilities” are fundamental objective chances in an indeterministic world—but it’s hard to see what could motivate acceptance of that and non-acceptance of the above).
What to give up? Lewis’s lawbreaking account of closeness? The safety set as a similarity sphere? The probability-safety connection? The safety constraint on knowledge? Or some kind of reformulation of one of the above to make them all play nicely together. I’m presently undecided….
If you read ‘maximize exact match’ as ‘maximize the largest continuous region of exact match’ then this worry doesn’t seem to be as pressing?
But if I had to give something up, it would be the idea that perfect match counts for more than approximate match, along the lines suggested by Kment in the paper I linked to in a comment on your previous post. If (perfect or approximate) match with the same explanation counts more than avoidance of even small miracles, then legal worlds can score as well or nearly as well on this point as illegal worlds, and the argument doesn’t go through.
I think there’s still difficulties, but they’re tricky to state.
On my temporal segment proposal, there are *no* legal worlds with any degree of exact match (other than duplicates of @). And provably so, given the determinism of @. On the “big region” proposal, we might have *some* legal worlds that have more exact match than the problematic counterfactual worlds where I get false beliefs on the same basis. If those are the only ones forced to be in the safety set, we’re fine.
But so long as such worlds overall have *low probability* the problem still arises. For then the negation has high probability. So there must be a world with lower degree of exact match than typical counterfactual worlds within the safety set. And so all the typical counterfactual worlds are forced to be in the set too.