# Barker on worlds-semantics for counterfactuals

In a forthcoming paper in Nous, Stephen Barker argues that embedding phenomena mean we should give up the ambition to give an account of the truth-conditions of counterfactuals in terms of possible worlds. Barker identifies an interesting puzzle: but it is obscured by the overly strong claims he makes for it.

Let’s suppose that we have a shuttered window, and Jimmy with a bunch of stones, which (wisely) he leaves unthrown). The counterfactual “Had Jimmy thrown the stone at the window, it would have hit the shutters” is true. But it isn’t necessarily true. For example, if the shutters had been opened a few seconds ago, then had Jimmy thrown the stone, it would have sailed through. Let’s write the first counterfactual as (THROW>SHUTTERS), and the claim we’ve just made as (OPEN>(THROW>SAILEDTHROUGH)). This is a true counterfactual with a counterfactual as consequent, capturing one way in which the latter is contingent on circumstance.

Here is Barker’s puzzle, as it impacts David Lewis’s account of counterfactuals. On Lewis’s account (roughly) a counterfactual is true iff all the closest worlds making the antecedent true also make the conclusion true. Therefore for the double counterfactual above to be true, (THROW>SAILEDTHROUGH) must be true at all the closest OPEN-worlds.

But Lewis also told us something about what makes for closeness of worlds. Cutting a long story short, for the “forward tracking” cases of counterfactuals, we can expect the closest OPEN worlds to exactly match the actual world up to a few seconds ago, whereupon some localized event occurs which is inconsistent with the laws of the actual world, which (deterministically) leads to OPEN obtaining. After this “small miracle”, the evolution of the world continues in accordance. Let’s pick a representative such world, W. (For later reference, note that both THROW and SAILEDTHROUGH are false in W—it’s just a world where the shutters are opened; but Jimmy’s stone doesn’t move).

Barker asks a very good question: is THROW>SAILEDTHROUGH true at W, on Lewis’s view? For this to be so, we need to look at the THROW world nearest to W. What would such a world be? Well, as before, we look for a world that exactly matches W up until shortly before THROW is supposed to occur—which then diverges from W by a small miracle (by the laws of nature in W) in such a way as to bring about THROW, and which from then on evolves in accordance with W’s laws.

But what are W’s laws? Not the actual laws—W violates those. Maybe it has no laws? But then all sorts of crazy evolutions will be treated as “consistent with the laws of W”, and so we’d have no right to assume that in all such evolutions SAILEDTHROUGH would be true. Maybe it has all of W’s laws except the specific one that we needed to violate? But still, if e.g. the actual law tying gravitational force to masses and square separation is violated at W, then removing this from the books allows in all sorts of crazy trajectories correlated with tailored gravitational forces—and again we have no right to think that on all legal evolutions SAILEDTHROUGH will be true. Net result, suggests Barker: we should assume absent further argument that THROW>SAILEDTHROUGH is false at W; and hence OPEN>(THROW>SAILEDTHROUGH) is false. The same recipe can be used to argue for the falsity of all sorts of intuitively true doubly-embedded counterfactuals.

I’ve presented Lewis’s theory very loosely. But the problem survives translation into his more precise framework. The underlying trouble that Barker has identified is this: Lewis’s account of closeness relies on two aspects of similarity to the relevant “base” world: matching the distribution of properties in the base world, and fitting the laws of the base world. But because the “counterfactually selected” worlds violate actual laws, there’s a real question-mark over whether the second respect of similarity has any teeth, when taken with respect to the counterfactually selected world, instead of actuality

This is a nice puzzle for one very specific theory. But does it really show that the worlds approach is doomed? I think that’s very far from the case. Note first that giving a closest-worlds semantics is separable from giving an analysis of closeness in terms of similarity; let alone the kind of analysis that Lewis favoured. So Barker’s problem simply won’t arise for many worlds-theorists. So, I think, Barker’s point is best presented as a problem for one who buys the whole Lewisian package. My second observation is that Lewis himself has the resources to avoid Barker’s case, thanks to his Humean theory of laws. And if we’ve gone so far as to buy the whole Lewisian package on counterfactuals, it won’t be surprising if we’ve gotten ourselves committed to some of the other Lewisian views.

The simple observation that lies behind the first point is that the bare closeness-semantics for counterfactuals does not get us anywhere near talk of miracles, violations of law and the rest. Indeed, to generate the logic of counterfactuals (and thus get predictions about the coherence of various combinations of beliefs with counterfactual content) we do not even need to appeal to the notion of an “intended interpretation” of closeness—we could treat it purely algebraically. If we do believe that one interpretation gives the actual truth conditions of counterfactuals, there’s nothing in principle to stop us treating closeness as primitive (Stalnaker, for example, seems to adopt this methodological stance) or even to give an explicit definition of closeness in counterfactual terms. Insofar as you wanted a reduction of counterfactuality to something else, this’d be disappointing. But whether such reduction is even possible is contentious; and even if it is, it’s not clear that we should expect to read it off our semantics.

So the algebraists, the primitivists, the reverse analyzers can all buy into worlds-semantics for counterfactuals without endorsing anything so controversial as Lewis’s talk of miracles. Likewise, it’s really not clear that anyone going in for a reduction of closeness to something else needs to follow Lewis. Lewis’s project is constrained by all sorts of extrinsic factors. For example, it’s designed to avoid appeal to de jure temporal asymmetries; it’s designed to avoid mention of causation, and so forth and so on. Connectedly, the laws Lewis considers are micro-laws, allowing him to focus on the case of determinism as the paradigm. But what about the (indeterministic) laws of statistical mechanics? If fit with the probabilistic laws of statistical mechanics as well as fit with determinsitic laws play a role in determining closeness, the game changes quite markedly. So there are all sorts of resources for the friend of illuminating reductions of closeness–it’d be a positive surprise if they ended up using only the same sparse resources Lewis felt forced to.

Can we get a version of the dilemma up and running even without Lewis’s particular views? Well, here’s a general thought. Supposing that the actual micro-laws are deterministic, every non-duplicate of actuality is either a universal-past-changer (compared to actuality) or is a law-breaker (with respect to actuality). For if the laws are deterministic, if worlds W and @ match on any temporal segment, they’ll match simpliciter. Now consider the “closest OPEN worlds to actuality” as above. We can see a priori that either this set contains past changers, or law breakers (or possibly both). Lewis, of course, set things up so the latter possibility is realized, and this is what led to Barker’s worries. But if universal-past-changers can’t be among the closest OPEN-worlds, then we’ll be forced to some sort of law-breaking conception.

What’s wrong with past-changing, then? Well, there are some famous puzzle cases if the past is wildly different from the present. Bennett’s “logical links” worry, for example, concerns counterfactuals like the following: “If I’d’ve been fitter, I’d have made it to the top of the hill where the romans built their fort many years ago”. Here, the intuitively true counterfactual involves a consequent containing a definite description, and the description singles out an individual via a past-property. If we couldn’t presuppose that in the counterfactually nearest world, individuals now around retained their actual past properties, these kinds of counterfactuals would be dodgy. But they’re totally smooth. And it’s pretty easy to see that the pattern generalizes. Given a true counterfactual “If p, then a is G”, and some truth about the past Q, we construct a definite description for a that alludes to Q (e.g. “the hill which is located a mile from where, 1000 years ago, two atoms collided on such-and-such trajectories”). The logical links pattern seems to me the strongest case we have against past-changing. If so, then independently of a Lewisian analysis of closeness, we can see that the closest worlds will be lawbreakers not past-changers.

Two things to note here. Universal-past-changers need not be Macro-past-changers. For all we’ve said, there past-changing worlds in the closest OPEN-set might coincide with actuality when it comes to the distribution of thermodynamic properties over small regions of spacetime (varying only on the exact locations etc of the microparticles). For all we’ve said, there may be worlds with same macro-past as actuality in the closest set that are entirely legal. If we have to give up logical-links style cases, but only where the descriptions involve microphysics, then that’s a surprise but perhaps not too big a cost—ordinary counterfactuals (including those with logical links to the macro-past) would come out ok. I’m not sure I’d like to assert that there are such past-changing macro-past duplicators around to play this role; but I don’t see that philosophers are in a position to tell from the armchair that there aren’t any—which is bad news for an a prioristic argument that the worlds-semantics is hopeless. Second, even if we could establish that lawbreaking is the only way to go, that doesn’t yet give enough to give rise to Barker’s argument. Once we have that W (our representative closest OPEN world) is a lawbreaker, the argument proceeds by drawing out the consequences of this for the closest THROW worlds to W. And that step of the argument just can’t be reconstructed, as far as I can see, without appeal to Lewis’s analysis itself. The primitivists, reverse-analysists, alternative-analysists and the like are all still in the game at this point.

So I think that Barker’s argument really can’t plausibly be seen as targetting the worlds-semantics as such. Indeed, I can’t see that it has any dialectical force against Stalnaker, who to say the least looms large in this literature.

But what if one insisted that the target of the criticism is not these unspecified clouds of worlds-theorists, but Lewis himself, or at least those who (perhaps unreflectively) go along with Lewis? To narrow the focus in this way would mean we have to cut out much of the methodological morals that Barker attempts to draw from his case. But it’s still an interesting conclusion—after all, many people do hand wave towards the Lewisian account, and if it’s in trouble, perhaps philosophers will awaken from their dogmatic slumbers and realize the true virtues of the pragmatic metalinguistic account, or whatever else Barker wishes to sell us.
I think this really is where we should be having the debate; the case does seem particularly difficult for Lewis. What I want to argue, however, is that the fallback options Barker argues against, though they may convince others, have little force against Lewis and the Lewisians.

The first fallback option that Barker considers is minimal law-modification. This is where we postulate that W has laws that look very similar to those of the actual world—except they allow for a one-off, specified exception. Suppose that the only violation of @’s laws in W occurs localized in a tiny region R. Then if some universal generalization (for all x,y,z, P) is a law of @, the corresponding W-law will be: (for all x,y,z that aren’t in region R, P). If we want to get more specific, we could add a conjunct saying what *should* happen in region R.

Barker rightly reminds us of Lewis’s Humeanism about laws—what it is to be a law in a world is to be a theorem of the best (optimally simple/informative) system of truths in that world. In the case at hand, it seems that the sort of hedged generalizations just given will be part of such a best system—what’s the alternative? The original non-hedged generalization isn’t even true, so on the official account isn’t in the running to be a law. the hedged generalization is admittedly a little less simple than one might like—though not by much—and leaving it out gives up so much information that it’s hard to see how the system of axioms without the hedged generalization could end up beating a system including the hedged axiom for “best system”. Whether to go for the one that is silent about the behaviour in R, or one that spells it out, turns on delicate issues about whether increased complexity is worth the increased strength that I don’t think we have to deal with here (if I had to bet, I’d go for the silence over specification). Lewis’s account, I think, predicts that minimal mutilation laws are the laws of W. If this is right, then Barker’s argument against this fallback option is inter alia an objection to Lewis’s Humean account of laws. So what is his objection?

Barker’s objection to such minimal mutilation laws is that they don’t fit with “the basic explanatory norms of physical science”. Either we have no explanation for why regularity has a kink at region R, or we have an explanation, but it alludes to region R itself. But if there’s “no explanation” then our law isn’t a real law; and if we have to allude to region R, we are assigning physical (explanatory) significance to particularity, which goes against practice in physical science.

As a Humean, Lewis is up to his ears in complaints from the legal realists that his laws don’t “explain” their instances. So complaints that the laws of W aren’t “real” because they’re not “explanatory” should send up danger-signals. Now, on the option where we build in what happens at R into the W-laws, we at least have a minimal kind of explanation—provability from a digested summary of what actually happens. That’s not a terribly interesting form of explanation, but it’s all we have even in the best case. If the W-laws simply leave a gap, we don’t even have that (though of course, we do have the minimal “explanation” for every other region of space). But who’s worried? It’s no part of Lewis’s conception of laws that they explain things in this sense. Indeed, if you look through his overall programme of vindicating Humean supervenience (exactly the programme, by the way, that motivates the severe constraints on the analysis of counterfactuals) then you’ll see that the theoretical role for laws is essentially exhausted by fixing the truth-conditions of standard counterfactuals—-of course, counterfactuals are used all over the place (not least in articulating Lewis’s favoured view of what it is for one proposition to explain another). And the W-laws seem to play that job pretty well. Another way to put this: for Lewis’s purposes of analyzing closeness, verbal disputes about what is or isn’t a “law” aren’t to the point. Call them Lewis-laws, if you like, stipulatively defined via the best system analysis. If Lewis-laws, Lewis-miracles and the rest successfully analyze closeness, then (assuming that the rest of Lewis’s account goes through) we can analyze closeness ultimately in terms of the Humean supervenience base.

So I think Lewis (and other Humeans) wouldn’t find much to move him in Barker’s account. And—to repeat the dialectical point from earlier—if we’re not Humeans, it’s not clear what the motivation is for trying to analyze closeness in the ultra-sparse way Lewis favours.

But the points just made aren’t entirely local to Lewis and the Lewisians. Suppose we bought into a strong theses that laws must explain their instances, and on this basis both reject Humeanism, and agree with Barker that the hedged generalizations aren’t laws in W. Lewis’s analysis of counterfactual closeness now looks problematic, for exactly the reasons Barker articulates. One reaction, however, is simply to make a uniform substitution in the analysis. Whereever Lewis wrote “law”, we now write “Lewis-law”. Lewis-laws aren’t always laws, we’ll now say, since they don’t explain their instances. Indeed, the explanation for why we have the regularities constitutive of Lewis-laws in the actual is because the real laws are there, pushing and pulling things into order. But Lewis-laws, rather than genuine laws, are what’s needed to analyze closeness, as the Barker cases show us.

To conclude. We should distinguish the general family of worlds-semantics, from the particular analytic project that Lewis was engaged in. Among the family are many different projects, of varying ambition. Barker’s arguments are particular to Lewis’s own proposal. And once we’re dealing with Lewis himself, Barker’s “fallback” position of minimal mutilation laws turns out to be predicted from Lewis’s own Humeanism—and Barker’s objections are simply question-begging against the Humean. Finally, since even a non-Humean can avail themselves of “Lewis laws” for theoretical purposes whenever they prove useful, there is an obvious variant of the Lewis analysis that, for all that’s been said, we can all agree to. Worlds accounts of counterfactuals, in all their glorious variety, are alive and well.

### 3 responses to “Barker on worlds-semantics for counterfactuals”

1. There is another option that’s close in spirit to the Humean one you mention. We could follow David Braddon-Mitchell and say the laws are ‘lossy’. That would mean that even at the world in which OPEN is true, the very same laws are still the laws. Of course, they aren’t all *true* laws in that world, but that’s neither here nor there for the purposes of this example. On that picture of laws, we can run the Lewis story more or less exactly as it is told.

2. [edited to remove some typos]

Hi Brian—that’s an interesting option. Barker actually mentions it, and has a few things to say against it (I’m not convinced by them, but there we go).

I reckon we do have to tread a bit carefully when allowing lossy laws into the analysis of closeness. On the lossy law account, I take it that a world W can have small, and even big, widespread miracles by the lights of its own laws (the alternatives is to try to reconstrue “fit” in some way so violations of law don’t count as miracles, but that seems pretty unpromising). And when we bring this feature to Lewis’s account of closeness, one worry is that weak centering might fail at W. If W contains a big miracle by the lights of W’s laws, and U doesn’t, then U is closer to W than is W itself.

Another prediction of using the lossy laws in this way—suppose we lived in a world which has pretty frequent, though unpredictable, small violations of the lossy laws that govern it. The counterfactually closest worlds, however, contain no violations of law in the future, since once we’re not duplicating the world we’re told to minimize even small violations.

I guess there are a number of patches to try here, but it’s the sort of thing that makes me a bit wary… (I wonder whether a version of that second point applies to minimal mutilation laws too…)

3. If there is anything we have learned from path dependency or chaos theory, the horseshoe nail or the butterfly wing, it is that there is no such thing as a “small miracle”. Change the slightest thing about laws or initial conditions, and all bets are off.
Under determinism, worlds with different laws or initial conidtions tell us little, no, zero, about our world. If laws and IC are the same, then the worlds are as identical and tell us zero about our world. All the hand waving in the world (or any world) wil not change this, they boil down to the fact that explanation and causation can never gain new information (as Benaceraff, Paul, 1973, notes) from iaginary other worlds. Our causal arguments about other worlds can ONLY be based on our causal assumptions from this world, never the other way around.
Barker is correct: RIP Counterfactuals