Monthly Archives: August 2010

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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.

Events and processes

I’ve been reading up on Lewis on causation, and in particular on the account of events he uses. The big thing that his metaphysics of events delivers is a way of getting rid of spurious causal dependence. I say hello, in the course of saying hello abruptly and loudly. I go for a walk, in the course of myself and my girlfriend going for a walk. These patterns of events arise not because one event causes the other, but because one event is part of the other. Pairs of events like these can stand in relations of counterfactual dependence. Suppose I say hello abruptly and loudly. Had I not said hello, then I wouldn’t have said hello abruptly and loudly. So Lewis says: causal dependence between events is not just a matter of counterfactual dependence: it’s counterfactual dependence between distinct events (events that don’t share a part).

When you dig into his metaphysics of events, you see that two notions of parthood are in play. One is broadly logical: event E is a logical part of event F if E occurring in region R entails that F occurs in region R.

A second notion of parthood he uses is spatio-temporal: event E is a st-part of event F if, necessarily, if F occurs in region R, then E occurs some subregion of R. Saying hello abruptly is an l-part of saying hello; my walking is an st-part of myself and my girlfriend walking.

But this still doesn’t cover all cases. Consider Trafalgar and the Napoleonic war. Intuitively, that battle is part of the wars (and not caused by the war, though caused by its earlier parts). But it’s not an l-part, since the region in which the war occurs is more extensive than the region in which the battle occurs. And it’s not an st-part, since the war could have been completed before Trafalgar happened. So Lewis defines up an “accidental” variant of spatio-temporal parthood between occurrent events: E is an a-part of F iff E and F are occurrent, and there’s an occurrent l-part x of E, which is a st-part of some occurrent y which is an l-part of F. I take it the idea is that as well as the Napoleonic wars, there’s another event, the Napoleonic-war-as-it-happened, that is a l-part of the Napoleonic wars; and there’s also Trafalgar-as-it-happened, which is an l-part of Trafalgar. And the latter is an st-part of the former; hence, derivatively, Trafalgar is an st-part of the war.

(Some notes on the interrelation of these notions: if E and F are occurrent, and E is an l-part of F, then it’s an a-part of F (take x=E, y=E). And if E is an st-part of F, then it’s an a-part of F (take x=E, y=F). Rather weirdly, note that when E is an l-part of F, then wherever E occurs, F occurs in an (improper) subregion. Hence F is an st-part of E. And so by the above, if they’re occurrent, we’ll have F an a-part of E. That is, when E is an l-part of F, and both are occurrent, then E and F are a-parts of each other (though of course they may still be distinct).)

Lewis’s requirement that events be “distinct” in order to be candidates for causing one another is that they don’t share a common part in any of these senses.

Lewis notes several times that this would be way too strong a constraint if we allowed events with very rich essences—I’m interested in what this tells us about what sorts of events we can think are hanging around.

Ok: so here is my puzzle. Here’s a first shot—an objection which is plausible but mistaken. Right now, a ball drops, and hits the floor. Consider the conjunctive event or “process” of the ball dropping and hitting the floor. Now (here comes the fallacy) doesn’t this event imply that the ball drops? And so doesn’t that mean the process is an l-part of the ball dropping, and likewise of the ball hitting the floor? But if so, then these two events wouldn’t be distinct, and so couldn’t stand in causal relations. It would be impossible to have a conjunctive process, whose constituents were causally interrelated.

That worried me for a bit but I reckon it’s not a problem. Necessarily, the region *in* which the dropping-and-hitting the floor is a region *within* which the dropping occurs; but it’s not a region *in* which the dropping occurs. “in” is like exact location; an event is then *within* any region it is “in”. But it’s only when every region in which the first occurs is a region *in* which the second occurs, that we have implication or l-parthood. What we have here is just st-parthood, running in the direction you’d have imagined—from constituents to process rather than vice versa.

So that exact puzzle isn’t an objection to Lewis; but I suspect he’s escaped on a technicality, and the underlying trouble with processes will rearise if we tweak the example. Lewis allows for colocated events—and allows they may stand in causal relations. He contemplates a battle of invisible goblins having causal influence on the progress of the AAP conference with which it’s colocated. More seriously, he thinks the presence of an electron in a electric field might cause its acceleration. But the location of the electron, and its acceleration, are colocated events. But in examples of this kind, we really are in trouble if we allow for the conjunctive “process”—the electron-being-so-located-and-accelerating. For necessarily, wherever we have that process in a given region, we have the accelaration *in that region*. So the process is an l-part of the acceleration. Likewise for the locatedness of the electron. But then the two events share a part, and are not distinct—so they couldn’t cause one another!

The trouble for Lewis will arise if we both allow (i) cause and effect to be located in the same region; and (ii) the existence of a “process” encompassing both cause and effect. Lewis says he wants to allow (i); and denying the existence of conjunctive events/processes in (ii) looks unprincipled if we allow them in parallel cases (where the ball drops to the floor). So I conclude there’s pressure on Lewis to rule out conjunctive events/processes across the board.

Loewer on laws

In “Laws and Natural Properties” (Philosophical Topics 2007—I can’t find an online copy to link to) Barry Loewer argues we should divorce Lewis’s Humean account of laws from its appeal to natural properties.

The basic Lewisian idea is something like this. Take all the truths about world w describable in a language NL whose basic predicates pick out perfectly natural properties. There are various deductive systems with true theorems, formulated in this language. Some are simpler than others, some are more informative. The best system optimizes simplicity and strength. The laws are the generalizations, equations, or whatever, entailed by this best system. (This is the basic case—his distinctive treatment of chance requires some tweaks to the setup).

Why the focus on NL? Why not look at any old system in whatever language you like, and pick the simplest/most informative? Lewis worries that the account would then trivialize. Consider the language with a basic predicate F that is interpreted as “being such that T is true”. The single axiom “(Ex)Fx” is then, thinks Lewis, maximally simple, and since its entailments are the same as T, it’s just as informative as T. So simplicity would be no constraint at all, with an appropriate choice of language. What NL does is provide a level playing field: we force the theories to be presented in a common base language, which allows us fairly to compare their complexity.

Loewer notes that the above argument seems pretty questionable. Sure, “informativeness” might be understood just as the modal entailments of the theory—roughly, a theory is more informative the smaller the region of logical space it is true at. But is that the right way to understand informativeness? After all, a sensible seeming physical theory could be applied to some description of a physical situation and produce specific predictions—we can extract a whole range of syntatic consequences of the deductive system relevant to individual situations. Isn’t something like this what we’re after?

Loewer thinks that the right way to extend the Humean project is to take Lewis’s “simplicity and strength” as placeholders for whatever those virtues are that the scientific tradition does in fact value. So he thinks that minimally, if we’re evaluating theories for informativeness, “the information in a theory needs to be extractable in a way that connects with the problems and matters that are of scientific interest”.

I’m not quite sure I understand the next move in the paper. Loewer moves on to say: “Lewis’s argument does show that [Humeanism about laws] requires a preferred language”. That’s a bit surprising, given the above! He goes on to identify the language of scientific English, SL, or its proper successors, SL+. Now, one way to read this is that Loewer is here restricting the languages in which the competing theories can be formulated, not to NL as Lewis did, but to SL or any of the SL+. If we took this line, we can stick with Lewis’s original modal understanding of informativeness I guess–trivialization is ruled out by the same basic Lewisian strategy.

There’s a different way of understanding what’s going on though (and maybe its what Loewer intends). This is to think that the way that we should evaluate informativeness of T is in terms of “truths” that are extractable (logically entailed, for example) from T—the truths that constitute the answers to “problems and matters of scientific interest”. But these truths have to be formulated in a particular language—that’s the cost of the shift from modal characterizations of informativeness to broadly linguistic ones. So as well as the question of what language the theory is in, there’s also the question of the language for presenting the data against which the theory’s virtues are evaluated. There’s nothing that requires the two languages to coincide, and we could insist on a particular formulation of the data-language, while leaving open the theory-language (of course, if the data is to be extractable from the theory in a syntactical sense, then we probably need to add a bunch of coordinative definitions to the theory to link the two vocabularies).

One nice thing about the second way of going is that we don’t have to build in the assumption that the One True system of laws is humanly understandable, or that scientific English or its successors will be adequate to formulate the laws. The first way (where laws are to be formulated in SL+) requires a certain kind of optimism about the cognitive tractability of the underlying explanatory patterns in a world. Lewis’s original theory didn’t require this optimism—NL immediately picks out the fundamental structure of whatever world we’re concerned with, whether or not inhabitants of that world are in a position to figure out what those fundamentals are. Maybe we feel entitled to be optimistic about the actual world—but the Humean account is supposed to apply to arbitrary possible worlds, and surely there are some possible situations out there where SL+ won’t cut it, and some other vocabulary would be called for.

So I prefer the second interpretation of Loewer’s proposal, on which SL+ is the data-language, but the language of theory could be quite different. This suffices, I think, to rebut Lewis’s worry about trivialization. But it allows that in some scenarios, the best system explaining homely facts, is itself quite alien.

A halfway-house between this version of Humeanism and Lewis’s would have the data-language be NL rather than SL+, but allow the language of the final theory to vary. The obvious advantage of this is that it removes the dependence on the contingencies of our scientific language in fixing the laws of arbitrary worlds—strange alien possibilities filled with protoplasm or whatever just might not have a very interesting description in the terms of a language developed in response to our actual situation. Appealing to NL for the data-language tailors informativeness to a description of the world appropriate to the basic features of that world, rather than using one developed in response to the world we happen to find ourselves in.

Let’s consider an example. Suppose that the natural properties are Fieldian, rather than Lewisian. The fundamental features of the world are relations like congruence and betweenness (and similar) that fix the spatio-temporal structure of the world and the mass distribution across it. Now, Field’s “nominalized physics” aims to articulate versions of the standard Newtonian equations in this setting—without appeal to standard resources such as the relation of “having mass of x kg” which brings in appeal to abstracta. Field thinks this “synthetic” formulation should appeal even to those who do not share his qualms about the existence of numbers. Let’s suppose we take his proposal in this spirit, so whatever other problems there may be with the mathematized physics, the worry isn’t that it’s false.

Are the usual mathematized Lagrangian formulations of Newtonian mechanics laws in this Fieldian world? On the original Lewisian proposal about laws, the best system should be formulated in perfectly natural terms—which here means the Fieldian synthetic relations. The natural thought is that the Fieldian nominalistic formulation wins this competition, and its deductive consequence won’t include the usual mathematized equations. So, presumably, the mathematized Lagrangian equation won’t be a law. On the other hand, if we go for either of the tweaked versions above, our candidates for “best theory” needn’t be given in this metaphysically privileged vocabulary. Given appropriate coordinating links between the vocabulary, standard mathematical definitions will entail all the data about mass-congruence and the rest, and so count as informative about the Fieldian data (whether formulated in the Fieldian NL or SL). And (you might argue) going this way enables gains in simplicity, making it the winner in the fight for best theory. So the usual, mathematically laden, Lagrangian may yet be a law. Likewise, a Hamiltonian formulation of mechanics could still be the winner in the race for best theory, and the Hamiltonian equation a law, without us having to claim that it is the simplest around when formulated in the perfectly natural, synthetic terms. More generally, we’re liberated to argue that the basic principles of statistical mechanics should feature in the winning theory, even if its terms are a long way from perfectly natural—so long as they add enough information about (for example) the synthetic perfectly natural truths to justify the extra complexity of adding them in.

Some of the use that the Lewisian account of laws is put to goes over more smoothly, I think, if the data-language is NL rather than SL.  Lewis famously wanted to use the Humean framework to help understand chance. His underlying metaphysics had no primitive chances—simply a distribution of particular outcomes (e.g. there’s an atom at one location, and the results of atomic theory at the next, and a particular statistical distribution among events of this type across space-time, but no primitive “propensity” relating the tokens) On the original account, Lewis liberalized his requirements for the vocabulary of candidate theories, allowing an initially uninterpreted chance operator. Given an appropriate understanding of the “fit” between a chancy theory and a non-chancy world, he thought that chancy theories would win the battle of simplicity and informativeness, grounding chancy laws and thereby the truth of chance talk.

It becomes somewhat tricky to replicate this idea if the data-language is construed to be SL+, as Loewer suggests. Take a world that’s set up with GRW quantum mechanics, with primitive chancy evolution of the wave function. Now, presumably SL+ contains chance talk, and so the data against which theories are to be measured for informativeness includes truths about chance. The original idea was that we could characterize, non-circularly, what made a chance-invoking scientific theory “selected”. But now it turns out that one of the ingredients to selection—informativeness—require appeal to chance. If the data-language in question were NL rather than SL, we wouldn’t face this obstacle.

Overall, I’m not attracted to the version of Humeanism where competitors for best theory must be formulated in SL or SL+—it seems excessively optimistic to think that the laws of a wide enough range of worlds will be formulated in these terms. The version where we appeal to SL+ only in evaluating theories for informativeness looks much more promising. Even so, I’m not sure what we gain from appealing to SL+ rather than NL in the evaluation. Sure, if you were sceptical about appeal to the perfectly natural in the first place, you might be attracted to this as a decent fallback. But I don’t see otherwise what speaks in favour of that.