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.