Accept no substitutes:  Against best-system theories without naturalness

1. The best-system theory

To determine what the laws are, Lewis said, you hold a competition.² The competitors are “systems”: sets of true sentences that are closed under logical implication. Systems compete for being strong—saying a lot about the world—and being simple—having a syntactically simple axiomatization. As Lewis points out, simplicity without strength is easy to achieve (for instance, by saying nothing more than that “everything is self-identical”), as is strength without simplicity (for instance, by including an exhaustive point-by-point specification of field values); but the goal of the competition, which is nontrivial to achieve, is to be as strong as possible without sacrificing too much simplicity. The winner of the competition is the system that strikes the best balance between strength and simplicity—the “best” system. The members of this system (or perhaps only its universal generalizations) are the laws of nature.

The best-system theory is schematic until a particular trio of measures of strength, simplicity, and balance—a “measure”, for short—is introduced. The measure is to be akin to the standards that scientists (or physicists, at least) actually use in theory choice. But flat-footedly defining the measure in terms of scientific practice (viz, “to be a law is to be a member of the system that best balances the virtues of strength and simplicity by the measure operative in physics departments on Earth in the year 2025”) would imply that there would have been no laws of nature if there hadn’t been any physics departments. One might wriggle using familiar tricks, for instance by replacing the description ‘the measure operative in physics departments on Earth in the year 2025’ with a proper name of a specific measure whose reference is fixed by that same description; but as Lewis says, a maneuver like this “doesn’t make the problem go away, it only makes it harder to state” (Lewis, 1994, p. 479).

In my view, the residual problem is that sentences about lawhood would fail to be objective, in the sense of objectivity discussed in Sider (2011, chapter 4).³ Even though those sentences would express propositions that are mind-independent, different linguistic communities employing different measures would use those same sentences to express propositions with different truth values. There would be no more objectivity than if different subcultures in Brooklyn used the predicate ‘cool song’ to express different mind-independent sonic properties of sounds, each reflecting the tastes of that subculture. This diagnosis of the problem meshes with what Lewis said in response to it: that if nature is kind, then in the actual world, lawhood coincides for all measures. For given his claim, lawhood is objective in the current sense: although there are multiple admissible measures we might have used to define ‘law’, sentences containing ‘law’ have the same truth value on all such measures.⁴

The systems in the competition are, we said, sets of sentences. But sentences in what language? This is where naturalness comes in. For as Lewis pointed out, if we are allowed to pick predicates that have any meanings whatsoever, we could choose a language in which the sole nonlogical predicate $F$ is true of all and only objects in possible worlds that are exactly like the actual world in every way, and formulate a bogus system that consists solely of the sentence $\forall xFx$ together with all of its logical consequences in that language. This system is extremely simple (it is axiomatized by the single sentence $\forall xFx$). Moreover, Lewis understands the strength of a system as its modal strength, which is defined as being inversely proportional to the “size” (under some suitable measure) of the set of worlds in which the system is true. Thus the bogus system is as strong as can be (since it is true only in possible worlds that are exactly like ours). So it would seem to count as the best system. As a result, every true proposition would count as nomologically necessary—as being a modal consequence of the laws of nature.

Lewis’s solution to the problem was to require that all systems in the competition be formulated in a language in which every predicate refers to some “perfectly natural” property or relation. The notion of perfect naturalness was one that Lewis had come to see as indispensable to systematic philosophy (Lewis, 1983a). In my view, ‘perfectly natural’ should be seen as an undefined posit, whose reference is fixed (though not in any particularly rigid way) by the totality of ways we appeal to it in theorizing, much as a theoretical predicate in physics lacks reductive necessary and sufficient conditions but nevertheless has its reference fixed by its theoretical role (Sider, 2011, chapter 2). But the intuitive core of the notion of naturalness is this: the perfectly natural properties and relations are the “fundamental” properties and relations whose distribution constitutes the fundamental facts about the world.

Although Lewis doesn’t quite say it explicitly,⁵ it’s clear that the “language” in which systems must be formulated isn’t meant to be an “embodied” language—a language that is actually used by people. And predicates in this language aren’t meant to “refer” to natural properties in the sense of “embodied” reference—the sense in which actual physicists refer to mass when they use the term ‘mass’. (There could have been laws about mass even if no physicists had ever referred to mass!) Rather, he is using ‘language’ and ‘reference’ as they are used in mathematical logic: “languages” and other syntactic entities are simply abstract entities, with stipulated syntactic forms; an “interpreted” language is simply a pair of a language and a function that maps “words” in the language to appropriate worldly entities (such as entities and properties and relations); and to say that a predicate in such a language “refers to” a natural property means nothing more than that the function paired with the language maps that predicate to the natural property.

2. A gap between science and metaphysics?

Many authors have argued that Lewis’s reliance on naturalness opens up an unacceptable gap between metaphysics and science. According to this worry, “metaphysical” laws could come apart from “scientific” laws. The former are given by the Lewisian best system, which is constrained by the “metaphysical” facts about naturalness; the latter are what scientists regard as laws; and there is no guarantee that the two will coincide. Bas van Fraassen (1989, p. 53) pressed this worry in an influential early discussion; and Jonathan Cohen and Craig Callender (2009, p. 12) vividly describe a specific scenario of this sort:⁶

…consider the case of Murray Gell-Mann and others organizing mesons and baryons into octets—now seen as representations of SU(3) symmetry—in what became famous as the Eightfold Way. The theory relies on the positing of new fundamental properties, in particular, fractional charge. The Eightfold Way seems to scientists the on balance strongest and simplest systematization of the relevant phenomena. Is it a law (or at least a corollary of a law)? Of course, it might fail to be because further experiments might reveal more phenomena that demand a better system, or because someone keener than Gell-Mann might come along and systematize the field even better. Stipulate for the sake of argument, however, that Gell-Mann reasoned impeccably and had all the facts available to him; given the kinds he [chose], the Eightfold Way really is the best systematization of the relevant facts. Even granting this much, [Lewis] cannot guarantee that the Eightfold Way is a law (law corollary); for there is nothing to guarantee that fractional charge is one of the properties enshrined as perfectly natural.

Now, a natural response is that it just is possible that physicists follow the best scientific methodology and yet are wrong in their conclusions, so it is unproblematic that Lewis’s theory commits us to this possibility. What Cohen and Callender are imagining is akin to the scenarios on which traditional skeptical arguments in epistemology are based. It is a scenario in which physicists’ evidence is misleading—a scenario in which (by stipulation) all available evidence supports a theory that isn’t true. If the argument is based on the premise that such a scenario is impossible—that the truth about laws could not possibly come apart from ideal scientific practice—then no “scientific realist” should accept the premise.⁷

It’s also unclear whether Cohen and Callender’s scenario can be adequately fleshed out. Suppose, for example, that although the Gell-Mannian properties, such as fractional charge, are not perfectly natural, they nevertheless have reasonably simple definitions in terms of the perfectly natural properties.⁸ Then Gell-Mann’s system presumably does consist of Lewisian laws of nature after all (or at least, of “law corollaries”); the imagined gap between metaphysics and science wouldn’t exist. For if we begin with Gell-Mann’s system $\varphi (T_1,\mathrm{\dots },T_n)$, and then substitute, for each of his terms $T_i$, its definition $d(T_i)$ in terms of perfectly natural properties, the resulting system $\varphi (d(T_1),\mathrm{\dots },d(T_n))$ would surely win the Lewisian competition: it is nearly as syntactically simple as the original system (since the definitions are simple) and just as strong. The original system $\varphi (T_1,\mathrm{\dots },T_n)$ would be analogous to Newton’s equation $F=ma$ under the assumption that position and time rather than acceleration are fundamental: although the winning entry in the Lewisian competition will not include the very sentence ‘$F=ma$’ (since the predicates in competitors must stand for perfectly natural properties and relations), it will include a definitional equivalent of that sentence, namely, the result of replacing the expression for acceleration in that sentence with an expression for the second time-derivative of position.⁹ But if, on the other hand, the putative Gell-Mannian properties have only extremely complex definitions (or infinite definitions, or no definitions at all) in terms of perfectly natural properties and relations, then the example threatens to be incoherent. The description of the example stipulates that the eightfold way is the best systematization of the facts stated in terms of its chosen kinds. Thus that theory is true; so its terms such as ‘fractional charge’ refer to (instantiated) properties. But these are theoretical terms, which are introduced to stand for whatever fundamental, or nearly fundamental, physical properties satisfy the theory (near enough).¹⁰ If no such physical property fits the theory associated with such a term, the term doesn’t successfully refer to some (instantiated) nonphysical property, or to some (instantiated) wildly complex physical property. Rather, the term is a failed natural-kind term, like ‘phlogiston’: it is semantically empty (or uninstantiated). But according to Lewis’s metaphysics (as I’ll explain in a moment), “physical properties” just are natural properties (whether perfectly natural or merely very natural) that play a role in the laws of physics. So there must be some intermediate way of understanding what the Gell-Mannian properties are supposed to be, distant enough from the natural properties so that Gell-Mann’s theory doesn’t give the Lewisian laws after all, but close enough to allow the Gell-Mannian terms to refer to them.

These two responses to the argument are made from a broader Lewisian point of view, which doesn’t only include Lewis’s best-system theory of lawhood, but also includes some further claims about the relationship between natural properties and science: that to be a physical property is to be a natural property that plays a role in the laws of physics—a role in the laws of that science whose job is (to use a common delimiter) to account for matter in motion;¹¹ and that the aim of physics is to discover the physical laws and physical properties as understood in the Lewisian way. (Of course, physicists may not recognize the objects of their endeavors under these descriptions.) Given the interconnections between lawhood, the notion of physical property, and the aims of physics, a coherent Lewisian account of physics must include these further claims.¹²

To be sure, Cohen and Callender reject this broader Lewisian point of view; and that rejection may inform how they conceive of their example. Only from a Lewisian point of view is the example akin to “skeptical scenarios” (from Cohen and Callender’s point of view, Lewis’s theory of laws is incorrect and Gell-Mann’s theory gives the laws despite their mismatch with the natural properties). Only from a Lewisian point of view does the fact that a property lacks a reasonably simple definition in terms of natural properties imply that it is nonphysical (from Cohen and Callendar’s point of view, being a physical property has nothing to do with naturalness). But if the argument is meant to be dialectically effective, the anti-Lewisian point of view cannot be assumed. Rather, the argument ought to attempt to show that Lewis’s view, understood in its own terms, leads to unacceptable predictions.

But even if this conception of the dialectical situation is correct as far as it goes, there is more to be said about the big picture. Behind such objections there is, I think, a widespread, primordial, vague sentiment: namely, that Lewisian lawhood cannot be the subject matter of physics, because naturalness is a metaphysical notion.¹³ The writings of the naturalness-skeptics are full of rhetoric that makes clear that they are indeed conceptualizing naturalness as “metaphysical”—“arcane”, “scholastic”, “theological” even—in a sense that is opposed to “physical” or “scientific”. It is because naturalness is metaphysical in this sense that Lewis’s account is insufficiently scientific in status.

I believe the sentiment to be fundamental to the thinking of many skeptics about Lewis’s account (and indeed, to the thinking of many skeptics of metaphysics itself). But at the risk of sounding too grand, I think it is based on a misunderstanding of what metaphysics is.

Critics sometimes have a picture of metaphysics deriving from Plato: that metaphysics is about some distinctive realm, separate from (and perhaps superior to) the ordinary material realm that we experience and which science studies. But metaphysics isn’t like this at all; or at any rate, it needn’t be. There is just one realm; and metaphysics is trying to say something about it. We encounter various parts of this realm in daily life and in science, such as objects, properties, time and space. In metaphysics, we try to say something systematic about those parts—to embed them in general theories. I don’t mean to say that metaphysicians can’t posit completely new stuff or even a new realm; they can (and have). But it isn’t the job description of a metaphysician to do so.

The notion of physically fundamental properties plays a central role in the philosophy of physics (sometimes under the guise of a distinction between “physical” and “conventional” phenomena). These are the properties we are trying to discover in physics; we often guess that there are new ones (such as spin) if theories using the properties we already posit aren’t adequate; we use the notion to characterize genuine from coordinate-dependent features of space and time; and so on. This isn’t some alien concept; it is a concept rooted in the practice of physics.

There is then a question of how to think about the notion metaphysically—how to embed it in a more general account of the nature of reality. Lewis’s posit of the notion of naturalness is, in part, an answer to this question. What are physicists talking about, when they talk about “physical” properties? Lewis has an answer: to be a physical property is just to be a natural property that plays a role in the laws of physics. Obviously physicists wouldn’t put it that way; and Lewis says much more about natural properties than physicists would say about physical magnitudes. But the point is that Lewis isn’t introducing a different subject matter—a new kind of stuff, or a different realm—when he brings in the notion of naturalness. He is, in part, giving an account of what physicists have been talking about all along.

My response here does not consist of an acceptance of the critics’ notion of “metaphysical”, followed by an argument that naturalness fails to be metaphysical in that sense. The response, rather, consists of rejecting the critics’ notion of “metaphysical” (because it embeds a false presupposition about the nature of metaphysics), followed by an argument that the theory of naturalness, while of course being a part of metaphysics, is (in part) a philosophical account of notions that physics has been concerned with all along.

Repudiating the mistaken picture of metaphysics won’t pull the rug out from under all of the critics, since for some, the ultimate target may simply be naturalness itself, or even metaphysics itself. More sweeping critiques of naturalness or metaphysics are certainly worthy of consideration; and a truly comprehensive defense of Lewis’s best-system-theory ought to include defenses—book-length, presumably—of the metaphysics of natural properties, and of metaphysics in general. But such sweeping critiques are very different from an objection that grants the metaphysics of naturalness, and tries to show that it doesn’t succeed on its own terms (because it opens up a gap between metaphysics and science).

Because this issue is so important, let’s pursue it further. Consider the following analogous dialectic. Some philosophers of physics—notably, Tim Maudlin—think that specifying an ontology—specifying what entities exist—is an essential part of providing a physical theory. These philosophers think, for example, that a desideratum on any acceptable quantum theory is that it give some kind of answer to the question “what kinds of entities does the physical world contain, and what are they like?”, rather than simply giving a mathematical model which generates correct predictions (Maudlin, 2018, introduction). And they regard questions such as “Is the correct ontology of quantum mechanics that of three-dimensional particles, or is it instead that of a single ‘marvelous particle’ moving in a high dimensional space?” (Albert, 1996) as being genuine. Of course, if the meaning of some predicate $F$ is unclear then the question of whether $F$s exist will itself be unclear; compare the question of whether points of spacetime exist in a general-relativistic setting.¹⁴ But assuming that $F$ itself is clearly defined, the question of whether there exist $F$s is in good standing.

Now, within metaphysics proper there is a lively debate over whether ontological questions are genuine. The status quo, I suppose, is that they are; but there is an important group of renegades who think that ontology is not an important part of inquiry into the nature of reality. For instance, Eli Hirsch and Amie Thomasson argue, in different ways, that many of the ontological questions that metaphysicians have traditionally asked are ill-posed: there is no well-defined question of whether the world “really” contains a statue that is distinct from the quantity of matter from which it is made, as opposed to merely containing the quantity of matter. And there is an analogous position in the philosophy of physics, which has been most clearly articulated by David Wallace, according to which there is no well-defined question of whether the physical world is “really” three-dimensional or high-dimensional. These apparently conflicting descriptions are in fact just notational variants.¹⁵ Just as Lewis’s critics argue that facts about natural properties are not part of the aim of science, Wallace argues that specifying an ontology is not part of the aim of science.

The debate between the Maudlins and Wallaces of the world is an important one, and is absolutely worth having. But it would be bizarre to accuse the Maudlin camp of, on their own terms, introducing a bifurcation between “metaphysical ontology” and “scientific ontology”, or to argue that their account is unacceptable because it implies that physicists who follow correct methodology might posit the existence of quarks (say) and yet be mistaken because there don’t “really”, or “metaphysically” exist quarks. For Maudlin’s view is that physics just is about ontology. That is the subject matter of physics. There is no gap between metaphysical and physical existence. There is just: existence, which physics strives to discover, just as for Lewis there is no gap between “metaphysically” and “physically” fundamental properties; there are just: fundamental (a.k.a. natural) properties, which physics strives to discover.

For a final illustration of this point, consider the most primordial metaphysical debate of them all, the debate over whether there exists an external world which exists independently of us. Imagine an external-world antirealist, who refuses to distinguish the real existence of the external world from appearances due to the deceptions of a Cartesian demon, accusing an external-world realist of opening an unacceptable gap between science and metaphysics. “According to your view”, so the accusation goes, “even if certain laws are scientifically true, they might not be ‘really’ or ‘metaphysically’ true, since the alleged ‘real world’ might differ from the appearances”. The accusation is mistaken, and not just because the accuser is wrong to reject external-world realism. It’s mistaken because according to the external-world realist, “being scientifically true” just is “being true in the external world”, which is to say, being true.

Even a critic who isn’t in the grip of the two-realms picture might nevertheless press Lewis to explain why naturalness is significant for physics. “Give me an account of naturalness that shows us why physicists ought to care about natural properties”, the critic might say.

Here again it’s instructive to consider the parallel with ontology: imagine pressing Maudlin to say why physicists ought to care about what objects exist. When it comes to very basic philosophical concepts such as existence and property (in the “sparse” sense of ‘property’ (Lewis, 1986a, pp. 59–69) that Lewisian naturalness is meant to capture), it’s hard to say in other terms why they matter. Lewis and Maudlin are “primitivists” about these notions, and can’t respond by giving them analyses from which we can just see why physicists should care about them.

What they can do, instead, is explain the overarching role they take these concepts to play, and claim that their proposals deliver a conception of cognitive life that is recognizably our own. The role, for both existence and naturalness, is partly a “world-making” one. For Lewis, our world, ultimately speaking, is a world of objects having natural properties. The space of possibilities is generated combinatorially (more or less) from objects and natural properties.¹⁶ All facts hold in virtue of the distribution of natural properties over objects.¹⁷ (In Lewis—and Sider (2011)—we hear more about the role that natural properties play, such as that facts about natural properties determine when things are exactly intrinsically like, and what the intrinsic structure of spacetime is.) The role also includes a component about lawhood (a reductive account for Lewis; a primitivist account, for Maudlin, but one that involves the notion of existence/ontology). But the role also includes the claim that the aims of science are centered on these notions of existence and property.¹⁸ Seen in this light, it isn’t at all strange to think of the aims of physics as being tied up with ontology and naturalness: it’s simply a matter of physics being concerned with the ultimate nature of reality. In sum, what Lewis and Maudlin are offering is a sort of ramsey sentence: that there are notions of existence and property that are central to metaphysics and the aims of science in certain proposed ways. Their argument for the truth of that ramsey sentence is that it is better supported by the overall weight of philosophical evidence than its competitors.

3. The package-deal account

Lite best-system theories are therefore not forced on us. Lewis’s “full-fat” best-system theory does not introduce a bifurcation between science and metaphysics. It succeeds on its own terms, in that we can recognize, in its proposed conception of laws, what science has all along been trying to discover.¹⁹

But even if they are not forced on us, lite best-system theories might be preferable, if they can be made to work. They have the dialectical advantage of neutrality, since they can be accepted even by skeptics about naturalness. And they might be preferable on grounds of parsimony.

In fact there are many different possible varieties of the lite best-system theory. In the remainder of the paper, I’ll critically examine some of them. Rather than following the letter of extant proposals, I will examine several views that are in their vicinity but with certain details filled in. My hope is that moving the debate into a more fine-grained phase will be useful to both partisans and critics.

Let’s begin with Barry Loewer’s idea that the choice of which properties and relations are to be the meanings of predicates should be incorporated into the Lewisian competition.²⁰ Thus an entry into the competition is a “package deal”, to use Loewer’s phrase: a system together with an interpretation of its language. That is (to spell it out a bit), a package deal is a pair $⟨S,I⟩$, where:

• $S$ is a system (set of sentences closed under entailment in a certain language, $L$);

• $I$ is an interpretation of $L$ (an assignment of referents to names, and properties and relations to predicates);

• Every member of $S$ is true under $I$;

• No constraints on $I$ are made (predicates needn’t stand for natural properties and relations since we are avoiding the use of naturalness);

• The strength of the package is inversely proportional to the “size” of the set of possible worlds in which $S$ is true as interpreted by $I$; and

• The simplicity of the package is a syntactic feature of $S$ alone—it is unaffected by $I$.

(As with Lewis’s theory, ‘language’, ‘interpretation’, and ‘reference’ here are understood as in mathematical logic—as having nothing to do with natural language.)

If no restrictions are placed on the packages, every true statement will again turn out to be nomically necessary, since a package $⟨\{\forall xFx,\mathrm{\dots }\},I⟩$, where $F$ is a one-place predicate, $\{\forall xFx,\mathrm{\dots }\}$ is the set consisting of the sentence $\forall xFx$ and all its logical consequences (in the language whose only nonlogical expression is $F$), and where $I$ is an interpretation assigning to $F$ a property had only by objects in worlds exactly like ours, will be a winning package.

Since scientists never consider sentences with such a simple syntax as $\forall xFx$ to be serious candidates for lawhood, one might think to place some sort of syntactic constraint on systems in the competition. But this wouldn’t help if the interpretations remain unconstrained, since we can still choose interpretations of the predicates and names in $S$’s language so as to again make $S$ true only in the actual world. This can always be done, despite any imposed syntactic constraints, so long as $S$ is consistent and satisfies a couple of other conditions (see the next paragraph). The argument for this conclusion uses some elementary model theory in a way that is familiar from various philosophical contexts; and I will present it in detail since I’ll be using the same argumentative strategy again below.

Suppose (first condition) that there exists a model of $S$ whose domain has the same size (cardinality) as the set $D_{\mathrm{@}}$ of actual concrete objects. By a “model” I mean the standard notion of a model from model theory: a domain that is a set of any objects whatsoever, including abstract entities, together with an assignment of referents and extensions drawn from this domain to names and predicates in $S$; and by a model of $S$ I mean a model in which every member of $S$ is true. Whether such a model exists depends only on purely syntactic features of the sentences in $S$ (holding fixed the set-theoretic universe!); it is a question, so to speak, of whether their logical form is consistent with the size of $D_{\mathrm{@}}$. This is a very weak condition; no scientifically serious system would be logically incapable of truth in a universe the size of ours. Now, since the model’s domain has the same size as $D_{\mathrm{@}}$, there exists some one-to-one function from its domain onto $D_{\mathrm{@}}$; we can then “image” the model through the function to obtain a “concrete” model $M_{\mathrm{@}}$ of $S$: a model in which every member of $S$ is true whose domain is the set of actual concrete objects.²¹ Next suppose (second condition) that for no cardinality is $S$ true in every model whose domain has that cardinality. This too is a very weak condition. No scientifically serious theory would be so weak as to have its truth logically guaranteed by some possible proposition about the universe’s cardinality. Then for each possible world $w$ other than the actual world, there is a model $M_w$ of $S$’s language in which $S$ is not true (that is, in which not every sentence in $S$ is true), whose domain is the set of concrete objects at $w$. Now let $I$ be the interpretation which assigns, to any predicate $\mathrm{\Pi }$ in $S$’s language, a relation whose extension in any world $w$ is the extension of $\mathrm{\Pi }$ in $M_w$.²² Under $I$, $S$ is true in $\mathrm{@}$ but not in any other world. (If strength is so measured that being true only in the actual world doesn’t automatically make a system maximally strong, the extensions could presumably be converted to intensions in some other way, resulting in a theory that is as strong as one likes, given the proposed measure.)

We can then use these mathematical facts to argue against the syntactic solution, as follows. Suppose for reductio that the current version of the package-deal account yields some “reasonable” winning package deal $⟨S,I⟩$. $S$ must surely satisfy the two constraints; so there is another package deal in the competition, $⟨S,I^{\prime }⟩$, such that $S$ under $I^{\prime }$ is true only in the actual world. No reasonable package would be as strong as $⟨S,I^{\prime }⟩$ (under it, every truth is nomically necessary); thus $⟨S,I^{\prime }⟩$ is stronger than $⟨S,I⟩$. But the simplicity scores of $⟨S,I⟩$ and $⟨S,I^{\prime }⟩$ are identical (since simplicity is a function purely of the system of a package, and these packages share the same system); therefore $⟨S,I^{\prime }⟩$ is a better package than $⟨S,I⟩$; and therefore, $⟨S,I⟩$ isn’t a winning package after all.

This sort of argument isn’t particularly tied to the assumption that the theories in the competition are first-order. If they are stated in a second- (or higher-) order language, then the argument is pretty much the same (although the first condition isn’t quite as weak). It isn’t even deeply tied to the assumption that the systems are stated using predicate logic. Without going into details, even if physical theories are understood as being about “quantities” rather than properties and relations, one can still construct interpretations corresponding to “quantity-theoretic models”, so long as we assume appropriate principles of plenitude for quantities, analogous to the principles of plenitude for sets that are built into standard set theory (and analogous to principles of plenitude for “abundant” properties and relations)—principles such as “for any $n$-place function, $f$, from concrete objects to real numbers, there exists some quantity $q$ such that for any concrete entities $x_1,\mathrm{\dots },x_n$, $q(x_1,\mathrm{\dots },x_n)=f(x_1,\mathrm{\dots },x_n)$”. Rejecting such principles would amount to treating quantities as being “sparse” rather than “abundant”, and thus would seem to be a tacit appeal to naturalness.

This style of argument—imaging an interpretation through an arbitrary one-to-one function to yield a bizarre interpretation in which the same sentences are true—is most familiar from its use by Hilary Putnam (1978, part IV; 1980; 1981, chapter 2) in his model-theoretic argument against realism. But there is an important difference between that context and the present one. Putnam’s question was one of “metasemantics”. It was about embodied language and embodied reference—about when an interpretation of a flesh-and-blood linguistic community is “correct” in the sense of giving the truth-conditions of sentences as they are used by speakers of that community.²³ (His argument used claims like this one: “the only constraint on when an interpretation is correct is that every sentence in a certain set must come out true in that interpretation”.) But the present argument has nothing to do with metasemantics, or embodied language and reference, or correctness. Its target is a proposal (namely, the package-deal account with no restriction on packages except for some syntactic constraint on systems) which (like all proposals I will discuss) is solely about the nature of lawhood; it takes no stand on metasemantics. ‘Language’ and ‘reference’, as used in the statement of that proposal, are intended only in their mathematical-logic senses. The “interpretations” $I$ in package deals $⟨S,I⟩$ are simply functions that map words to objects, properties, and relations; they have nothing to do with embodied reference in linguistic communities.²⁴ (One could, I suppose, consider a version of the package-deal account that defines the laws as being given by the best system of all competitors $⟨S,I⟩$ in which the language of $S$ is used by some linguistic community, and in which $I$ is a correct interpretation of that language as used by that community. But in addition to reversing the intuitive order of dependence between lawhood and metasemantics, it has the apparently absurd implication that there could not exist laws of nature without language users.)

4. Relativism

The problem we have been discussing is one of choice: how to choose the vocabulary in systems? We could avoid having to make that choice by treating lawhood as being relative. For any set $S$ of properties and relations, define a system-relative-to-$S$ (“system_S”) as a set of true sentences all of whose predicates express properties and relations in $S$; and define a law-relative-to-$S$ (“law_S”) as any logical consequence of the system_S that best balances strength and simplicity (or: best balances_S strength_S and simplicity_S, insofar as these three notions were understood, in the original theory, in terms of naturalness). According to lawhood relativism, there is no such property as being a law—that is, being a law simpliciter. There is only the relation of being a law-relative-to, which holds between sentences and sets of properties and relations. This is Cohen and Callender’s (2009) preferred view.³⁷

Relativism can be attractive in other contexts in which we face analogous choices. How to choose which aesthetic standards determine what is beautiful? There is no need to choose, according to aesthetic relativism, because there is no such property as being beautiful. Rather, there is a relation, being beautiful-relative-to, holding between aesthetic objects and standards.

Relativism about lawhood will lead to relativism about a great many other matters.³⁸ For example, it is usually assumed that counterfactuals are closely connected to lawhood. The exact nature of the connection is debated, but whatever it is, relativism about lawhood will induce relativism about counterfactuals.

One simplistic argument for this conclusion runs as follows. Counterfactuals are defined in terms of lawhood. That is, there is some true sentence of the form:

• If it had been that $A$ then it would have been that $B$ ${=}_{\text{df}}$ $\varphi$

where $\varphi$ contains the predicate ‘is a law’. So if lawhood on the right-hand-side is relative to sets of properties and relations, so are counterfactuals.

This is too simplistic. If lawhood and counterfactuals are relative, then the displayed sentence could not be true since it would be ill-formed; both its left-hand-side and every occurrence of ‘is a law’ on its right-hand-side would have unfilled argument places.

A better argument is this: whatever our evidence was for accepting the original definition, that evidence will support, when combined with the information that lawhood is relative, the conclusion that i) counterfactuals are also relative, and ii) are defined by the obvious modification of the original definition, namely:

• (If it had been that $A$, it would have been that $B$)_S ${=}_{\text{df}}$ ${\varphi }_S$

where $S$ is a free variable, and where ${\varphi }_S$ is the result of beginning with the original definiens, $\varphi$, and replacing all references to lawhood with references to lawhood_S (as well as, perhaps, making certain other correlated replacements).

One might object that if lawhood really is relative, then since counterfactuals are obviously not relative, counterfactuals are not after all defined in terms of lawhood. But how else would they be defined?

As stated, the argument relies on a notion of “definition”. But similar arguments could be given in other terms. For instance: “We have a certain body of evidence, which we normally take to support a class of statements of the form ‘if the laws are such-and-such then so-and-so counterfactuals are true’; combined with the information that lawhood is relative, this body of evidence supports correspondingly relativized conditional statements.”

One way or another, then, if lawhood is relative, so are counterfactuals.

This dialectic recurs for all notions that are appropriately connected to lawhood. The list of such notions is not short, because of the conceptual centrality of one of its key members, namely, causation, which is normally thought to be closely connected to, well, pretty much everything: rationality, action, freedom, personal identity, morality, consciousness, etc.³⁹ Not to mention more mundane concepts like being a chair. Practically every concept would be relative to sets of properties and relations. This is hard to stomach.

It wouldn’t be hard to stomach if the relativity were merely around the edges. Most of us already accept something similar, that ‘cause’ is vague around the edges—“around the edges” in the sense that although some sentences about causation are vague, the everyday sentences about causation that form the backbone of our conceptual lives, sentences such as ‘my being hungry caused me to eat’, retain their truth values under any sharpened meaning of ‘cause’ that is consistent with that predicate’s conventional meaning. But the proposed relativity in lawhood is not “around the edges” in this sense. No interesting sentences about lawhood will be true relative to every set $S$. The reason is that there is to be no restriction on the sets of properties and relations $S$ to which lawhood is relativized. These sets don’t only include rival “scientific” conceptions of what properties and relations are to be utilized in physics. They also include, for instance, the set $\{F\text{-ness}\}$, where $F$ is Lewis’s predicate; relative to this set, every true sentence is a law. There is also the empty set, relative to which there are no systems with a reasonable amount of strength. Therefore no laws, no causation, no anything.⁴⁰

The proposed relativity is thus drastic as well as pervasive. Practically nothing we ever say about the physical world—that grass is green, that the sky is blue, that we ourselves exist—is true simpliciter. All such statements are true relative to some sets of properties and relations and false relative to others.

Some will have heard enough. But for those who wish to linger until the end of the show…

The relativist might argue that even though lawhood (and practically everything else) is relative, so that there is no cosmic standard of correctness for judgments about these matters, there is nevertheless a local standard. There is a certain set of properties, $S$, determined (in some sense) by our local situation, such that judgments about lawhood (and other things) made in that situation can be considered correct when they are true relative to $S$. Thus our discourse about laws (and everything else) enjoys a sort of local objectivity. All that is missing is a cosmic objectivity whose existence is doubtful anyway, since it is objectionably metaphysical in the same way that Lewisian naturalness is.

Two obstacles confront this sort of story. First, it isn’t obvious how to characterize which properties go into $S$—what it is about our “local situation” that determines the local standards for correctness. We cannot say, for example, that $S$ is the set of properties that have in fact been expressed in physics journals, since physicists can make mistakes. We justifiably hope that physicists employ predicates for properties that figure in statements correctly judged to be laws, but there is no constitutive guarantee that they do. (Thus the story is less analogous to typical forms of relativism than it might have seemed. The local aesthetic standards, for example, are constitutively determined by something like the shared aesthetic feelings of the members of the local community.) Less flat-footed construals are available: perhaps $S$ contains the properties that would be most useful (in some specified sense) for us to employ. This would move the account further toward the “pragmatic” end of the spectrum.⁴¹ Too far, in my view—what’s the point of the notion of law, if it’s not a somewhat objective conception of what physics strives for?—but reasonable people may disagree.

Second, there is a question of “circularity”. The traditional reductionist project was to “analyze” (to use Lewis’s word) lawhood using non-nomic notions—notions that are not themselves analyzed in terms of lawhood. In particular, neither counterfactuals nor causation could be used in the analysis; otherwise the analysis would be “circular”. But on the current account, the notions used to characterize what properties go into $S$, such as usefulness-to-us, seem to be themselves analyzed in terms of causation or counterfactuals. Perhaps the relativist will reimagine the traditional project in some way.

5. Lawhood by list

Pretend that the laws are Newton’s. According to Lewis, this is because i) Newton’s laws are the best systematization of the facts about mass and the spatiotemporal relations, and ii) mass and the spatiotemporal relations are all and only the natural properties and relations. But one might wonder why ii) is needed. Why not i) alone?

Freed from the pretense that Newton’s view is correct, the proposal is this: there is a certain set of properties and relations, $S$, such that what it is to be a law is to be the best systematization of the properties and relations in $S$. How do we figure out which properties and relations these are? Follow the same advice that a Lewisian would give for figuring out which properties and relations are natural: look to predicates of well-confirmed scientific theories.

If I myself accepted a “lite” best-system theory, it would be this one.

Cohen and Callender (2009, section 3) discuss a related view, and it may even be the same view, except that they describe it as saying that we “stipulate” which properties and relations the laws must systematize. It’s unclear what that means, but in any case, the view I mean to discuss has nothing to do with stipulation or any other speech act. It is a purely metaphysical view: to be a law just is to be entailed by the best systematization of $S$.

Two objections come immediately to mind. First, the view is inconsistent with the modal status of lawhood: it is metaphysically possible that there exist properties and relations other than those in $S$ (properties that are “alien” to the actual world), which figure in the laws of physics. Second, the view is inconsistent with the epistemic status of lawhood. Supposing one of the members of $S$ to be the property of spin, Newton believed that it was a law that $F=ma$, but he didn’t believe that “$F=ma$” is part of the best systematization of any set that includes spin.⁴² Both objections are weighty, but neither is decisive.

The modal objection will be decisive for many. It isn’t for me, since in my view, modal considerations don’t carry weight in this domain.⁴³ But let’s set this issue aside.

The epistemic objection is problematic because epistemic contexts are “opaque”. One can believe that one is in pain without believing that one’s C-fibers are firing, even if pain $=$ C-fibers firing. To be sure, it is an open question how to theorize this opacity. But it is much too quick to reject the view solely on this basis. If the argument were correct, it could be used to refute any reductive theory of lawhood: “I could believe that such-and-such is a law without believing that such-and-such is a $\varphi$; therefore lawhood $\ne \varphi$”.

There is a third, more powerful, objection: that there must be some explanation for why the particular properties in $S$ play the role that they do in an account of the laws of nature. Such an explanation can be provided by Lewis: those properties play the role that they do because they are the natural properties. But for the view under discussion, no explanation can be given.

Not every fact has an explanation. Lewis himself has no answer to the question of why naturalness is part of a constitutive account of lawhood. That is simply what lawhood is, Lewis must presumably answer; but one could make a similar answer on behalf of the members of $S$.

Thus the objection must rest on a judgment about the comparative explanatory merits of two theories. Each is a theory of a range of phenomena centered on lawhood: facts about lawhood, causation, counterfactuals, and other matters. One theory explains all of these phenomena by reference to the members of $S$; the other theory explains them by reference to naturalness; and the judgment behind the objection is that the latter theory is explanatorily better.

One could regard the view under discussion as being, not a way of doing without naturalness, but rather, a way of reductively defining naturalness: to be natural is to be a member of $S$—to be mass or charge or spin or spatiotemporal separation, perhaps. Naturalness is given by list.⁴⁴ Now, Lewis’s view was that naturalness is the basis for theorizing about an even wider range of phenomena, encompassing not only lawhood, causation, and counterfactuals, but also semantic and mental content, intrinsicality, materialism, and practically everything else he thought about after the early 1980s.⁴⁵ For a fan of Lewis’s program, the third objection rests on a judgment about the relative explanatory merits of the following two theories: Lewis’s own, which is centered on a non-list-like notion of naturalness, and a theory like Lewis’s but in which ‘natural’ is replaced by ‘is mass or charge or spin or spatiotemporal separation’ (or whatever the members in $S$ happen to be).

“Explanations by list” typically aren’t explanations at all. One cannot explain the fact that everyone attending my party was a philosopher by citing the fact that my party was either party₁ or party₂ or …[listing all and only parties that are in fact attended only by philosophers], and the fact that each of these was attended only by philosophers. But perhaps the fact that the very same list recurs throughout the Lewisian constellation of explanations—lawhood is a matter of mass or charge or spin or spatiotemporal separation, and so is causation, and so is mental content, and so is…—turns what otherwise would be a bad explanatory strategy into a good one.⁴⁶

It is no accident where our discussion has landed. A great many metaphysical debates, especially those between more and less “inflationary” outlooks, turn on whether a given posit is explanatory. Indeed, the debate over the best-system theory itself is an instance. Lewis’s critics (like Armstrong (1983, chapter 5)) regard his view of laws as being too anemic, and argue that without positing robust laws, genuine explanation is impossible, whereas Lewis regards the extra posits as having no explanatory added value. And the natural home for opposition to naturalness is the view that it, too, has no explanatory added value. Perhaps it can be replaced, in Lewisian explanations, by a list, or perhaps the Lewisian program itself lacks explanatory value. But for fans of the Lewisian program who reject these list-like explanations, Lewis’s original best-system theory, the one with naturalness, the OG BST, remains the best reduction of lawhood.⁴⁷