What, ontologically speaking, are best system laws? Should the laws of the best system account be identified with sentences, or propositions, or perhaps something else? These questions differ from more familiar questions about which criteria we should use to pick out the best system laws: regardless of whether laws are picked out using simplicity, informativeness, tractability, predictability, perfect naturalness, or whatever, there remains the question of what, ontologically, the best system laws are. But very little has been written on this issue; and what has been written is often ambiguous. So in this paper, we present and evaluate three ontological options for best system laws: taking them to be sentences, unstructured propositions, and structured propositions. Surprisingly, the two most cited options—sentences and unstructured propositions—face significant problems. Defenders of the best system account might want to take seriously the idea that best system laws are structured propositions.
There is a substantive ontological ambiguity lurking within the best system account of laws: it is unclear what sorts of objects the laws of the best system are. There is much discussion about how the best system account should be formulated—about the criteria, regarding simplicity and informativeness and so on, used to determine inclusion in the best system.
To help clarify the issue, we assume that the best system is a set of sentences, and then we ask the further question of how those sentences are related to the laws of nature. In other words, are the best system sentences to be
We argue that these different ontologies of laws have surprisingly different costs and benefits, and so authors who endorse a best system account should be circumspect about which ontology to adopt. For instance, proponents of the best system account who take laws to be largely representational and pragmatic (
In section 3, we formulate and evaluate three distinct options for the ontology of laws. The first option, which we explore in section 3.1, is that the laws are sentences. We argue that this view has serious problems—so serious, in fact, that we conclude it is untenable. In section 3.2, we consider laws as unstructured propositions, that is, sets of possible worlds. This view also faces a serious problem: the sets lack the lawlike structures that characterize the best system. In section 3.3, we develop the idea that the laws are structured propositions. Because the additional structure of these propositions can mirror that of the best system, it seems to us that, with some caveats, this view is stronger than the other two. Furthermore, some theories take structured propositions to be comprised of worldly entities rather than linguistic entities, which might appeal to those who think that laws are patterns in the world, rather than our representations of patterns in the world. In this section, we also briefly consider the related idea that laws are those worldly states which make best system sentences true.
Strikingly, the two views which appear most often in the best system literature—taking laws to be sentences and to be (unstructured) propositions—are weaker, we argue, than the view that laws are structured propositions. But regardless of whether readers agree with us about the relative merits of the different accounts, we hope to demonstrate that the ontological choice about best system laws is a substantive one, and that different options come with surprisingly different costs and benefits.
In this section, we summarize a standard version of the best system account of laws. Then we show that defenders of the best system account are not always clear about what they take the laws to be: options range from sentences to propositions to regularities to generalizations.
To begin, we adopt a standard formulation of the best system account: the laws are given by the members of the true deductive systems which best balance theoretical virtues such as simplicity and strength (
Many different variants of the best system account have been proposed in the literature. Some variants invoke different theoretical virtues, or additional theoretical virtues, beyond simplicity and strength (
The issue on which we focus arises for all of these variants of the best system account. And that issue, to a first approximation, is this: what sorts of items are best system laws? In other words, supposing that some particular best system account is successful, and that some collection of statements best balances whatever virtues we take to be important for the laws, what should we say about the relationship between those statements and the laws themselves? Are the statements to be identified with the laws? Or do the statements merely express the laws? Or perhaps the statements are made true by the laws?
Lewis—by far the most influential author on best system accounts of laws—writes in a way that makes it difficult to tell exactly what he takes best system laws to be. In early work, Lewis claims that laws are theorems of each true deductive system that best balances simplicity and strength, where true deductive systems are “deductively closed, axiomatizable sets of true sentences” (
Ten years later, however, Lewis seems to suggest otherwise. He writes that “[d]ifferent ways to express the same content, using different vocabulary, will differ in simplicity. The problem can be put in two ways, depending on whether we take our systems as consisting of propositions (classes of worlds) or as consisting of interpreted sentences” (
Even so, in the same paper, Lewis offers a third interpretive option: namely, that best system laws are regularities. He writes that “[a] law is any regularity that earns inclusion in the ideal system” (
“If we adopt the remedy proposed, it will have the consequence that laws will tend to be regularities involving natural properties. Fundamental laws, those that the ideal system takes as axiomatic, must concern perfectly natural properties. Derived laws that follow fairly straightforwardly also will tend to concern fairly natural properties. Regularities concerning unnatural properties may indeed be strictly implied, and should count as derived laws if so. But they are apt to escape notice even if we someday possess a good approximation to the ideal system. For they will be hard to express in a language that has words mostly for not-too-unnatural properties, as any language must” (
A straightforward reading of the above passage indicates that Lewis takes best system laws to be regularities and not sentences. He draws a clear distinction between the regularities—which
An important aside: if Lewis does indeed think that best system laws are expressed by sentences, it is puzzling that he would use the term “regularities” rather than “propositions.” As is well known, in his other work, Lewis takes sentences to express propositions, and he takes propositions to be sets of worlds. Perhaps Lewis thinks of regularities as propositions, and simply did not say this explicitly. However, the above passage leaves open the possibility that laws could be something like states of affairs, and that those states of affairs make certain propositions—the ones expressed by best system sentences—true. In other words, on such a view, sentences express propositions that themselves are made true by best system laws.
Following Lewis, other authors also endorse a range of different accounts about what best system laws are. To illustrate, we now present a variety of claims by authors who have developed accounts in the Lewisian tradition. In addition to the familiar differences between competing best system accounts, there are also differences in the ontology of the laws. It is clear that there is no univocal “best system” view about what sorts of objects, ontologically speaking, the laws are. In Section 3, we will argue that different answers come with distinctive costs and benefits. For now, we merely present examples of this ambiguity in the literature; note that the examples are meant to be representative, not exhaustive.
To start, we present examples of authors who explicitly take best system laws to be sets of sentences.
Demarest describes the simplicity of best system laws in terms of sentences’ lengths (
Wheeler takes best system laws to be language-relative truths, so it is reasonable to assume Wheeler takes the laws to be sentences. (
Hicks claims that laws are truths which jointly maximize strength and simplicity (
Wilhelm describes some deductive systems, whose members are laws, as containing sentences (
Arguably, Bhogal and Perry also take each best system law to be a sentence, since they describe those laws as members of “axiom systems [formulated] in terms of the base language” (
Next, we present examples of authors who explicitly take best system laws to be propositions of some kind or other.
Loewer writes that “[a]ccording to the [best system account] laws are certain true propositions and equations that are entailed by the ideally best scientific systematization of the totality of fundamental truths of the world” (
Roberts writes that “[t]he laws of nature are those generalizations that belong to the best deductively closed system of propositions” (
Woodward writes that “[t]he best systemization (or systemizations) is (are) the one (or ones) that consist(s) only of true propositions and achieves an optimal balance of simplicity and strength” (
Wilhelm writes that a chancy law is a “proposition which…follows from the best deductive systems” (
Finally, here are examples of authors who explicitly take best system laws to be regularities or generalizations.
Dorst writes that laws of nature are “the regularities of the simplest and strongest systematization of the Humean base” (
Urbaniak and Leuridan write that laws of nature are “those regularities which earn inclusion in the ideal system(s) to which we aspire in science” (
Beebee writes that “laws are those generalizations which figure in the most economical true axiomatization of all the particular matters of fact that obtain” (
Cohen and Callender write that “laws are true generalizations that best systematize knowledge” (
Miller writes that physical laws are “special systematic generalizations” (
These authors do not say whether regularities and generalizations are sentences, propositions, or something else. All three options seem reasonable. Other options seem reasonable as well: perhaps laws are states of affairs, distinct from both sentences and propositions, for instance. We briefly discuss this later and think such an option is worth pursuing in future work.
To be clear: many of these authors do not have a univocal notion of a law even within their own work. Some describe best system laws as sentences and then later as regularities. Others describe best system laws as propositions and then later as generalizations. This lack of consistency provides even more evidence of just how pervasive the ambiguity is within the literature on best system laws. As we argue below, this is not a harmless ambiguity: different ontologies have different strengths and weaknesses. Defenders of the best system should be more careful in how they formulate their accounts of the laws.
In this section, we examine three different views of what best system laws are: sentences (Section 3.1); unstructured propositions (Section 3.2); and structured propositions, or perhaps something more akin to states or facts (Section 3.3). After presenting each view, we discuss its benefits and costs, focusing especially on those that we think will be of particular relevance to defenders of best system accounts in general.
The view that laws are sentences in a best system is a popular and plausible way to interpret Lewis, and is endorsed by many proponents of the best system account.
One benefit of Sentence Law is that sentences are relatively uncontroversial metaphysical posits. While some philosophers may be unwilling to posit abstract meanings which sentences express—that is, to posit propositions (
Unfortunately, there are three serious problems with Sentence Law. The first problem, discussed occasionally in the literature, is this: if best system laws are sentences, then arguably, assessments of those systems’ theoretical virtues must be relativized to languages (
The second problem for Sentence Law is, we feel, even worse. An example will illustrate it. Consider a best system law ‘
Demarest makes a version of this point. As she writes, “[i]t seems that, ‘gravity is inversely proportional to the square of the distance’, states the very same law as, ‘la gravité est inversement proportionnelle au carré de la distance’. But [given a view like Sentence Law] they would be different laws, since they are different sentence types” (
Note that this problem is distinct from the first problem we mentioned. Previous authors have objected to the idea that lawhood is language-relative. That objection might be described as a rejection of the idea that lawhood is, for instance, a two-place relation which obtains between a sentence and a chosen language. The problem we raise here is much worse than that. Given Sentence Law, ‘
This argument is based on the view that in philosophical theorizing, ontological posits should always be accompanied by clear, reasonable identity conditions: as Quine puts it in slogan form, “no entity without identity” (
So the issue, here, is related to—but ultimately different from—the fact that Sentence Law fails to respect certain intuitive locutions about law individuation, like “the law that
One might object by claiming that fans of the best system account should not care too much about whether sentences, taken to be laws, have the kinds of identity conditions which we intuitively associate with lawhood. Perhaps what matters most is whether (i) nomological necessity operators can be placed in front of certain true claims to get other true claims, (ii) certain counterfactual conditions can be truly asserted, and (iii) certain inferences count as valid. So long as an account of laws allows for that, all is fine—regardless of considerations concerning identity conditions.
We offer two different responses to this objection, based on two different ways of understanding the basic concern. Understood one way, this objection endorses an account that posits entities which themselves (i) license placing nomological necessity operators in front of certain true claims to get other true claims, (ii) underwrite true assertions of certain counterfactual conditionals, and (iii) render certain inferences valid. If so, then by Quine’s slogan, this account of laws should be accompanied by identity conditions for the entities posited. And if those identity conditions have unreasonable implications, then that is a cost of the associated theory; and if the identity conditions of other theories—positing other sorts of entities—have more reasonable implications, while still facilitating the use of nomological necessity operators and true assertions of counterfactual conditionals and valid inferences, then those other theories are preferable.
Understood another way, this objection remains neutral, or quietist, on whether any entities should be identified with laws at all. So understood, this objection takes a more pragmatic, linguistic, inferential approach to lawhood: rather than positing entities which are laws, it seeks to reproduce our law-practices. There is, of course, nothing wrong with developing this sort of approach. But this approach addresses questions different from those addressed in this paper. Our central question, which arises throughout the literature on best system accounts, concerns exactly what sorts of items the laws are. Developing a theory of lawhood along purely pragmatic, linguistic, inferentialist lines is certainly worth doing—but that is a project for another time.
The third problem for Sentence Law is that, given our standard formulation of the best system, the existences of multiple languages may preclude the existence of any laws at all. Recall that if there is more than one best system, each law is a member of each deductive system which best balances various theoretical virtues. But because they are written in different languages, ‘
One might respond to these problems by identifying laws with equivalence classes of sentences, rather than with sentences themselves. This response faces its own difficulties, however, stemming from the equivalence relation which it invokes. Is that equivalence relation defined over the class of all actual sentences in all actual languages, or is that equivalence relation defined over the class of all possible sentences in all possible languages? If the former, then this response implies that if we had spoken a non-actual but still possible language, then the laws would have been different. For if we had spoken such a language, then the class of actual sentences would have been different, and so the equivalence classes of sentences would have been different, and so the laws would have been different too; and that is implausible.
If the latter, then a separate issue arises. Basically anything could serve as a linguistic expression in some possible language or other. Any arbitrary squiggle on a page, any sequence of smoke signals, any random flags used in naval communication—even my coffee mug, or a patch of grass, or whatever, is a piece of vocabulary in some possible language. So the class of all possible sentences is just the class of all possible sequences of everything whatsoever. And so this response implies that laws are just sets of all possible sequences; again, an implausible result.
Overall, we take these to be strong reasons to reject Sentence Law. Fortunately, there are other views of best system laws for which these problems do not arise. That is what the next few sections are about.
The following view denies that best system laws are sentences, and claims instead that best system laws are what sentences mean.
According to the orthodox account, endorsed by Lewis, unstructured propositions are sets of worlds (
For starters, the benefits. Unstructured Proposition Law avoids the serious problems for Sentence Law. Unstructured propositions are not language-relative, so the laws are not language-relative either. Furthermore, recall that numerically distinct sentences like ‘
This also demonstrates how Unstructured Proposition Law avoids the third problem for Sentence Law. For two sets of distinct sentences can nevertheless pick out the same set of propositions. When there are ties for the best system, on Unstructured Proposition Law, it would be more accurate to require that a law,
Unstructured Proposition Law faces some problems, however. The first problem is that sets of possible worlds are more metaphysically committal than sentences. Lewis, for instance, famously defends the idea that possible worlds are real and concrete (
The second problem for Unstructured Proposition Law is particularly significant. To set the problem up, let
And of course, by invoking gerrymandered predicates, we can construct many more ways to express
This generates a problem for Unstructured Proposition Law. The problem—which is different from the predicate
Regarding the first interpretation: the problem stems from the bizarreness of the claim that what it is for an unstructured set of worlds—namely,
Here is another way to put the point. The best system account, according to the first interpretation, endorses substantive structural constraints on what it is to be a law, on the constitutive nature of lawhood. Those structural constraints are defined at the level of sentences: they concern the linguistic structures of the sentences which the best system includes. But Unstructured Proposition Law identifies laws with objects—sets of worlds—that are totally structureless. So there is a displeasing mismatch here: to be a law is to be expressible by structured sentences, according to the first interpretation of the best system account; but to be a law is also to be a structureless set, according to Unstructured Proposition Law. This is not a contradiction, but it does seem like an unattractive implication for an account of laws to have. If the structures of sentences are relevant to the constitutive nature of lawhood, it would be better to adopt an account which identified laws with objects whose structures correspond to the structures of the associated sentences.
In the next section, we consider the view Structured Proposition Law: that the best system laws are better thought of as structured propositions. This allows that the structures which the laws themselves have, namely their propositional structure, can ‘match’ the relevant sentential structure. However, there could be a position that occupies a middle ground between Unstructured Proposition Law and Structured Proposition Law. Suppose there are ways to recover ‘structure-like’ aspects of unstructured propositions, by relating those unstructured propositions to structured items which could serve as surrogates for them. For example, perhaps the unstructured proposition “Susie is tall” stands in some sort of structure-securing relation
Yet another way of conveying this issue with identifying laws with unstructured propositions begins by considering the relevant set of worlds. If that set is to be identified with the best system laws, one would think that our best system sentences are best because, among other things, they pick out the right worlds. Indeed, this judgment likely underlies the claim in the previous section that different linguistic items—equations like ‘
Regarding the second interpretation: the problem stems from how unambitious it is to merely seek an extensionally adequate account of laws. The best system account purports to offer more than just a way of picking out what the laws are. There are many ways of picking out the law
This is a significant dilemma for any defender of Unstructured Proposition Law. It is not, we feel, as bad as the problems which Sentence Law faces. But it is troubling enough to motivate looking for another view of what sorts of items the laws are.
Now we consider another view which takes laws to be propositions. But in this case, propositions have complex structure; they are not structureless sets of worlds.
There are several different accounts of structured propositions in the literature (
Structured Proposition Law enjoys several benefits. Structured Proposition Law, like Unstructured Proposition Law, avoids the serious problems for Sentence Law. First, structured propositions are not language-relative. Second, according to Structured Proposition Law, the sentences ‘
In addition, Structured Proposition Law does a nice job of respecting the claims, frequently made throughout the best system literature, that laws are regularities or generalizations. Some structured propositions have the structure of regularities, and so can be reasonably identified with regularities. Given the tuple account, the regularity that all electrons are massive, for instance, is
Moreover, Structured Proposition Law avoids the serious problem which Unstructured Proposition Law faces. The problem, recall, could be phrased as a dilemma: the best system account provides either (i) a constitutive theory of lawhood, or (ii) a merely extensionally adequate theory of lawhood. If (i), and if Unstructured Proposition Law holds, then the lawhood of a certain set is constitutively a matter of that set being expressible by sentences with certain linguistic structures—and that is bizarre. But if (ii), and if Unstructured Proposition Law holds, then the best system account is unambitious, since it fails to provide a theory of what it is to be a law.
Structured Proposition Law avoids all this. For given Structured Proposition Law, there is nothing bizarre about endorsing (i), and so the best system account provides more than a merely extensionally adequate theory of lawhood. And there is nothing strange about claiming that structured propositions are structured in ways which make them particularly well-suited for being expressed by sentences with various linguistic structures. That is, in large part, what structured propositions were posited to do. Much of the initial motivation for positing structured propositions in the first place, is that they are structured items “looking something like the sentences which express them” (
There are different accounts of structured propositions, so the defender of the best system account who wishes to identify best system laws with structured propositions will have to say which account they prefer. Some accounts, such as King’s (
Other accounts, such as Salmon’s (
Unfortunately, the most straightforward formulations of such structured propositions leaves them vulnerable to the Russell-Myhill argument. This argument shows that these accounts—as well as natural generalizations of them—are inconsistent with plausible principles of higher-order logic (
To illustrate, one of the key principles—related to lambda abstraction and beta equivalence—has implications like this: given a predicate ‘
Bacon formulates an account of structured propositions that avoids the Russell-Myhill argument (
Many higher-order resources of Bacon’s account are not needed, in order to represent the propositions which Structured Proposition Law identifies with laws. Just take a fragment of Bacon’s language containing only the constants, variables, and logical expressions which scientific theories need. Plausibly, this fragment will include first-order quantifiers, and maybe some higher-order quantifiers for regimenting claims about all functions: for instance, claims about all wavefunctions in a quantum theory, or claims about all tensor fields in a general relativistic theory. This fragment might also include propositional quantifiers, for regimenting claims about lawhood itself.
To illustrate, consider the claim that lawhood is factive. This claim could be regimented as follows. Let ‘
Other principles of lawhood, endorsed by fans of the best system account, could be formulated in this fragment of Bacon’s account too.
This relates to another benefit of Structured Proposition Law. When combined with an appropriate fragment of Bacon’s account, Structured Proposition Law facilitates the formulation of many different principles which fans of the best system account endorse. The higher-order vocabulary used to define the structured propositions which are identified with laws, in other words, can also be used to rigorously regiment the principles of the best system account more generally. For these reasons, we think that structured propositions provide a promising avenue for further research.
One more approach to lawhood is worth mentioning, although we cannot develop it in detail here. The idea is to accommodate the metaphysical structure of laws by taking them to be truthmakers for best system sentences. That is, identify laws with something like Armstrong’s states of affairs (
Such a proposal would take some worldly entities—namely, states—to be the laws which make best system sentences or propositions true. And so fans of the best system account who dislike identifying laws with representational items may find such an approach attractive. Because state laws would not have linguistic items as components, they would avoid the objections to Sentence Law, as well as the objections to a King-style version of Structured Proposition Law. Because state laws would contain the metaphysical structure required to make the best system sentences true, they could potentially avoid the objections to Unstructured Proposition Law. And because state laws are not propositions, they might avoid the logical inconsistencies of certain versions of Structured Proposition Law generated by higher-order resources.
Having said that, we are not sure what the details of such an account would look like. Armstrong, Fine, and Jago offer very different accounts of states, and each account faces its own unique challenges. For instance, on Fine’s view, states may lack the metaphysical structure required to avoid the kind of objections that face Unstructured Proposition Law. And Armstrong’s view of states, like his account of laws, posits necessary connections between distinct existences—namely, between states on the one hand, and the objects and properties that comprise those states, on the other. Note that while this is an unwelcome posit for those who defend Humean best system accounts, it may be acceptable to those who defend powers best system accounts (
Regardless, any defender of such a view will have to be precise about their ontology of states, and they will have to provide arguments that show why that ontology is the best thing to identify with the laws that make the best system sentences true. One cost that applies to any version of state law is that states are not the sorts of items which can be true or false: states obtain, or fail to obtain, instead. This conflicts with existing views of best system laws, which near-universally stipulate that the laws are true. We leave the further exploration of state laws to future work.
Defenders of the best system account have been unclear about exactly what sorts of items the laws are. After demonstrating this trend throughout the literature, we proposed and evaluated three different options—that the laws are sentences, unstructured propositions, or structured propositions. Sentence Law has some very serious costs; so serious, in fact, that we found it untenable. That is a significant result, since a version of Sentence Law appears frequently in the best system literature. Likewise, it was striking to find that Unstructured Proposition Law also has serious costs. These costs, though not decisive, motivate exploring alternative views of what the laws are, like Structured Proposition Law. We are cautiously optimistic that this latter view can be worked out in more detail with some success. We also suggested that a fourth view—based on states—may be worth developing, though that is left for future work.
Since each view has distinctive costs and benefits, the question of ontology is a substantive one for defenders of the best system. Significantly, the two most popular accounts—Sentence Law and Unstructured Proposition Law—fared the worst, which shows that those writing on the best system can no longer uncritically assert that their laws are sentences or propositions that best balance their preferred criteria. The question of the laws’ ontology should be taken seriously by proponents of the best system account.
For different approaches to best system accounts of lawhood, see (
Note that this way of framing the question assumes that it is sentences which are members of the best deductive system; this allows us to ask the further question of how those sentences are related to the laws. By contrast, one could reject that assumption, and ask which objects—sentences, propositions, or whatever—should be used to best balance simplicity and informativeness. For instance, one set of propositions might be stronger than another if the latter contains the former as a subset. Likewise, it would be interesting to see if structured propositions or states of affairs could be compared for simplicity or strength, or any of the additional virtues that have been proposed for best system laws. While we set this question aside, we encourage those who are drawn to such an approach to develop these different ontological options for laws—particularly in the context of balancing whatever virtues they think make for a best system. We think there is a rich landscape of possibilities here that should be explored.
Authors often add a further restriction on
Urbaniak and Leuridan also note that Lewis recognizes both as options, though their central concerns are different from ours (
Note, also, that these quotations exhibit another ambiguity, corresponding to the question raised and set aside in footnote
In sections 3.2 and 3.3, we will disambiguate the proposition view and explore two different options: propositions that are unstructured sets of worlds and propositions that are structured complexes individuated by the objects and properties which they are about.
The problem here arises regardless of whether sentences are taken to be sequences of symbol tokens or sequences of symbol types. One could take ‘
In response, one might relativize lawhood to languages: the laws, relative to a language
Another point worth flagging: in this paper, we assume that the target analysis of best system accounts has a certain form. In particular, we assume that laws are ontological items which the best system account seeks to analyze. Alternatively, one might think that the target analysis of the best system account is not an item at all, but rather, a locution of the form ‘sentence
Note that this problem does not arise for views which identify worlds with sets (
See Goodman (
More precisely, a version of the tuple account along the lines of a view developed by Soames (
The proof follows directly from the definitions in footnote 13.
According to King’s account, the sentence “Snow is white” expresses a structured proposition which includes, within it, existential generalizations over contexts and languages and lexical items. In particular, King’s account implies that the structured proposition expressed by “Snow is white” is the following fact: there exists a context
We would like to thank Chris Dorst, Katrina Elliott, Alison Fernandes, Jackie Han, Michael Townsen Hicks, Christian Löw, Elizabeth Miller, the audiences at the 2024 Laws and Quantities in Physics Conference in Glasgow and at the 2025 Society for the Metaphysics of Science Conference in Lugano, the participants in the 2024 seminar on laws and causation at the University of Colorado, and two anonymous referees for helpful feedback on this paper.