An adequate account of laws should satisfy at least five desiderata: it should provide a unified account of laws and chances, it should yield plausible relations between laws and chances, it should vindicate numerical chance assignments, it should accommodate dynamical and non-dynamical chances, and it should accommodate a plausible range of nomic possibilities. No extant account of laws satisfies these desiderata. This paper presents a non-Humean account of laws, the

This paper defends a new account of laws, the

The Nomic Likelihood Account satisfies all of these desiderata. In broad strokes, the nomic likelihood account proceeds as follows. First, it posits a single fundamental nomic relation—the “nomic likelihood” relation—which satisfies certain constraints. Then it characterizes laws and chances in terms of this relation. So on this account, laws and chances end up being things that encode facts about the web of nomic likelihood relations.

I’ll present the Nomic Likelihood Account in a largely theory-neutral manner. The main assumption I’ll make, following Lewis (

Here is a road map for the rest of this paper. In Section 2, I spell out the desiderata on an adequate account of laws sketched above. After presenting and motivating these desiderata (Section 2.1), I suggest that none of the extant accounts of laws satisfy these desiderata, and show how several popular accounts fail to do so (Section 2.2). In Section 3, I offer an intuitive sketch of the Nomic Likelihood Account. In Section 4, I present the nomic likelihood relation and the constraints I take this relation to satisfy. In Section 5, I present a representation and uniqueness theorem showing that the pattern of instantiations of the nomic likelihood relation can be uniquely represented by things that look a lot like laws and chances (Section 5.1). This theorem has some unique features that are of independent interest: it can distinguish between nomically forbidden events and chance

I’ll now present five desiderata that I think must be satisfied by any adequate account of laws. While I’ll briefly motivate these desiderata, I won’t engage in an extended defense of them here. Those who are inclined to contest some of these desiderata can understand my case for the Nomic Likelihood Account as taking conditional form:

An adequate account of laws should provide a unified account of laws and chances. It should allow for both probabilistic and non-probabilistic laws, and it should recognize non-probabilistic laws as a limiting case of probabilistic laws. That is, it should recognize that nomic requirements/forbiddings and chances are of a kind, differing only on where they lie on the spectrum of nomic likelihood, with nomic requirements at one end, nomic forbiddings at the other, and non-trivial chances in-between. Moreover, it should do this without conflating being nomically required/forbidden with having a chance of

An adequate account of laws should yield plausible relations between laws and chances, laws and other laws, and chances and other chances. For example, it should entail that nomically required events have a chance of

An adequate account of laws should provide a satisfactory explanation for why chance events deserve the numerical values we assign them. That is, it should provide an account of the metaphysical structure underlying chances that explains why these numerical assignments are a “good fit” with the underlying metaphysical reality.

To get a feel for what this desideratum requires, let’s consider an unsatisfactory attempt to meet this demand. Suppose one tried to satisfy this desideratum by stipulating that, as a primitive fact, the world has a nomic disposition of

At first glance, this would seem to amount to positing a fundamental “nomic disposition” relation between one state of affairs, another state of affairs, and the number

An adequate account of laws should be able to accommodate both dynamical chances—such as those of the GRW interpretation of quantum mechanics—and non-dynamical chances—such as those of statistical mechanics.

An adequate account of laws should be able to make sense of a plausible range of nomic possibilities. For example, it should be able to make sense of laws concerning particular locations, times, or objects, like the Smith’s garden case discussed by Tooley (

While this is a desideratum that many accounts of laws and chances fail to fully satisfy (see Section 2.2), it’s most notably violated by Humean accounts—accounts on which the laws and chances supervene on the distribution of local qualities. For example, such accounts cannot make sense of uninstantiated laws, nor can they distinguish between worlds which differ only with respect to their chance assignments. Humeans take this to be a bullet worth biting in order to avoid positing fundamental nomic properties or powers. As such, Humeans won’t take desideratum 5 to be a

To my knowledge, no existing account of laws satisfies the five desiderata described above. Due to space constraints, I won’t try to provide an exhaustive discussion of the existing accounts and why they fall short. Instead, I’ll just briefly discuss seven prominent accounts, and flag the desiderata that each fails to satisfy.

1. Carroll’s (

Likewise, simply stating that it’s a primitive fact that a certain event has a chance of

2. Lewis’s (

3. Armstrong’s (

4–5. Swoyer’s (

The preceding discussion suggests that most extant accounts of laws have particular trouble satisfying desiderata 2 and 3. This is likely because these accounts have largely focused on non-probabilistic laws, with probabilistic laws being something of a sideshow. So I’ll conclude by assessing two accounts of chances that do better with respect to desiderata 2 and 3. Since these accounts are only intended as accounts of chance, they won’t provide a unified account of laws and chances (desideratum 1), nor say everything we’d like about how laws/chances bear on other laws/chances (desideratum 2). But it’s worth seeing how they fare.

6. Suppes’s (

Suppes’s account fails to satisfy desiderata 1 and 2 for the reasons given above—since it only provides an account of chances, not laws and chances, it doesn’t provide a unified account of laws and chances, or the relationships between them. Moreover, Suppes’s account doesn’t provide a unified account of chances. For Suppes takes different probabilistic phenomena to impose different kinds of constraints, and goes on to provide different representation theorems for these different phenomena. Thus Suppes’s account of chances is highly heterogeneous.

Suppes’s account does better with respect to desideratum 3, making substantial progress with respect to explaining what, at the fundamental level, makes chance events deserve the numerical values they’re assigned. Unfortunately, it still falls short of providing a satisfactory justification. For while Suppes’s approach yields a representation theorem, it doesn’t yield the uniqueness theorem required to show that these numerical representations are unique. Thus this account doesn’t justify our assigning the particular numerical values that we do.

7. Konek’s (

But Konek’s account fails to satisfy desiderata 1 and 2 for reasons we’ve already seen—since it’s not an account of laws and chances, just chances, it doesn’t provide a unified account of laws and chances, or describe the relations that hold between them. Moreover, Konek’s account also doesn’t yield all of the relations between chances that one would like. For example, it doesn’t say anything about how dynamical chances at different times are related.

Finally, Konek’s account fails to recognize some plausible nomic possibilities (desideratum 5). It seems possible for there to be a world with only one chance event—a coin toss—with a chance of

Let’s start by sketching the intuitive picture behind the Nomic Likelihood Account.

It’s natural to think that laws and chances are of a kind. Deterministic laws tell us that if one state of affairs obtains, then another state of affairs is nomically required to obtain. Chances tell us that if one state of affairs obtains, then another state of affairs has a certain nomic likelihood of obtaining. And nomic requirements and nomic likelihoods seem to be instances of the same kind of thing. Nomic requirements are just what you get when you turn the nomic likelihood “all way up”.

Now, the nomic likelihood of one state of affairs given another is a quantitative feature of the world. You can have different degrees of nomic likelihood. And these degrees can be characterized in precise, numerical ways—one state of affairs can be twice as likely as another, for example. So what undergirds these quantitative features of the world? What’s the metaphysical structure underlying nomic likelihoods?

The view I propose takes its cue from a popular account of quantitative properties like mass.

The Nomic Likelihood Account adopts a similar approach to nomic likelihood. In the case of mass, what bears a quantity of mass is an object.

Now consider a triple that has a certain nomic likelihood—at this world, given this state of affairs, there’s such-and-such likelihood of this other state of affairs coming about. What undergirds the fact that this triple has

Of course, a satisfying account has to do more than just gesture at certain relations. Return to the case of mass. A satisfying account of quantities of mass has to do more than gesture at some mass relations. It has to tell us what these relations are, what these relations are like, and how these relations vindicate taking masses to be quantitative, i.e., vindicate assigning numerical values to these quantities in the way that we do. And this is what accounts of quantitative properties like mass do. They propose certain fundamental mass relations, present some “axioms” that describe how these relations behave, and provide a representation and uniqueness theorem showing that these relations vindicate our using numbers to represent the amount of mass things have in the way that we do.

Providing a satisfying account of nomic likelihood requires doing something similar. We need to spell out what the fundamental relations are, what these relations are like, and how these relations vindicate assigning numerical values to chances in the way that we do. This is what I’ll do in the next two sections. I’ll spell out the fundamental nomic likelihood relation, present some “axioms” describing how this relation behaves, and provide a representation and uniqueness theorem showing that these relations vindicate our using numbers to represent amounts of nomic likelihood in the way that we do. And with an account of nomic likelihood in hand, it’s straightforward to provide an account of laws and chances.

While proponents of the Nomic Likelihood Account can remain neutral about many metaphysical debates, it’s hard to sketch an intuitive picture of the view in a theory-neutral manner. So I’ve made some assumptions in this Section while presenting the picture; for example, I’ve appealed to things like Chisholm-style states of affairs. But these aren’t assumptions that the Nomic Likelihood Account is wedded to; we’ll return to discuss some alternative approaches in Section 7.

In this Section I’ll present the key posit of the Nomic Likelihood Account, the nomic likelihood relation. In Section 4.1 I’ll introduce the nomic likelihood relation. In Section 4.2 I’ll introduce some helpful terminology. In Section 4.3 I’ll describe the constraints (i.e., axioms) that I take the nomic likelihood relation to satisfy.

Two comments before we get started. First, in Section 3 I talked about nomic likelihoods in terms of states of affairs. As it turns out, it will be formally more convenient to characterize nomic likelihoods in terms of propositions instead of states of affairs. But this is purely for convenience—we could formulate everything in terms of states of affairs instead, albeit in a slightly clunkier way.

Second, it’s worth saying something about the representation and uniqueness theorem this approach employs in order to help the reader understand the motivation for some of the axioms. The measurement theory literature contains a number of representation and uniqueness theorems which take an ordering relation that satisfies certain constraints, and show that there’s a unique numerical representation that lines up with that relation. Given this, working out the axioms of the nomic likelihood relation and providing a representation and uniqueness theorem for it seems like a straightforward task. All that’s required to complete this project, it seems, is to take one of these formal results and change its interpretation.

Unfortunately, none of the results in the literature can do the work required, for two reasons. First, none of the results in the literature I’m aware of can distinguish between having a probability of

Second, all of the theorems in the literature I know of require strong “richness” assumptions in order to derive their result.

The framework I’ll present will allow us to distinguish between having a chance of

Here is the fundamental posit of the Nomic Likelihood Account:

The last clause ensures that the propositions the nomic likelihood relation holds of aren’t themselves about nomic facts. I take this constraint to be independently plausible, and it ensures that we won’t run into self-reference paradoxes. Now let’s turn to the 12 nomic axioms that the nomic likelihood relation is required to satisfy.

Let me start by introducing some terminology.

Let

At the risk of abusing notation, I’ll often express the nomic likelihood relation in terms of these triples. Thus I’ll use

Let

Let the

With this notation in hand, let’s turn to the 12 nomic axioms.

1. We haven’t imposed any constraints on which consequent propositions

If

If

Formally, this axiom ensures that for every non-empty

2. Nothing we’ve said so far requires all triples with nomic likelihoods to be comparable, or requires comparisons between triples to be transitive. For all we’ve said, it could be the case that it raining (given meteorological conditions

Formally, this axiom ensures that the nomic likelihood relation provides a weak ordering of

3. The previous axioms haven’t imposed any constraints on how the nomic likelihoods assigned to members of different

If

If

4. So far, nothing we’ve said requires there to actually

There is a pair of triples in

(a)

(b) For all

(c) For all

There are no

(a)

(b)

(c)

(d)

For any

This is an important axiom, so it’s worth talking through what it says in a bit more detail. This axiom posits the existence of a “rich” cluster,

The first clause entails that in this rich cluster there’s (i) a “next highest” rank of nomic likelihood, which sits below

It’s worth emphasizing that

In what follows, it will be convenient to have a name for triples

The second clause is the analog of the standard “atomless” assumption.

The third clause ensures that every degree of nomic likelihood is instantiated in

5. Intuitively, nomic likelihoods should satisfy something like a qualitative notion of additivity. For example, given meteorological conditions

It’s worth flagging two ways in which this qualitative additivity axiom differs from typical qualitative additivity axioms. First, typical qualitative additivity axioms don’t include conditions (i)–(iii). But the introduction of

6. The sixth axiom plays an important role in establishing the representation and uniqueness theorem, but it’s a bit harder to get an intuitive grip on than the other axioms. Consider a sequence of triples from some cluster that’s “expanding”, in the sense that the consequent proposition of each triple in the sequence is entailed by the consequent propositions of all the earlier members of the sequence. And suppose some other triple

Formally, this axiom ensures that the

7. So far we’ve said little about how the nomic likelihoods of triples on a par with

In particular, the seventh axiom entails that adding things on a par with

If

If

8. We haven’t yet imposed any requirements tying nomic likelihood to truth. For all we’ve said, it could be the case that if meteorological conditions

9. Nothing we’ve said so far has imposed conditions tying the fact that

10. Nothing we’ve said so far has imposed conditions tying the fact that

For all

11. The previous axioms haven’t imposed any constraints on what triples there are in different

12. Axiom 11 ensures that clusters at the same world have consequent propositions that line up with each other. But while axiom 11 ensures that these clusters will assign nomic likelihoods to the appropriate propositions, we haven’t yet said anything about what the magnitudes of these nomic likelihoods should be. Suppose that given meteorological conditions

Formulating the twelfth axiom precisely requires a little stage-setting. Let an

Intuitively,

This axiom ensures that if

Some key lemmas that follow from the axioms are described in appendix A.1. The proofs of these lemmas are given in appendix A.2.

In this Section I finish developing the Nomic Likelihood Account. In Section 5.1 I’ll present a representation and uniqueness theorem regarding the nomic likelihood relation. In Section 5.2, using these results, I’ll present the Nomic Likelihood Account of laws and chances. In Section 5.3 I’ll present some consequences of this account regarding laws and chances. And in Section 5.4 I’ll present a toy example of some complete laws given the Nomic Likelihood Account.

Before we proceed, it’s worth sketching the role that the representation and uniqueness theorem plays in this account. It’s helpful to start with an analogy. In the decision theory literature, people have offered representation and uniqueness theorems showing that if a subject’s preferences satisfy certain conditions, then there’s a (more or less) unique pair of functions that line up with these preferences in the way you’d expect rational credences and utilities to line up with them. One popular account of credences and utilities identifies them with the functions picked out by these theorems.

Similarly, the representation and uniqueness theorem described in Section 5.1 shows that if the nomic likelihood relation satisfies certain conditions, then there’s a unique function and pair of relations that line up with these nomic likelihood relations in the way you’d expect chances and nomic requirements/forbiddings to line up with them. The Nomic Likelihood Account identifies chances and nomic requirements/forbiddings with the function and relations picked out by the theorem. On this account, chances and nomic requirements/forbiddings are just things that encode facts about the web of nomic likelihood relations. And if we adopt this account, the theorem provides a straightforward explanation for why chances deserve the numerical values we assign them—because these are the only numerical assignments that line up with the nomic likelihood relations in the right way.

We can partition the space of worlds such that two worlds

The following theorem is shown in appendix B:

(a)

(b)

(c)

Furthermore, the function

This theorem shows that the nomic likelihood relation can be uniquely represented by a countably additive probability function

Given the representation and uniqueness theorem, we can provide an account of laws, chances, and nomic requirements and forbiddings, as follows.

It will be convenient to follow Lewis (

The Nomic Likelihood Account then identifies chances, nomic requirements and nomic forbiddings with the

The second desideratum discussed in Section 2.1 was that an adequate account should yield plausible connections among laws and chances. We can now show some of the ways in which the Nomic Likelihood Account satisfies this desideratum by describing some further lemmas that follow from the nomic axioms described in Section 4.3, and the account of laws and chances offered in Section 5.2. (The numbering of these lemmas starts at 10 because they follow the 9 lemmas given in appendix A.1. The derivations of these lemmas are given in appendix C.)

10. If (given

11. It seems like nomic requirements should be closed under entailment. For example, if (given

12. It seems nomic requirements and nomic forbiddings should be linked: if

13. It seems like nomic requirements and forbiddings should be tied to the truth. For example, if (given

If

If

14. It seems nomic likelihoods should be tied to chances. For example, if (given

If

If

If

15. It seems related chance distributions should assign chances to the same propositions. For example, suppose

16. It seems related chance distributions should have related chance assignments. For example, suppose

It can be helpful to see a concrete example of some complete laws

Let

The complete laws of

We can also consider a world

Now let’s turn to see how the Nomic Likelihood Account fares with respect to the five desiderata given in Section 2.1.

The Nomic Likelihood Account provides a unified account of laws and chances, characterizing both in terms of the nomic likelihood relation (cf. Section 5.2). Probabilistic and non-probabilistic laws are treated similarly, with the laws that impose nomic requirements just being stronger versions of the laws that impose chances. And the Nomic Likelihood Account is appropriately discriminating, distinguishing between propositions that are nomically required and propositions that have a chance of

The Nomic Likelihood Account yields the kinds of relations between laws and chances that one would expect (cf. Section 5.3). For example, it entails that nomically required propositions are not nomically forbidden, and vice versa; it entails that nomic requirements are closed under entailment;

The Nomic Likelihood Account provides a satisfactory explanation for why chance events deserve the numerical values we assign them. At the fundamental level we have various instantiations of the nomic likelihood relation which satisfy certain constraints (cf. Sections 4.1 and 4.3). And we have a representation and uniqueness theorem that shows that there is exactly one way of assigning numbers in the

The Nomic Likelihood Account itself doesn’t appeal to a distinction between “dynamical” and “non-dynamical” chances. But we can distinguish between different kinds of chances, and see what the Nomic Likelihood Account entails about them.

Here is one way to draw such a distinction. Let’s say that a world

Given this characterization of dynamical chances, the Nomic Likelihood Account will entail that dynamical chances will have the features they’re expected to have. For example, the Nomic Likelihood Account will entail that worlds with dynamical chances can’t have deterministic laws. If

Likewise, the Nomic Likelihood Account will entail that at worlds with dynamical chances, propositions about the past can only be assigned a chance of

By contrast, the Nomic Likelihood Account will allow worlds with deterministic laws to have non-dynamical chances, and so can accommodate classical mechanical worlds with statistical mechanical chances. For example, let the laws of

Likewise, the Nomic Likelihood Account doesn’t require non-dynamical chances to assign propositions about the past a chance of

The Nomic Likelihood Account can accommodate a wide range of plausible nomic possibilities. For example, since the only kind of consequent proposition the account can’t assign nomic likelihoods to are propositions concerning nomic facts (Section 4.1), the account allows nomic likelihoods to be assigned to propositions about particular locations, times, and objects. Thus the account allows for laws about particular locations, times, and objects, like the case of Smith’s garden discussed by Tooley (

Let’s turn to assess some worries one might raise for the Nomic Likelihood Account.

Second, although I’ve characterized the nomic likelihood relation as taking propositions and worlds as relata, one could characterize the relation in other ways to avoid these commitments. If one doesn’t like propositions, one could replace the appeal to propositions with an appeal to properties, i.e., the property of being a world at which the relevant proposition is true. Or one could replace the appeal to propositions with an appeal to Chisholm-style states of affairs.

Likewise, if one doesn’t like worlds, one could replace the appeal to worlds with an appeal to propositions, i.e., the maximally specific propositions describing that possibility. (On this approach, of course, one would not identify propositions with sets of worlds.) Or one could replace the appeal to worlds with an appeal to very detailed properties or states of affairs. These alternative characterizations of the nomic likelihood relation would require only superficial modifications to the details presented in Sections 4 and 5.

Likewise, following David Lewis (

There are three ways for the proponent of the Nomic Likelihood Account to reply to the worries raised above. These replies mirror the options available to the proponents of the popular measurement theoretic account of quantitative properties just described. They can (1) challenge the characterizations of duplication and intrinsicality given above, (2) modify the posits the theory makes, or (3) bite the bullet. I won’t discuss the third reply,

(1) Let’s start by distinguishing between two kinds of relations. First, there are relations that only hold between things located at the same possible world; call these

Intuitively, qualitative duplicates are perfectly alike “in and of themselves”. That is, duplicates must share their monadic fundamental properties. By contrast, duplicates need not be alike in how they are connected to other things—two copies of a book may differ in their spatiotemporal relations to me and still be duplicates. That is, duplicates can differ with respect to their fundamental connecting relations. But these two truisms leave open the question of whether duplicates should be alike with respect to their non-connecting relations. One thought is that duplicates must also be alike with respect to their fundamental non-connecting relations. So in order for two objects to be duplicates, they must not only share their monadic fundamental properties, they must also stand in the same kinds of fundamental non-connecting relations—e.g., they must bear the more-mass-than relation to the same things.

This suggests an alternative to Lewis’s account of duplication. Let’s say that a pair of objects

We saw above that given a popular measurement theoretic account of mass, the up quark and the charm quark will be

Likewise, on the Nomic Likelihood Account, two otherwise identical worlds with different laws will be

Turning to intrinsicality, let’s say that a property is

Likewise, proponents of the Nomic Likelihood Account can maintain that the property of being a world where the laws are

(2) Those who would prefer to keep Lewis’s characterizations of duplication and intrinsicality can respond to this objection in a different way.

As we saw above, according to a popular measurement theoretic account of quantitative properties, things that differ solely with respect to their quantitative properties (e.g., the up and charm quarks) will be

We can avoid the analogous worries for the Nomic Likelihood Account presented in Sections 3–5 by modifying it in a similar fashion. Namely, instead of positing one layer of fundamental nomic likelihood properties—fundamental nomic likelihood relations over worlds and propositions—we can posit two layers of fundamental nomic likelihood properties—fundamental monadic nomic properties instantiated by worlds, and fundamental second-order nomic likelihood relations that hold between these monadic properties and propositions. In this two-layer picture, the monadic properties will intuitively line up with the complete laws instantiated by that world,

I don’t have any strong intuitions about whether the holistic or the local picture is correct.

For example, it’s true that this account will take the dart landing on the

Likewise, if the probability measure representing the chances is absolutely continuous with respect to some other salient

I’ve suggested (Section 2.1) that an adequate account of laws should satisfy five desiderata: it should (1) provide a unified account of laws and chances, (2) yield plausible relations between laws and chances, (3) explain why we’re justified in assigning numerical values to chance events in the way that we do, (4) allow for both dynamical and non-dynamical chances, and (5) allow for an appropriately expansive range of nomic possibilities. I’ve argued (Section 2.2) that no extant account of laws satisfies these desiderata.

In this paper I’ve developed an account of laws, the Nomic Likelihood Account (sections 3–5), that satisfies all five desiderata (Section 6). On this account, the fundamental nomic property is a nomic likelihood relation. And laws and chances are things that encode facts about the web of nomic likelihood relations. As I’ve noted, there are various challenges one might raise for this account (Section 7). But I think this is ultimately the most attractive account of laws and chances on offer.

If

If

If

If

If

If

While the lemmas in section A.1 are ordered thematically, the proofs are presented in order of dependence (with later lemmas depending on earlier ones, but not vice versa). Most of these proofs implicitly appeal to axioms like 1 and 2 to discharge the existence assumptions of the other axioms they employ; to avoid needless clutter, I’ll leave such appeals implicit.

•

(2) The second part of the lemma follows from the first and the assumption that it’s also not the case that (iv)

First, let’s establish that if

Second, let’s establish that if

•

•

•

Second, let’s establish that if

Note that

Note also that

We’ve only shown this result, though, in cases where none of (i)–(iii), (i*)–(iv*) obtain. To establish the result in full generality, we need to show that in each of these cases lemma 8 will still hold. So suppose

(i) Suppose

(ii) Suppose

(iii) Suppose

(i*) Suppose

(ii*) Suppose

(iii*) Suppose

(iv*) Suppose

This representation and uniqueness theorem can be broken down into three steps. First, I’ll show that given the nomic likelihood relation, we can define a relation

•

Define

If

If

(2) If

(i) Suppose

(ii) Suppose

(iii) Suppose

(iv) Suppose

For all

(2) It follows from lemma 1 that for all

Now, suppose that

If both sides of niff are true, then since

What if both sides of this niff are false? For the left hand side of niff to be false, one of the following three possibilities must obtain: (a)

(a&a*): Suppose

(a&b*): Suppose that

Now, note that the fact that

Since

(a&c*): Suppose

(b&(a*)-(c*)): Suppose

(c&(a*)-(c*)): Suppose

(a)

(b)

(i) If for all

(ii) If for some

(iii) Finally, suppose that

Note that this is equivalent to the condition that

We’ve established that the nomic likelihood relation over

Now, strictly speaking

In what follows I’ll speak loosely of

1. Recall that in order for

It follows from part 3 of axiom 4 that every triple in _{k} with. Since part 3 of axiom 4 entails that there will be such a

2. Now let’s establish that this

The first probability axiom requires that every assignment in

The second probability axiom requires every

Let’s establish that the third probability axiom is satisfied in two steps, first (a) showing that

(a) Let’s start by showing that

By part 3 of axiom 4, the rich algebra

Since the left hand side of both biconditionals are true, it follows that

To derive this result, we assumed that none of the conditions (i)–(iii) obtained. Now let’s relax that assumption, and show that it will still be the case that

(i) Suppose that

(ii) Suppose that

We know

Now, since

It follows from lemma 2 and the fact that

(iii) Suppose that

(b) Now let’s establish that

It follows from a result by Villegas (

Since

It will also follow that

Together, these results entail (by lemma 1) that

Likewise, suppose

(2) The representation and uniqueness theorem entails that

(3) It follows from the representation and uniqueness theorem that

(i): Since

(ii): Since

Now, the members of an

It follows from the above that if

It follows from this that if we know the values of

Now, this only shows that we can identify the unique real number

• Given lemma 18, we can now prove lemma 16 as follows. Suppose

If

What if either of the denominators of

For helpful comments and discussion, I’d like to thank Maya Eddon, Alejandro Perez-Carballo, two anonymous referees, and participants of the Fall 2019 UMass Brown Bag group, the 2019 Rutgers Conference on the Philosophy of Probability, the 2021 Canadian Society for the History and Philosophy of Science conference, and the 2021 Formal Philosophy conference. I’d also like to thank Robert Mason for excellent proof-reading and typesetting advice.

Some have argued that instead of taking the distinction between fundamental and non-fundamental properties to be primitive, one should take something like a grounding relation to be primitive, and characterize the fundamental properties in terms of this grounding relation (e.g., see

I speak loosely here of chance

For further worries regarding such appeals to fundamental relations to numbers, see section 4 of Eddon (

The claim that an adequate account of laws should be able to accommodate non-dynamical chances is somewhat contentious, but it’s been defended by a number of people, including Loewer (

Some have suggested understanding non-dynamical chances, such as those of statistical mechanics, as measures of rational indifference. If one adopted this stance, then one could dispense with this fourth desideratum, since one would only need an account of laws to accommodate dynamical chances. But there are well-known reasons for being skeptical of this understanding of statistical mechanical chances. For some of these reasons, see Strevens (

Maudlin’s (

Though there are variants of Lewis’s proposal that allow for such chances; e.g., see Loewer (

For a discussion of this and other ways of understanding Armstrong’s account of probabilistic laws, see Jacobs and Hartman (

Tooley’s (

Lange’s (

See Suppes (

This is something Suppes takes to be a merit of his account. For he takes the expectation that there will be some unified account in the offing to be wrong-headed. Like much of the contemporary literature, I’m inclined to disagree.

Of course, it would be unfair to raise any of this as a criticism of Konek. Konek’s goal is simply to show that proponents of propensity accounts of chances can provide a principled story for why they expect propensities to satisfy the probability axioms. And just as it would be unfair to criticize Konek for presenting a view which doesn’t provide an account of laws (since Konek wasn’t trying to provide an account of laws), it would be unfair to criticize Konek for failing to yield relations between dynamical chances at different times (since Konek wasn’t trying to provide a comprehensive account of chances).

For a survey of different accounts of quantitative properties, see Eddon (

Assuming we’re taking objects to be world-bound. If we don’t, then since an object’s mass can vary from world to world, we might take the bearer of mass to be an object and world pair.

For those familiar with the literature on quantitative properties, the account of nomic likelihood described here is analogous to the version of the first-order relations account of quantitative properties discussed by Eddon (

For example, we can replace the role of states of affairs with properties or propositions (as I do in Section 4), or replace the role of worlds with states of affairs or properties.

For discussion of some different ways of characterizing the nomic likelihood relation, see Section 7.

I’ll use the term “set” here loosely to cover both sets and classes.

For some discussions of worries regarding these richness axioms in the context of standard theories of quantitative properties, see Melia (

An

Though this is not all it entails; it also entails that for any two disjoint triples in any cluster, there are two disjoint triples in

Where I’m assuming here that raining, snowing, and being sunny are mutually exclusive.

We need to add these restrictions because if any of (i)–(iii) obtain, we can construct counterexamples to the additivity claim (that

A third and more subtle way in which it differs from typical qualitative additivity axioms is that it doesn’t require

To see why the

That is, an

For classic presentations, see Savage (

As it turns out, only the first seven nomic axioms are required to obtain this result. The last five nomic axioms only come into play when deriving the lemmas regarding laws and chances given in Section 5.3.

One would typically express these relations as

That is, for all

For all

For any

For any sequence

Given this account of complete laws, how do we determine whether a given proposition (e.g., a statement of Newton’s gravitational force law) is a law? Presumably a necessary condition is that it should be entailed by the complete laws. One might take this to be a sufficient condition as well, or one might add various other requirements—that it express a regularity, be appropriately general, etc. From the perspective of the Nomic Likelihood Account, this is merely a terminological matter—what really matters, metaphysically speaking, are the complete laws. (In a similar vein, there won’t be an interesting distinction to draw between “fundamental laws” and “derived laws” on the Nomic Likelihood Account [

Assuming that the entailed propositions bear any likelihood relations at all.

Or at relativistic worlds, propositions describing a complete history up to some Cauchy slice.

Although this is one way to draw the distinction between “dynamical” and “non-dynamical” chances, it is not the only way. A different (and to my mind, equally reasonable) way to draw the distinction is to call these chances

On some ways of characterizing determinism, such as Lewis’s (

Following Albert (

I’m evaluating the claim that “all propositions about the past get a chance of

See Chisholm (

A related complaint is that the account described in Sections 3–5 commits one to taking propositions and worlds to be more fundamental than (say) chance events and lawful states of the world at a time. But, one might argue, the latter should be more fundamental than the former—for example, it’s natural to take chance events to be more fundamental than the propositions describing them. (I owe an anonymous referee for raising this concern.)

So far I’ve followed Lewis (

I’d like to thank an anonymous referee for bringing this worry to my attention.

Indeed, the two-layer version of the Nomic Likelihood Account discussed in worry 3—which posits both fundamental first-order “complete law” properties of worlds, and a fundamental second order nomic likelihood relation that holds of these properties and propositions— might naturally be classified as a form of primitivism (cf. footnote 50).

See Shumener (

For some early and influential accounts along these lines see Krantz, Luce, Suppes, and Tversky (

For a defense of this third reply, see Dasgupta (

Lewis’s (

Or, if one takes our ordinary notion of duplication to be vague, that

Or more or less our ordinary notion. For some remaining issues that crop up, see Eddon (

Since this two-layer version of the Nomic Likelihood Account posits a range of fundamental complete law properties in addition to the fundamental nomic likelihood relation, this account might be classified as a form of primitivism about laws. I have no objection to this classification, since I don’t think there’s anything inherently problematic about primitivist accounts. What matters is not whether an account is primitivist, but whether it satisfies the desiderata an adequate account of laws should satisfy.

I’d like to thank an anonymous referee for encouraging me to address this worry.

Some (like

A different response to this objection would be to develop a variant of the Nomic Likelihood Account whose axioms provide a representation and uniqueness theorem that yields nonstandard probability assignments (e.g., hyperreal valued-probabilities). Although this is an interesting avenue for future research, there are some

One might be tempted to construct a second and more fine-grained nomic likelihood relation in light of such facts, and take this to be the “real” nomic-likelihood relation. I think this would be a mistake. For these densities will only be defined with respect to a second measure; so at best they’re providing us with something like comparisons of nomic likelihood

A similar obstacle prevents us from skipping over having to posit the

Of course, this identification requires it to be the case that for all