The Nomic Likelihood Account of Laws

2.1. The Desiderata

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: if one takes these to be desiderata for an adequate account of laws, then we have reason to accept something like the Nomic Likelihood Account.

Desideratum 1. An adequate account should provide a unified (and appropriately discriminating) account of laws and chances.

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 $1/0.$ After all, there are events that have a chance of $0$ that aren’t nomically forbidden (e.g., infinitely many fair coin tosses landing heads), and events that have a chance of $1$ that aren’t nomically required (e.g., infinitely many fair coin tosses not all landing heads).²

Desideratum 2. An adequate account should yield plausible connections between laws and chances, laws and other laws, and chances and other chances.

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 $1.$ It should entail that something can’t be nomically forbidden and nomically required at the same time. And it should say something about how the dynamical chances at one time are related to the dynamical chances at another.

Desideratum 3. An adequate account should describe what, at the fundamental level, makes it the case that chance events deserve the numerical values they’re assigned.

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 $0.6$ strength to bring about one state of affairs given some other state of affairs. What, at the fundamental level, does this posit amount to?

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 $0.6.$ But it’s implausible to think that, at the fundamental level, the chance facts boil down to relations to numbers of this kind. After all, the choice to assign chances values between $0$ and $1$ is purely conventional; we could assign chances using values between $0$ and $2,$ or $0$ and $0.5,$ just as well.³ A more plausible story would provide some non-numerical relations whose structure justifies these numerical assignments. But this would, of course, require doing more than simply stipulating the existence of a nomic disposition of a certain numerical strength.

Desideratum 4. An adequate account should be able to accommodate both dynamical and non-dynamical chances (like those of statistical mechanics).⁴

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.⁵ Since statistical mechanical chances are macrostate-relative and compatible with determinism, it follows that an adequate account of laws should be able to make sense of macrostate-relative chances and non-trivial chances at deterministic worlds.⁶

Desideratum 5. An adequate account should be able to accommodate plausible nomic possibilities.

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 (1977). It should be able to make sense of uninstantiated laws, such as worlds where $F=ma$ is a law but there are no massive objects. It should be able to make sense of world in which there is only one chance event—a coin toss, say—with a chance of $0.6$ of landing heads and a chance of $0.4$ of landing tails. And it should be able to distinguish such a world from an otherwise identical world in which the chance of heads is $0.7$ and the chance of tails is $0.3.$

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 requirement on an adequate account of laws, even though they might concede that failing to accommodate plausible nomic possibilities is a mark against their view. The debate between Humeans and non-Humeans is a long one, and I won’t attempt to settle it here. Instead, I’ll simply side with the non-Humeans, and assume that desideratum 5 is a requirement on an adequate account of laws.

2.2. Other Accounts

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 (1994) primitivist account fails to satisfy desiderata 2 and 3. Carroll’s account takes what the laws and chances are to be primitive. But simply stating that such-and-such laws and chances hold doesn’t suffice to tell us what relations can hold between laws/chances and other laws/chances (desideratum 2). For example, it doesn’t tell us anything about how the dynamical chances at one time should be related to the dynamical chances at another.

Likewise, simply stating that it’s a primitive fact that a certain event has a chance of $0.6$ doesn’t provide a plausible story for what, at the fundamental level, makes this event deserve this numerical assignment (desideratum 3). At first glance, the claim that it’s a fundamental fact that a certain event has a chance of $0.6$ seems to be asserting that some kind of fundamental relation holds between that event and a number. But as we saw in Section 2.1, this story is deeply implausible. Alternatively, one might understand claims about numerical chance assignments as concise ways of describing some more fundamental non-numerical structure that underlies these numerical assignments. But such a story requires a description of what this more fundamental non-numerical structure is, and Carroll’s account doesn’t provide us with these details.⁷

2. Lewis’s (1994) best system account of laws fails to satisfy desiderata 4 and 5. Lewis’s account requires all chances to be dynamical chances, and so fails to satisfy desideratum 4.⁸ And as a Humean account—an account which takes the laws to supervene on the distribution of local qualities—it fails to satisfy desideratum 5, since it’s unable to accommodate a plausible range of nomic possibilities. For (as we saw in Section 2.1) there are plausible nomic possibilities—such as pairs of worlds that are identical with respect to the distribution of local qualities but different with respect to the chances—that Humean accounts cannot recognize.

3. Armstrong’s (1983) universalist account fails to satisfy desiderata 2, 3 and 5. On one natural reading of Armstrong’s account, it takes the nomic facts to be entailed by infinitely many fundamental necessitation relations—each intuitively corresponding to a different chance value—which hold between pairs of fundamental properties (universals) $F$ and $G.$ ⁹ Armstrong’s account fails to satisfy desideratum 3 because it doesn’t provide these necessitation relations with any structure that would justify one numerical assignment over any other. For example, nothing about the account tells us whether the necessitation relation $N_a$ is stronger than the necessitation relation $N_b,$ or whether $N_a$ is closer in strength to $N_b$ than $N_c,$ or whether $N_a$ is twice as strong as $N_b.$ In a similar vein Armstrong’s account fails to satisfy desideratum 2, since it doesn’t say enough about these necessitation relations to determine what, for example, the relation between dynamical chances at different times is. Finally, Armstrong’s account rules out plausible nomic possibilities (desideratum 5), since it rules out the possibility of worlds with uninstantiated laws or chances (such as a world where Newton’s gravitational force law holds but there are no masses).¹⁰

4–5. Swoyer’s (1982) necessitarian account and Lange’s (2009) counterfactual account both fail to satisfy desiderata 1, 2 and 3. While these accounts differ in a number of ways, they are similar in that they both don’t take chances to be part of the laws. Instead, they take chances to be just another quantitative property like mass or charge, and their accounts say little more about what chances are like. As a result, these accounts fail to satisfy the first three desiderata: they fail to provide a unified account of laws and chances (desideratum 1), they fail to yield plausible relations between laws/chances and other laws/chances (desideratum 2), and they fail to explain what, at the fundamental level, makes chance events deserve the numerical values they’re assigned (desideratum 3).¹¹

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 (1973) propensity account of chances fails to satisfy desiderata 1, 2 and 3, though it does better with respect to desideratum 3 than the other accounts we’ve considered. Suppes takes an “at least as probable than” relation as primitive, imposes certain constraints on this relation, and then uses these constraints to provide representation theorems for various kinds of probabilistic phenomena, such as radioactive decay and coin tosses.¹² These representation theorems show, roughly, that one can assign numerical values to chance events that will line up with the “at least as probable than” relation and satisfy the probability axioms.

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.¹³

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 (2014) propensity account of chances fails to satisfy desiderata 1, 2 and 5. Konek’s account employs a primitive “comparative propensity ordering” that satisfies certain constraints, and then uses these constraints to provide a representation and uniqueness theorem. Thus we finally have an account which fully satisfies desideratum 3—an account that explains what, at the fundamental level, makes chance events deserve the numerical values we assign them.

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.¹⁴

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 $0.6$ of landing heads (cf. Section 2.1). And this possibility seems distinct from an otherwise identical world where the chance of heads is $0.7.$ But on Konek’s account neither of these worlds are possible—the comparative propensity ordering facts that line up with these numbers will be too weak to yield a precise numerical chance assignment, so Konek’s account will take such worlds to have imprecise chances. And since the comparative ordering facts that line up with these numbers will be the same in both worlds, Konek’s account can’t recognize these possibilities as distinct.

4.1. The Nomic Likelihood Relation

Here is the fundamental posit of the Nomic Likelihood Account:

The Nomic Likelihood Relation: There exists a fundamental six-place nomic likelihood relation, $\succcurlyeq (C,A,w,C^{\prime },A^{\prime },w^{\prime })$ $(“C$ given $A$ at $w$ is at least as nomically likely as $C^{\prime }$ given $A^{\prime }$ at $w^{\prime }”),$ that satisfies the 12 nomic axioms (cf. Section 4.3), where $w,$ $w^{\prime }$ are worlds, and $A,$ $A^{\prime },$ $C,$ $C^{\prime }$ are propositions that supervene on the fundamental properties and relations other than $\succcurlyeq .$

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.

4.2. Terminology

Let me start by introducing some terminology.

Let ${\mathit{\text{C}}}_{A,w}$ be an ordered triple consisting of a pair of propositions $A,C\subseteq \Omega$ and a world $w\in \Omega .$ I’ll call $A$ and $C$ the antecedent and consequent propositions of the triple, respectively. When expressing such triples, everything that’s bolded should be understood as describing the consequent proposition of the triple. E.g., ${(\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }})}_{A,w}$ is a triple whose consequent proposition is $C\cap C^{\prime },$ whose antecedent proposition is $A,$ and whose world is $w.$ When talking about triples which share the same indices, I’ll leave the indices implicit.

At the risk of abusing notation, I’ll often express the nomic likelihood relation in terms of these triples. Thus I’ll use $“{\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}”$ as shorthand for $“\succcurlyeq (C,A,w,C\prime ,A\prime ,w\prime )”.$ Using this notation, we can define the “more nomically likely than” relation $\succ$ as follows: $\mathit{\text{C}}\succ {\mathit{\text{C}}}^{\mathbf{\prime }}$ iff $\mathit{\text{C}}\succcurlyeq {\mathit{\text{C}}}^{\mathbf{\prime }}$ and ${\mathit{\text{C}}}^{\mathbf{\prime }}\cancel{\succcurlyeq }\mathit{\text{C}}.$ Likewise, we can define the “nomically on a par” relation $~$ as follows: $\mathit{\text{C}}~{\mathit{\text{C}}}^{\mathbf{\prime }}$ iff $\mathit{\text{C}}\succcurlyeq {\mathit{\text{C}}}^{\mathbf{\prime }}$ and ${\mathit{\text{C}}}^{\mathbf{\prime }}\succcurlyeq \mathit{\text{C}}.$

Let $\mathrm{NS}$ (for “nomic space”) be the set of all triples ${\mathit{\text{C}}}_{A,w}$ such that $C,A,$ and $w$ are either the first three or last three arguments of some instantiation of $\succcurlyeq .$ Intuitively, $\mathrm{NS}$ is the set of all triples that have nomic likelihoods.

Let the $\mathrm{(A},\mathrm{w)-cluster}$ be the subset of $\mathrm{NS}$ containing all the triples with $A$ and $w$ as their second and third members. Intuitively, $A$ and $w$ pick out a situation, and the $(A,w\text{)-cluster}$ identifies the consequent propositions that nomic likelihoods are assigned to in that situation. For example, if $A$ and $w$ pick out a chance distribution, the $(A,w\text{)-cluster}$ will consist of the triples whose consequent propositions are assigned chances by this distribution. Note that clusters can be “gappy”, in the sense that for some propositions $C,$ the $(A,w\text{)-cluster}$ won’t contain ${\mathit{\text{C}}}_{A,w}.$ This is because, holding $A$ and $w$ fixed, there can be nomic constraints on some consequent propositions but not others. For example, $A$ and $w$ might pick out a chance distribution which assigns chances to propositions about the behavior of particles, but not to propositions about the behavior of incorporeal spirits. Likewise, note that clusters can be empty. For example, if $w$ is a lawless world, then the $(A,w\text{)-cluster}$ will be empty, since no triples of the form ${\mathit{\text{C}}}_{A,w}$ are assigned nomic likelihoods.

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

4.3. The Nomic Axioms

1. We haven’t imposed any constraints on which consequent propositions $C$ are assigned nomic likelihoods in an $(A,w\text{)-cluster.}$ For example, as it stands, it could be the case that $C$ is assigned a nomic likelihood but $\overline{C}$ is not; or that $C$ and $C^{\prime }$ are assigned nomic likelihoods but $C\cup C^{\prime }$ is not. The first axiom ensures that the consequent propositions that are assigned nomic likelihoods are closed under natural operations like negation and disjunction. E.g., it ensures that if given certain meteorological conditions $A$ at world $w$ there’s some nomic likelihood of it raining $(C),$ then there’s also some nomic likelihood of it not raining $(\overline{C});$ and if there’s some nomic likelihood of it raining $\left(C\right)$ and some nomic likelihood of it snowing $(C^{\prime }),$ then there’s also some nomic likelihood of it raining or snowing $(C\cup C^{\prime }).$

Axiom 1 $(\mathit{\sigma }\mathbf{\text{-algebra):}}$

1. If $\mathit{\text{C}}$ is in $\mathrm{NS},$ then $\overline{\mathit{\text{C}}}$ is in $\mathrm{NS}.$

2. If ${\mathit{\text{C}}}_1,{\mathit{\text{C}}}_2,\dots$ are in $\mathrm{NS},$ then ${\displaystyle {\bigcup }_{i=1}^{\infty }{\mathit{\text{C}}}_i}$ is in $\mathrm{NS}.$

Formally, this axiom ensures that for every non-empty $(A,w\text{)-cluster,}$ the consequent propositions in that cluster form a $\mathit{\sigma }\text{-algebra.}$

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 $A$ at $w)$ is more nomically likely than it snowing (given $A^{\prime }$ at $w^{\prime }),$ and it snowing (given $A^{\prime }$ at $w^{\prime })$ is more nomically likely than it being sunny (given $A^{\prime \prime }$ at $w^{\prime \prime }),$ but it raining (given $A$ at $w)$ is neither more nomically likely than, less nomically likely than, or on a par with, it being sunny (given $A^{\prime \prime }$ at $w^{\prime \prime }).$ The second axiom rules this out, by ensuring that all triples with nomic likelihoods are comparable, and that these comparisons are transitive.

Axiom 2 (Weak Order):

1. $\succcurlyeq$ is connected: for all ${\mathit{\text{C}}}_{A,w},{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}$ in $\mathrm{NS},$ either ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}$ or ${\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}\succcurlyeq {\mathit{\text{C}}}_{A,w}.$

2. $\succcurlyeq$ is transitive: for all ${\mathit{\text{C}}}_{A,w},{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }},$ ${\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}$ in $\mathrm{NS},$ if ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }},$ and ${\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}.$

Formally, this axiom ensures that the nomic likelihood relation provides a weak ordering of $\mathrm{NS}.$

3. The previous axioms haven’t imposed any constraints on how the nomic likelihoods assigned to members of different $(A,w\text{)-clusters}$ line up with each other. For example, as it stands, it could be that all the triples in one cluster are more nomically likely than all the triples in another. The third axiom ensures that triples whose consequent propositions are trivially true $(\Omega )$ or trivially false $(\varnothing )$ have the same nomic likelihoods in all $(A,w\text{)-clusters.}$ Intuitively, this ensures that the “ceiling” and “floor” of nomic likelihoods is the same at all clusters.

Axiom 3 (Cross-algebra Comparisons):

1. If ${\mathbf{\Omega }}_{A,w}$ and ${\mathbf{\Omega }}_{A^{\prime },w^{\prime }}$ are in $\mathrm{NS}:$   ${\mathbf{\Omega }}_{A,w}~{\mathbf{\Omega }}_{A^{\prime },w^{\prime }}.$

2. If ${\mathbf{\varnothing }}_{A,w}$ and ${\mathbf{\varnothing }}_{A^{\prime },w^{\prime }}$ are in $\mathrm{NS}:$ ${\mathbf{\varnothing }}_{A,w}~{\mathbf{\varnothing }}_{A^{\prime },w^{\prime }}.$

4. So far, nothing we’ve said requires there to actually be any triples with nomic likelihoods. For all we’ve said, it could be the case that all $(A,w\text{)-clusters}$ are empty. And even if we assume there are non-empty clusters, nothing we’ve said requires them to be fine-grained. E.g., it could be the case that every triple which has a nomic likelihood is on a par with (say) one of three triples, entailing that there are effectively only three degrees of nomic likelihood. And even if we assume there is a cluster with a rich range of nomic likelihoods, nothing we’ve said requires these nomic likelihoods to be fine-grained enough to distinguish between consequent propositions that are nomically required and ones which are “just” overwhelmingly likely (e.g., that at least one of infinitely many fair coin tosses lands heads). The fourth axiom imposes “richness” requirements that ensure there’s an appropriately fine-grained range of nomic likelihoods.

Axiom 4 (Rich Algebra): There exists a particular cluster, call it $“R”$ (for “rich”), with the following features:

1. There is a pair of triples in $R,$ call them $“\mathbf{\varnothing +}”$ and $“\mathbf{\Omega }\mathbf{-}”,$ such that:

   (a) $\mathbf{\varnothing }\prec \mathbf{\varnothing +}\prec \mathbf{\Omega }\mathbf{-}\prec \mathbf{\Omega }.$

   (b) For all $\mathit{\text{C}}$ such that $\mathit{\text{C}}\nsim \mathbf{\varnothing },\mathit{\text{C}}\succcurlyeq \mathbf{\varnothing +}.$

   (c) For all $\mathit{\text{C}}$ such that $\mathit{\text{C}}\nsim \mathbf{\Omega },\mathit{\text{C}}\underset{¯}{\prec }\mathbf{\Omega }\mathbf{-}.$

2. There are no $\mathit{\text{C}}\succ \mathbf{\varnothing +}$ in $R$ such that, for any ${\mathit{\text{C}}}^{\mathbf{\prime }}$ in $R$ such that $C\prime \subset C,$ either:

   (a) ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathit{\text{C}}.$

   (b) ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing }.$

   (c) ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing +}.$

   (d) ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\Omega }\mathbf{-}$ and $\mathit{\text{C}}~\mathbf{\Omega }.$

3. For any ${\mathit{\text{C}}}_{A,w}$ and ${\mathit{\text{C}}}_{A,w}^{\mathbf{\prime }}$ in $\mathrm{NS}$ such that $C\cap C^{\prime }=\varnothing ,$ there’s some ${\mathit{\text{C}}}_R^{\mathbf{\prime \prime }}$ and ${\mathit{\text{C}}}_R^{\mathbf{\prime \prime \prime }}$ in $R$ such that ${\mathit{\text{C}}}_{A,w}~{\mathit{\text{C}}}_R^{\mathbf{\prime \prime }},$ ${\mathbf{C}}_{A,w}^{\mathbf{\prime }}~{\mathbf{C}}_R^{\mathbf{\prime \prime \prime }}$ and $C^{\prime \prime }\cap C^{\prime \prime \prime }=\varnothing .$

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, $R.$ The three clauses of this axiom ensure that $R$ is rich in three different ways. (This axiom is compatible with there being multiple clusters which satisfy these clauses. But $“R”$ is a name for a particular one of them.)

The first clause entails that in this rich cluster there’s (i) a “next highest” rank of nomic likelihood, which sits below $\mathbf{\Omega }$ but above every other rank, and (ii) a “next lowest” rank of nomic likelihood, which sits above $\mathbf{\varnothing }$ but below every other rank. I use the names $“\mathbf{\Omega }\mathbf{-}”$ and $“\mathbf{\varnothing +}”$ for some particular triples in $R$ that have these ranks. (This clause is compatible with there being multiple triples in $R$ which have these ranks. But $“\mathbf{\Omega }\mathbf{-}”$ and $“\mathbf{\varnothing +}”$ are names for a particular pair of them.)

It’s worth emphasizing that $“\mathbf{\Omega }\mathbf{-}”$ and $“\mathbf{\varnothing +}”$ are names for two particular triples in $R,$ not names for the consequent propositions of some triples whose indices have been left implicit. (E.g., I’m not using $“\mathbf{\Omega }\mathbf{-}”$ as shorthand for $“\mathbf{\Omega }{-}_R”;$ $“\Omega \text{-}”$ is not the name of a proposition.) Thus $\mathbf{\Omega }\mathbf{-}$ and $\mathbf{\varnothing +}$ will never be expressed with indices; the second and third elements of these triples are fixed.

In what follows, it will be convenient to have a name for triples $\mathit{\text{C}}$ whose rank is such that $\mathbf{\varnothing +}\prec \mathit{\text{C}}\prec \mathbf{\Omega -}.$ I’ll say that such triples have a middling rank.

The second clause is the analog of the standard “atomless” assumption.²² Roughly, it ensures that in this rich cluster, any triple $C$ of at least middling rank can be always be decomposed into smaller triples of middling rank.

The third clause ensures that every degree of nomic likelihood is instantiated in $R.$ That is, it entails that $R$ is rich enough to be such that every triple in $\mathrm{NS}$ is nomically on a par with some triple in $R.$ ²³

5. Intuitively, nomic likelihoods should satisfy something like a qualitative notion of additivity. For example, given meteorological conditions $A$ at world $w,$ if it raining $\left(C\right)$ is more nomically likely than it snowing $(C^{\prime }),$ then it raining or being sunny $(C\cup C^{\prime \prime })$ should be more nomically likely than it snowing or being sunny $(C\prime \cup C^{\prime \prime }).$ ²⁴ The fifth axiom ensures that nomic likelihoods will satisfy this kind of additivity requirement.

Axiom 5 (Restricted Cross-algebra Additivity): Suppose that ${(\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }})}_{A,w}~{({\mathit{\text{C}}}^{\mathbf{\prime \prime }}\cap {\mathit{\text{C}}}^{\mathbf{\prime \prime \prime }})}_{A^{\prime },w^{\prime }}~{\varnothing }_{A,w},$ that ${\mathit{\text{C}}}_{A,w}~{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime }},$ and that none of the following three conditions hold: (i) ${\mathit{\text{C}}}_{A,w}~\mathbf{\Omega -},$ ${\mathit{\text{C}}}_{A,w}^{\mathbf{\prime }}\text{~\hspace{0.17em}\hspace{0.17em}}{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime \prime }}~\varnothing +,$ (ii) ${\mathit{\text{C}}}_{A,w}~\mathbf{\varnothing +},$ ${\mathit{\text{C}}}_{A,w}^{\mathbf{\prime }}~{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime \prime }}~\mathbf{\Omega -},$ (iii) ${\mathbf{\Omega }}_{A,w}\succ {\mathit{\text{C}}}_{A,w}\succ {\mathbf{\varnothing }}_{A,w},$ ${\mathit{\text{C}}}_{A,w}^{\mathbf{\prime }}~{\mathbf{\varnothing }}_{A,w},$ ${\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime \prime }}~\mathbf{\varnothing +}.$ Then ${\mathit{\text{C}}}_{A,w}^{\mathbf{\prime }}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime \prime }}$ iff ${(\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\prime })}_{A,w}\succcurlyeq {({\mathit{\text{C}}}^{\prime \prime }\cup {\mathit{\text{C}}}^{\prime \prime \prime })}_{A^{\prime },w^{\prime }}.$

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 $\mathbf{\Omega }\mathbf{-}$ and $\mathbf{\varnothing +}$ requires the additivity claim to be restricted to cases where none of conditions (i)–(iii) hold.²⁵ Second, typical qualitative additivity axioms effectively only apply within a single cluster. But in order to “import” richness facts from other clusters, we need the additivity claim to apply to triples belonging to different clusters.²⁶

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 $\mathit{\text{C}}$ is more nomically likely than any triple in this sequence. Then it’s natural to think that $\mathit{\text{C}}$ should also be more nomically likely than a triple whose consequent proposition is the disjunction of all of the consequent propositions in this sequence. This is what the sixth axiom requires.

Axiom 6 (Continuity): If for all $i,$ $\mathit{\text{C}}\succcurlyeq {\mathit{\text{C}}}_i$ and $C_i\subseteq C_{i+1},$ then $\mathit{\text{C}}\succcurlyeq {\displaystyle {\bigcup }_{i=1}^{\infty }{\mathit{\text{C}}}_i}.$

Formally, this axiom ensures that the $\succcurlyeq$ relation is monotonically continuous.

7. So far we’ve said little about how the nomic likelihoods of triples on a par with $\mathbf{\varnothing +}$ and $\mathbf{\Omega }\mathbf{-}$ behave. For example, given conditions $A$ at world $w,$ suppose the nomic likelihood of a certain coin landing heads $\left(C\right)$ is middling, the nomic likelihood of an independent sequence of infinitely many coins all landing heads $\left(C^{\prime }\right)$ is on a par with $\mathbf{\varnothing +},$ and the nomic likelihood of at least one coin in this infinite sequence landing tails $\left(\overline{C\prime }\right)$ is on a par with $\mathbf{\Omega }\mathbf{-}.$ How does the nomic likelihood of the coin landing heads $\left(C\right)$ compare to that of the coin landing heads or the infinite sequence of coins all landing heads $(C\cup C^{\prime })?$ Likewise, how does the nomic likelihood of the coin landing heads $\left(C\right)$ compare to that of the coin landing heads and at least one of an independent infinite sequence of coins landing tails $(C\cap \overline{C\prime })?$ The seventh axiom settles the answer to these questions, holding in both cases that the likelihoods are the same.

In particular, the seventh axiom entails that adding things on a par with $\mathbf{\varnothing +}$ can only result in a change of likelihood in extremal cases, when it’s added to something on a par with $\mathbf{\varnothing }$ or $\mathbf{\Omega }\mathbf{-}.$ Likewise, it entails that intersecting things on a par with $\mathbf{\Omega }\mathbf{-}$ can only result in a change of likelihood in extremal cases, when it’s intersecting something on a par with $\mathbf{\varnothing +}$ or $\mathbf{\Omega }.$

Axiom 7 $\mathbf{(}\mathbf{\varnothing }\text{+/}\mathbf{\Omega }\mathbf{\text{-}}$ Differences):

1. If $\mathbf{\varnothing +}\underset{¯}{\prec }\mathit{\text{C}}\prec \mathbf{\Omega -},$ and ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathit{\varnothing }+,$ then $\mathit{\text{C}}~\mathit{\text{C}}\cup \mathit{\text{C}}\prime .$

2. If $\mathbf{\varnothing +}\prec \mathit{\text{C}}\underset{¯}{\prec }\mathbf{\Omega -},$ and $\mathit{\text{C}}\prime ~\mathbf{\Omega -},$ then $\mathit{\text{C}}\cap \mathbf{C}\prime ~\mathit{\text{C}}.$

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 $A$ hold at world $w$ then it’s maximally likely that it will rain $(C),$ and meteorological conditions $A$ do hold at $w,$ and yet it doesn’t rain at $w.$ The eighth axiom ensures that nomic likelihood is tied to truth in the way we’d expect.

Axiom 8 (Ω Instantiation): If ${\mathit{\text{C}}}_{A,w}~{\mathbf{\Omega }}_{A,w},$ and $w\in A,$ then $w\in C.$

9. Nothing we’ve said so far has imposed conditions tying the fact that if $A$ obtained at $w$ then $C$ would have a certain likelihood to the possibility of $A$ obtaining. Consider the set of worlds $L_w$ containing all the worlds that assign the same nomic likelihoods as world $w.$ (I.e., if $w^{\prime }\in L_w,$ then for all ${\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}$ in $\mathrm{NS},$ ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}$ iff ${\mathit{\text{C}}}_{A,w\prime }\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}.)$ And suppose that given meteorological conditions $A$ at a world in $L_w,$ there’s a certain nomic likelihood of rain $(C).$ As it stands, this could be true even though there’s no world in $L_w$ at which conditions $A$ hold. One might take this to be implausible. If there’s a certain likelihood of rain given certain meteorological conditions at $w,$ then there should be some nomically similar world where those meteorological conditions obtain. The ninth axiom ensures that this is the case.

Axiom 9 (Antecedent Instantiation): If ${\mathit{\text{C}}}_{A,w}$ is in $\mathrm{NS},$ then there exists a $w^{\prime }\in A$ such that for all ${\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}$ in $\mathrm{NS},$ ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}$ iff ${\mathit{\text{C}}}_{A,w\prime }\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime },w^{\prime \prime }}^{\mathbf{\prime \prime }}.$

10. Nothing we’ve said so far has imposed conditions tying the fact that if $A$ obtained at $w$ then $C$ would have a middling likelihood to the possibility of $C$ obtaining. Suppose that given meteorological conditions $A$ at a world in $L_w,$ there’s a middling nomic likelihood of rain $(C).$ As things stand, it could be the case that it rains at every world in $L_w$ where $A$ obtains, even though it only has a middling likelihood of doing so. Likewise, it could be the case that it doesn’t rain at any world in $L_w$ where $A$ obtains, even though it has a middling nomic likelihood of doing so. Both scenarios are implausible: if there’s a middling likelihood of rain, then there should be some $A\text{-worlds}$ in $L_w$ where it rains, and some where it does not. The tenth axiom ensures that this is the case.

Axiom 10 (Chancy Instantiation): If  ${\mathbf{\varnothing }}_{A,w}\prec {\mathit{\text{C}}}_{A,w}\prec {\mathbf{\Omega }}_{A,w},$ then there exists a $w^{\prime }$ and $w^{\prime \prime }$ such that:

1. For all ${\mathit{\text{C}}}_{A^{\prime \prime \prime },w\prime \prime \prime }^{\mathbf{\prime \prime \prime }}$ in $\mathrm{NS},$ ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime \prime },w\prime \prime \prime }^{\mathbf{\prime \prime \prime }}$ iff ${\mathit{\text{C}}}_{A,w\prime }\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime \prime },w\prime \prime \prime }^{\prime \prime \prime }$ iff ${\mathit{\text{C}}}_{A,w\prime \prime }\succcurlyeq {\mathit{\text{C}}}_{A^{\prime \prime \prime },w\prime \prime \prime }^{\mathbf{\prime \prime \prime }}.$

2. $w^{\prime }\in A$ and $w\prime \prime \in A.$

3. $w^{\prime }\in C$ and $w\prime \prime \cancel{\in }C.$

11. The previous axioms haven’t imposed any constraints on what triples there are in different $(A,w\text{)-clusters}$ indexed to the same world. Suppose that given meteorological conditions $A$ at $w,$ there’s a middling likelihood of it raining the next day $\left(C\right)$ and a middling likelihood of it raining the day after that $(C^{\prime }).$ And consider the nomic likelihoods that might obtain at $w$ given those meteorological conditions and that it rains the first day $(A\cap C).$ For all we’ve said so far, it could be that given $A\cap C$ at $w$ there’s a maximal likelihood assigned to it raining the first day $(C),$ but no likelihood at all—whether high or low—assigned to it raining the second day $(C^{\prime }).$ That is, it could be that the $(A\cap C,w\text{)-cluster}$ is simply silent about the likelihood of it raining the second day. This is odd. If the $(A,w\text{)-cluster}$ assigns a nomic likelihood to $C^{\prime },$ it seems the $(A\cap C,w\text{)-cluster}$ should as well. The eleventh axiom ensures this, by requiring clusters at the same world to have consequent propositions that line up with each other.

Axiom 11 (Same Algebra): Suppose that $A\supset A\prime ,$ that ${\mathit{\text{A}}}_{A,w}^{\mathbf{\prime }}\succ \mathbf{\varnothing +},$ and that the $(A\prime ,w\text{)-cluster}$ is not empty. Then ${\mathit{\text{C}}}_{A\prime ,w}$ is in $\mathrm{NS}$ iff ${\mathit{\text{C}}}_{A,w}$ is in $\mathrm{NS}.$ ²⁷

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 $A$ at $w,$ there’s a middling likelihood of it raining the next day $(C),$ a middling likelihood of it raining the day after that $(C^{\prime }),$ and a smaller but still middling likelihood of it raining both days $(C\cap C^{\prime }).$ Given those meteorological conditions and that it rains the next day $(A\cap C),$ what should the likelihood of it raining both days be? For all we’ve said so far, it could be anything, including on a par with the trivially true proposition $\mathbf{\Omega }$ or the trivially false proposition $\varnothing .$ This is implausible: the likelihood of it raining both days should be middling. The twelfth axiom ensures this, by requiring the nomic likelihoods assigned by same-world clusters to line up in the way you’d expect.

Formulating the twelfth axiom precisely requires a little stage-setting. Let an n-equipartition $P$ of a cluster be a set of $n$ triples ${\mathit{\text{P}}}_i$ which are all nomically on a par with each other, and whose consequent propositions are mutually exclusive and exhaustive.²⁸ Let $f:\mathbb{N}\times \mathrm{NS}\to \mathbb{N}$ be a function such that: $f(n,{\mathit{\text{C}}}_{A,w})=x$ iff for any $n\text{-equipartition}$ $P$ of the rich cluster $R,$ and any ${\mathit{\text{C}}}_{A,w}$ in $\mathrm{NS}:$

• $f(n,{\mathit{\text{C}}}_{A,w})$ = $n$ if ${\mathit{\text{C}}}_{A,w}~{({\cup }_{i=1}^{i=n}{\mathit{\text{P}}}_i)}_{A,w}.$

• $f(n,{\mathit{\text{C}}}_{A,w})$ = $m$ if $n>m>0$ and ${({\cup }_{i=1}^{i=m+1}{\mathit{\text{P}}}_i)}_{A,w}\succ {\mathit{\text{C}}}_{A,w}\succcurlyeq {({\cup }_{i=1}^{i=m}{\mathit{\text{P}}}_i)}_{A,w}.$

• $f(n,{\mathit{\text{C}}}_{A,w})$ = $0$ if ${\mathit{\text{P}}}_1{}_{A,w}\succ {\mathit{\text{C}}}_{A,w}.$

Intuitively, $f$ takes a natural number $n$ and a triple $\mathit{\text{C}},$ and spits out a natural number $x$ indicating that the nomic likelihood of $\mathit{\text{C}}$ is at least $\frac{x}{n}$ that of $\mathbf{\Omega },$ but less than $\frac{x+1}{n}$ that of $\mathbf{\Omega }.$ Thus if $f(n,\mathit{\text{C}})=0,$ we know the nomic likelihood of $\mathit{\text{C}}$ is less than $\frac{1}{n}$ of $\mathbf{\Omega };$ if $f(n,\mathit{\text{C}})=1,$ we know the nomic likelihood of $\mathit{\text{C}}$ is at least $\frac{1}{n}$ but $n$ less than $\frac{2}{n}$ of $\mathbf{\Omega };$ and so on; and if $f(n,C)=n,$ we know the nomic likelihood of $C$ is at least $\frac{n}{n}$ of $\mathbf{\Omega },$ i.e., is exactly that of $\mathbf{\Omega }.$

Axiom 12 (Algebra Coordination): Suppose that $A_{A,w},$ $A_{A\prime ,w},$ ${(\mathit{\text{C}}\cap A)}_{A,w},$ and ${(\mathit{\text{C}}\cap A)}_{A\prime ,w}$ are in $\mathrm{NS}.$ If $A\subseteq A\prime ,$ and if it’s not the case that there’s some $m$ such that for all $n>m,$ $f(n,{\mathit{\text{A}}}_{A,w})=0$ or $f(n,{\mathit{\text{A}}}_{A\prime ,w})=0,$ then:

$\underset{n\to \infty }{\lim }\frac{f(n,{(\mathit{\text{C}}\cap \mathit{\text{A}})}_{A,w})}{f(n,{\mathit{\text{A}}}_{A,w})}=\underset{n\to \infty }{\lim }\frac{f(n,{(\mathit{\text{C}}\cap \mathit{\text{A}})}_{A\prime ,w})}{f(n,{\mathit{\text{A}}}_{A\prime ,w})}.$

This axiom ensures that if $A\subseteq A\prime ,$ the $(A,w\text{)-cluster}$ and the $(\mathit{\text{A}}\prime ,w\text{)-cluster}$ agree on the proportion of $A’\text{s}$ nomic likelihood that contributes to $\mathit{\text{C}}’\text{s}$ likelihood.

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.

5.1. The Representation and Uniqueness Theorem

We can partition the space of worlds such that two worlds $w^{\prime }$ and $w^{\prime \prime }$ are in the same cell of the partition iff, for all $C^{\prime },$ $A^{\prime },$ and all ${\mathit{\text{C}}}_{A,w}$ in $\mathrm{NS}:$ ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}$ iff ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A\prime ,w\prime \prime }^{\mathbf{\prime }}.$ Intuitively, two worlds are in the same cell of this partition iff the same nomic facts hold at both worlds. I’ll call this the nomic partition. I’ll use $L,$ $L^{\prime },$ $L^{\prime \prime },$ etc., to denote different cells of this partition, and $L_w$ to denote the cell $w$ is in.

The following theorem is shown in appendix B:³⁰

The Representation and Uniqueness Theorem: If $\succcurlyeq$ satisfies the nomic likelihood axioms, then there’s a unique function $c{h}_{A,L}(C)$ (that takes three propositions $C,$ $A,$ and $L$ as arguments, and spits out a real number between $0$ and $1),$ and a unique pair of three-place relations $\mathrm{NR}({\mathit{\text{C}}}_{A,w})$ and $\mathrm{NF}({\mathit{\text{C}}}_{A,w})$ (that hold between a pair of propositions $C$ and $A$ and a world $w),$ ³¹ such that:

1. $c{h}_{A,L}(C)\ge c{h}_{A\prime ,L\prime }(C\prime )$ iff for any $w\in L$ and $w^{\prime }\in L\prime ,$ either:

   (a) ${\mathit{\text{C}}}_{A,w}\succcurlyeq {\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}.$

   (b) ${\mathit{\text{C}}}_{A,w}\cancel{\succcurlyeq }{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }},$ and ${\mathit{\text{C}}}_{A,w}~\mathbf{\Omega -}$ and ${\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}~{\mathbf{\Omega }}_{A^{\prime },w^{\prime }}.$

   (c) ${\mathit{\text{C}}}_{A,w}\cancel{\succcurlyeq }{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }},$ and ${\mathit{\text{C}}}_{A,w}~{\mathbf{\varnothing }}_{A,w}$ and ${\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime }}~\mathbf{\varnothing +}.$

2. $\mathrm{NR}({\mathit{\text{C}}}_{A,w})$ iff ${\mathit{\text{C}}}_{A,w}~{\mathbf{\Omega }}_{A,w}.$

3. $\mathrm{NR}({\mathit{\text{C}}}_{A,w})$ iff ${\mathit{\text{C}}}_{A,w}~{\mathbf{\varnothing }}_{A,w}.$

Furthermore, the function $c{h}_{A,L}(\cdot )$ will be a countably additive probability function.³²

This theorem shows that the nomic likelihood relation can be uniquely represented by a countably additive probability function $\mathrm{ch}$ which assigns numbers that line up with the nomic likelihood relation, and a pair of relations $\mathrm{NR}$ (nomically required) and $\mathrm{NF}$ (nomically forbidden) that hold between the members of a triple ${\mathit{\text{C}}}_{A,w}$ iff it’s maximally or minimally nomically likely, respectively.

5.2. The Account of Laws and Chances

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

Complete Laws of Nature: A world $w$ has complete laws of nature $L$ iff $L=L_w.$ ³³

It will be convenient to follow Lewis (1979) and identify properties with the set of possible individuals that instantiate them. Since the property $ℒ$ of being a world with laws $L_w$ picks out the same set of worlds as the proposition $L$ that laws $L_w$ obtain, it follows that $ℒ=L.$ Thus we can refer to the laws as both properties and propositions, since they’re both.

The Nomic Likelihood Account then identifies chances, nomic requirements and nomic forbiddings with the $\mathrm{ch}$ function and $\mathrm{NR}$ and $\mathrm{NF}$ relations provided by the representation and uniqueness theorem:

Chances: The chance of $C$ given complete laws $L$ and antecedent $A$ is $x$ iff $c{h}_{A,L}(C)=x.$

Nomic Requirements: If $A$ holds at $w$ then $C$ is nomically required to hold at $w$ iff $\mathrm{NR}({\mathit{\text{C}}}_{A,w}).$

Nomic Forbiddings: If $A$ holds at $w$ then $C$ is nomically forbidden from holding at $w$ iff $\mathrm{NF}({\mathit{\text{C}}}_{A,w}).$

5.3. Some Lemmas Regarding Laws and Chances

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 $A$ at $w)$ there’s some likelihood of $C,$ and $A$ entails $C,$ then it seems $C$ should be nomically required. E.g., suppose there’s some nomic likelihood of rain $\left(C\right)$ given that it’s raining hard $\left(A\right)$ at $w.$ Then, given that it’s raining hard at $w,$ it should be nomically required that it rains. This is what the tenth lemma shows.

Lemma 10: If ${\mathit{\text{C}}}_{A,w}$ is in $\mathrm{NS},$ and $A$ entails $C,$ then $\mathrm{NR}({\mathit{\text{C}}}_{A,w}).$

11. It seems like nomic requirements should be closed under entailment. For example, if (given $A$ at $w)$ it’s nomically required that it be rainy $(C),$ and nomically required that it be windy $(C^{\prime }),$ then it should be nomically required that it be rainy and windy $(C\cap C^{\prime }).$ This is what the eleventh lemma says.

Lemma 11: For all $\mathit{\text{C}}$ in $\mathrm{NS}:$ If $C_1,\dots ,C_n,$ entail $C,$ and $\mathrm{NR}({\mathit{\text{C}}}_1),\dots ,\mathrm{NR}({\mathit{\text{C}}}_n),$ then $\mathrm{NR}(\mathit{\text{C}}).$

12. It seems nomic requirements and nomic forbiddings should be linked: if $C$ is nomically required, then $\overline{C}$ should be nomically forbidden, and vice versa. For example, if (given $A$ at $w)$ it’s nomically required that it rain $(C),$ then it should be nomically forbidden that it not rain $(\overline{C}),$ and vice versa. This is what the twelfth lemma states.

Lemma 12: $\mathrm{NR}(\mathit{\text{C}})$ iff $\mathrm{NF}(\overline{\mathit{\text{C}}}).$

13. It seems like nomic requirements and forbiddings should be tied to the truth. For example, if (given $A$ at $w)$ rain is nomically required, and $A$ obtains, then it should rain. Likewise, if (given $A$ at $w)$ rain is nomically forbidden, and $A$ obtains, then it shouldn’t rain. This is what the thirteenth lemma asserts.

Lemma 13:

1. If $\mathrm{NR}({\mathit{\text{C}}}_{A,w})$ and $w\in A,$ then $w\in C.$

2. If $\mathrm{NF}({\mathit{\text{C}}}_{A,w})$ and $w\in A,$ then $w\in \overline{C}.$

14. It seems nomic likelihoods should be tied to chances. For example, if (given $A$ at $w)$ the nomic likelihood of rain is on a par with $\mathbf{\Omega }\mathbf{-}$ or $\mathbf{\Omega },$ then the chance of rain (given $A$ and the laws that hold at $w)$ should be $1.$ Likewise, if the nomic likelihood of rain is on a par with $\mathbf{\varnothing +}$ or $\mathbf{\varnothing },$ then the chance of rain should be $0.$ And if the nomic likelihood of rain is middling, then the chance of rain should be greater than $0$ but smaller than $1.$ This is what the fourteenth lemma says.

Lemma 14:

1. If ${\mathit{\text{C}}}_{A,w}\succcurlyeq \mathbf{\Omega -},$ then $c{h}_{A,L_w}(C)=1.$

2. If ${\mathit{\text{C}}}_{A,w}\underset{¯}{\prec }\mathbf{\varnothing +},$ then $c{h}_{A,L_w}(C)=0.$

3. If $\mathbf{\Omega -}\succ {\mathit{\text{C}}}_{A,w}\succ \mathbf{\varnothing +},$ $c{h}_{A,L_w}(C)\in (0,1).$

15. It seems related chance distributions should assign chances to the same propositions. For example, suppose $A$ and $L$ yield a well-defined chance distribution over a sequence of fair coin tosses. And suppose the conjunction of $A$ and the first fair coin toss landing heads (call this conjunction $A^{\prime })$ and $L$ also yield a well-defined chance distribution. It would be strange if $c{h}_{A\prime ,L}$ assigned chances to coin tosses that $c{h}_{A,L}$ did not assign chances to, or vice versa. Rather, it seems $c{h}_{A,L}$ and $c{h}_{A\prime ,L}$ should assign chances to the same propositions. This is what the fifteenth lemma says.

Lemma 15: If $A\supset A\prime ,$ $c{h}_{A,L}(A\prime )>0,$ and $c{h}_{A\prime ,L}(\Omega )$ is well-defined, then for all $C,$ $c{h}_{A\prime ,L}(C)$ is well-defined iff $c{h}_{A,L}(C)$ is well-defined.

16. It seems related chance distributions should have related chance assignments. For example, suppose $A$ and $L$ yield a well-defined chance distribution over a sequence of independent coin tosses, and this distribution assigns a chance of $0.5$ to the first coin landing heads $(C),$ and chance of $0.25$ to the first two coin tosses landing heads $(C\cap C^{\prime }).$ And suppose the conjunction of $A$ and the first fair coin toss landing heads—i.e., $A\cap C—\text{and}$ $L$ also yield a well-defined chance distribution. What chance should $c{h}_{A\cap C,L}$ assign to the first two coin tosses landing heads? Given the chances $c{h}_{A,L}$ assigns, it seems the right answer is $0.5.$ This is what the sixteenth lemma entails.

Lemma 16: If $A\supseteq A\prime ,$ and $c{h}_{A,L}(C\mid A\prime )$ and $c{h}_{A\prime ,L}(C)$ are well-defined, then $c{h}_{A\prime ,L}(C)=c{h}_{A,L}(C\mid A\prime ).$

5.4. A Toy Example

It can be helpful to see a concrete example of some complete laws $L_w$ on the Nomic Likelihood Account. But it’s hard to do so concisely for realistic physical theories. So I’ll instead present a toy example corresponding to a pair of cases discussed in Section 2.1: a pair of worlds in which there’s only one chance event, a coin toss, where the chance of heads is $0.6$ in one world, and $0.7$ in the other.

Let $A$ be a proposition describing the state of a world at $t$ consisting of a certain coin toss set-up, and let $C$ be a proposition stating that the outcome of this coin toss was heads. Let $w$ be a world such that there are only four triples indexed to $w$ in $\mathrm{NS}:{\mathbf{\varnothing }}_{A,w},{\mathit{\text{C}}}_{A,w},{\overline{\mathit{\text{C}}}}_{A,w},$ and ${\mathbf{\Omega }}_{A,w}.$ Let ${\mathit{\text{C}}}_{A,w}$ be on a par with the triples in the rich cluster that are assigned a value of $0.6$ by the representation and uniqueness theorem.

The complete laws of $w,$ $L_w,$ will consist of the set of worlds in $w’\text{s}$ cell of the nomic partition. And these laws describe a world in which there’s almost nothing of nomic interest going on: there’s only a single non-trivial chance event—a coin toss—which has a chance of $0.6$ of landing heads.

We can also consider a world $w^{\prime }$ such that the only triples indexed to $w^{\prime }$ in $\mathrm{NS}$ are: ${\mathbf{\varnothing }}_{A,w\prime },$ ${\mathit{\text{C}}}_{A,w\prime },$ ${\overline{\mathit{\text{C}}}}_{A,w\prime }$ and ${\mathbf{\Omega }}_{A,w\prime }.$ And in this case, ${\mathit{\text{C}}}_{A,w\prime }$ is on a par with the triples in the rich cluster that are assigned a value of $0.7.$ The complete laws $L_{w\prime }$ will consist of the set of worlds with the same nomic facts as $w^{\prime },$ and these laws describe a world in which there’s only a single non-trivial chance event—a coin toss—which has a chance of $0.7$ of landing heads.

A.1. Some Key Lemmas

Lemma 1: For all $\mathit{\text{C}}$ in $\mathrm{NS},$ $\mathbf{\Omega }\succcurlyeq \mathit{\text{C}}\succcurlyeq \mathbf{\varnothing }.$

Lemma 2: For all $\mathit{\text{C}},$ ${\mathit{\text{C}}}^{\mathbf{\prime }}$ in $\mathrm{NS},$ if $C\subseteq C\prime ,$ then $\mathit{\text{C}}\underset{¯}{\prec }{\mathit{\text{C}}}^{\mathbf{\prime }}.$

Lemma 3: For all ${\mathit{\text{C}}}^{\mathbf{\prime }}$ in $\mathrm{NS}:$

1. If $\mathit{\text{C}}~\mathbf{\varnothing },$ then $\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing }.$

2. If $\mathit{\text{C}}~\mathbf{\Omega },$ then $\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\Omega }.$

Lemma 4: For all ${\mathit{\text{C}}}^{\mathbf{\prime }}$ in $\mathrm{NS}:$

1. If $\mathit{\text{C}}~\mathbf{\varnothing },$ then $\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime }}.$

2. If $\mathit{\text{C}}~\mathbf{\Omega },$ then $\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime }}.$

Lemma 5: $\mathit{\text{C}}~\mathbf{\varnothing }$ iff $\overline{\mathit{\text{C}}}~\mathbf{\Omega }.$

Lemma 6: $\mathit{\text{C}}~\mathbf{\varnothing +}$ iff $\overline{\mathit{\text{C}}}~\mathbf{\Omega -}.$

Lemma 7: If $\mathbf{\varnothing +}\prec \mathit{\text{C}}\prec \mathbf{\Omega -},$ then $\mathbf{\varnothing }\text{+}\prec \overline{\mathit{\text{C}}}\prec \mathbf{\Omega -}.$

Lemma 8: If $\mathit{\text{C}}\succcurlyeq {\mathit{\text{C}}}^{\mathbf{\prime }},$ then $\overline{{\mathit{\text{C}}}^{\mathbf{\prime }}}\succcurlyeq \overline{\mathit{\text{C}}}.$

Lemma 9: For all $\mathit{\text{C}}$ in $\mathrm{NS}:$

1. If $\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }}~\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing },$ and none of the following three conditions hold: (i) $\mathit{\text{C}}~\mathbf{\Omega -},$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing +},$ (ii) $\mathit{\text{C}}~\mathbf{\varnothing +},$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\Omega -},$ or (iii) $\mathbf{\Omega }\succ \mathit{\text{C}}\succ \mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing +},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\succcurlyeq {\mathit{\text{C}}}^{\mathbf{\prime \prime }}$ iff $\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime }}\succcurlyeq \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$

2. If $\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }}~\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing },$ and none of the following four conditions hold: (i) $\mathit{\text{C}}~\mathbf{\Omega -},$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing +},$ (ii) $\mathit{\text{C}}~\mathbf{\varnothing +},$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~{\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\Omega -},$ (iii) $\mathbf{\Omega }\succ \mathit{\text{C}}\succ \mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing +},$ or (iv) $\mathbf{\Omega }\succ \mathit{\text{C}}\succ \mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing +},$ ${\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing },$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }}$ iff $\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime }}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$

A.2. Proofs

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.

• Proof of Lemma 9: (1) The first part of the lemma is a special case of axiom 5, where all of the relevant propositions belong to the same cluster, and $C=C^{\prime \prime }.$ (Note that while axiom 5 imposes the condition that ${\mathit{\text{C}}}_{A,w}~{\mathit{\text{C}}}_{A^{\prime },w^{\prime }}^{\mathbf{\prime \prime }},$ which entails that ${\mathit{\text{C}}}_{A,w}$ is in $\mathrm{NS},$ lemma 5 doesn’t have such a clause since $C=C^{\prime \prime }.$ Thus lemma 5 needs to explicitly add the existence assumption “For all C in $\mathrm{NS}”.)$

(2) The second part of the lemma follows from the first and the assumption that it’s also not the case that (iv) $\mathbf{\Omega }\succ \mathit{\text{C}}\succ \mathbf{\varnothing },$ ${\mathit{\text{C}}}^{\mathbf{\prime }}~\mathbf{\varnothing +},$ ${\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing }.$ To see this, suppose that the relevant triples are on a par with the emptyset, and none of conditions (i)–(iv) hold.

First, let’s establish that if ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ If ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ and since none of (i)–(iii) hold, the first part of the lemma entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succcurlyeq \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ Furthermore, ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }}$ entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\cancel{\underset{¯}{\prec }}{\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ and since none of (i), (ii) or (iv) hold, the first part of the lemma entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\cancel{\underset{¯}{\prec }}\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ (The conditions change because $C^{\prime }$ and $C\prime \prime$ switch places. Conditions (i) and (ii) treat $C^{\prime }$ and $C\prime \prime$ symmetrically, but condition (iii) does not; condition (iv) is what you get when you swap $C^{\prime }$ and $C\prime \prime$ in condition (iii).) So if ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$

Second, let’s establish that if ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ If ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succcurlyeq \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ and since none of (i)–(iii) hold, the first part of the lemma entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ Furthermore, ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }}$ entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\cancel{\underset{¯}{\prec }}\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ and since none of (i), (ii) or (iv) hold, the first part of the lemma entails that ${\mathit{\text{C}}}^{\mathbf{\prime }}\cancel{\underset{¯}{\prec }}{\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$ So if ${\mathit{\text{C}}}^{\mathbf{\prime }}\cup \mathit{\text{C}}\succ \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ then ${\mathit{\text{C}}}^{\mathbf{\prime }}\succ {\mathit{\text{C}}}^{\mathbf{\prime \prime }}.$

• Proof of Lemma 2: Suppose that $A\supseteq B.$ Let $C=B,$ $C\prime =A-B$ and $C\prime \prime =\varnothing .$ Note that $\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime }}~\mathit{\text{C}}\cap {\mathit{\text{C}}}^{\mathbf{\prime \prime }}~\mathbf{\varnothing }.$ Note also that none of conditions (i)–(iii) of lemma 9 obtain (since in all of them ${\mathit{\text{C}}}^{\mathbf{\prime \prime }}\nsim \mathbf{\varnothing }).$ Thus by lemma 9, ${\mathit{\text{C}}}^{\mathbf{\prime }}\succcurlyeq {\mathit{\text{C}}}^{\mathbf{\prime \prime }}$ iff $\mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime }}\succcurlyeq \mathit{\text{C}}\cup {\mathit{\text{C}}}^{\mathbf{\prime \prime }},$ i.e., $(\mathit{\text{A}}-\mathit{\text{B}})\succcurlyeq \mathbf{\varnothing }$ iff $\mathit{\text{B}}\cup (\mathit{\text{A}}-\mathit{\text{B}})\succcurlyeq \mathit{\text{B}}\cup \mathbf{\varnothing }$ iff $\mathit{\text{A}}\succcurlyeq \mathit{\text{B}}.$ Since the left-hand side is true, the right-hand side must be true as well.

• Proof of Lemma 1: (1) Since $\varnothing$ is a subset of every proposition, lemma 2 entails that $\mathbf{\varnothing }\underset{¯}{\prec }$ every proposition. (2) Likewise, since every proposition is a subset of $\Omega ,$ lemma 2 entails that $\mathbf{\Omega }\succcurlyeq$ every proposition.