1.1 Humean Laws and Ontology

David Lewis made famous the combination of a neo-Humean ontology he called ‘Humean Supervenience’ and the best-system analysis of laws (BSA). I will briefly discuss these elements in turn.

Lewis refers to the ‘vast mosaic of local matters of particular fact’ as the Humean Mosaic, and I shall use this terminology in subsequent discussion.

The basic properties countenanced by Humean supervenience are categorical, which means that no causal role is essential to them. The property charge, in our world, occupies a certain role; it confers upon its bearers a disposition to exert a force on other charged bodies in accordance with Coulomb's law. However, according to the categoricalist about properties, this role occupancy is thoroughly contingent. There are worlds in which charge confers no causal role at all and worlds in which it plays the role that we at the actual world associate with mass. It is these categorical properties of, or instantiated at, points and their spatiotemporal relations that make up Lewis's neo-Humean ontology.

Now imagine that God wanted to convey to us all the facts about the Humean Mosaic. To this end he might give us a big book that listed the spatiotemporal location of every fundamental property instance. But this would not be very useful for us insofar as we were interested in having the information readily accessible to our finite intellects. A better option might be to provide us with fewer, more general statements about the distribution of qualities throughout the Humean Mosaic, from which we could deduce additional information not explicitly given. Such a more informative systematization will have more basic statements, more axioms if you like. A simpler systematization will have fewer axioms but will sacrifice informativeness. Hence, the virtues of informativeness and simplicity of a system compete. According to the BSA, the fundamental laws are the axioms of this system that strikes the optimal informativeness-simplicity balance.

Balancing the virtues of informativeness and simplicity in this way will involve a collective consideration of the mosaic as a whole: ‘An adequate analysis must be collective. It must treat regularities not one at a time, but rather as candidates to enter into integrated systems’ (Lewis Reference Lewis1983: 367, my emphasis).

Adding to the system a statement such as ‘all electrons are negatively charged’ might increase complexity at little to no informative gain if this regularity followed from some more general statement of, say, quantum theory. The point is that the BSA treats regularities collectively as candidates to enter into an integrated system because matters of fact far beyond those concerning any given regularity or the participants in a regularity, considered in isolation, will be relevant to that regularity's status (or lack thereof) as a law. This is an important feature of the BSA to which I shall return in section 2 when I consider how a best-system account of laws in an unHumean world might be deserving of the name ‘BSA’.

1.2 unHumean Laws and Ontology

Any ontology that admits modal properties or necessary connections that do not reduce to some nonmodal features of the ontology is unHumean. To illustrate the idea, I will briefly consider two quite different examples of unHumean ontologies: dispositional essentialism, for example, Bird (Reference Bird2007), and the nomic necessitation view, for example, Armstrong (Reference Armstrong1983).

According to dispositional essentialism, at least some fundamental properties are not categorical because they are constituted by their causal roles. The property charge, for example, just is, in all possible worlds, the disposition to exert a force on other charged objects in accordance with Coulomb's law—there is no possibility of charge switching roles with mass on this view. Hence, dispositional essentialists maintain that at least some properties are irreducibly modal.

Another quite different way in which Humean supervenience has been rejected is by Armstrong (Reference Armstrong1983; see also Dretske Reference Dretske1977; Tooley Reference Tooley1977), who maintains categoricalism about fundamental properties but introduces primitive necessitation relations between universals to account for laws. On this account, laws of the form ‘all Fs are Gs’ are analyzed in terms of a necessitation relation, N, that in this case holds between the universals F and G. The fact that the higher-order universal, N, connects the universals F and G is what makes it a law that all Fs are Gs, on this account.

In each case, giving up Humean supervenience is closely connected to the provision of a non-Humean account of laws. Dispositional essentialism grounds the laws in irreducibly modal properties and the nomic necessitation view analyses the laws in terms of primitive necessary connections between universals.

Among the positions outlined, two broad conceptions of natural law have been employed: a governance conception and a codification conception. Armstrong's nomic necessitation view is a governance conception. It conceives of the laws as imposed ‘pushers and pullers’ of the stuff in the world. By contrast, the BSA conceives of the laws as merely codifying matters of fact. The laws, according to the BSA, have no prescriptive power over events, rather they describe, in a particularly efficient way, what goes on in the universe. Dispositional essentialism might also be understood as a codification conception; on this view, the laws describe the dispositional essences of properties and what this implies for the behaviors of their bearers.

3.1. A Skeptical Worry

Call a world, w*, relevant to the laws at a distinct world, w, iff some element of w* partly determines w’s laws. Thus, if the distribution of, say, mass at a world w1 is systematized by the laws of w2 because mass is instantiated at both, then w1 is relevant to the laws at w2.

To highlight the skeptical worry, I will consider which worlds Demarest's potency-BSA deems relevant to the laws at a given world. Consider a simple world, w0, at which just mass and charge are instantiated. I will denote the situation like this: w0(mass, charge). According to the potency-BSA, the laws of w0 systematize all w0-potency distributions, where a w0-potency distribution is a possible distribution of only those potencies appearing in w0 (Demarest Reference Demarest and Jacobs2017: 49). The laws of w0 are thus partly determined by the distributions of mass and charge at worlds other than w0. Hence, worlds other than w0 are relevant to w0’s laws. For all that has been said, four options for the range of worlds relevant to the laws of w0(mass, charge) can be discerned:

Option 1: worlds with all the potencies found at w0. This would include worlds with some potencies alien to w0 and would omit worlds lacking potencies instantiated at w0. For example, w1 would be included w1(mass, charge, schmass), but not w2(mass).

Option 2: Worlds with only the potencies found at w0. This would rule out worlds with alien potencies and include worlds absent some potencies instantiated at w0. For example, w2 would be included, but not w1.

Option 3: Worlds with all and only those potencies found at w0. This rules out worlds with potencies that are alien to w0 and worlds absent any potencies instantiated at w0. For example, w3 (which has the same potency instances as w0, though those potency instances might be differently distributed) would be included w3(mass, charge), but w2 and w1 would not.

Option 4: Worlds with some of those potencies found at w0. This just rules out worlds that are absent all of the potencies instantiated at w0. For example, w1, w2, and w3 would be included, but not w4(schmass, schmarge).

I suggest that Demarest may be interpreted as endorsing either option 2 or option 4. Option 4 seems to follow from Demarest's explicit statement of the potency-BSA (2017: 49) as well as perhaps from her response to the impoverished world objection (2017: 51). Saddling Demarest with option 4, however, might seem less charitable, since radical skepticism about the laws quickly follows from this option. Hence, the potency-BSA could at least benefit from clearer articulation to avoid this interpretation. As it happens, however, option 2 also faces a skeptical worry via a subtler route. I will discuss these different interpretations (and the skeptical threat to each) in turn before proposing a revision to the potency-BSA that avoids these problems and satisfies the desiderata identified in section 2.

Demarest is clear that the laws of a world, w, are unconcerned with possible distributions of potencies alien to w: ‘The basic laws of nature at w are the axioms of the simplest, most informative, true systematization of all w-potency-distributions, where a w-potency-distribution is a possible distribution of only potencies appearing in w’ (2017: 49, my emphasis). However, it cannot be inferred from this that only those worlds containing just the same potencies as w0(mass, charge) are relevant to w0’s laws because among the possible distributions of mass are those distributions of mass at worlds where, for example, schmarge is instantiated too.

Moreover, consider Demarest's response to the impoverished world objection: ‘Consider, again, a world with a single massive particle, traveling inertially for all time. The laws of this world will systematize not just this world, but all worlds that contain mass’ (2017: 51, my emphasis). Relative to the impoverished world, potencies found at the actual world: charge, spin, etc., are alien, but Demarest seems to imply that the laws at the impoverished world nonetheless concern the distribution of mass at the actual world because ‘the laws of this world will systematize . . . all worlds that contain mass’ and the actual world contains mass. On this reading, it seems that for Demarest all worlds with at least some of the potencies found at a world, w, are relevant to the laws at w, which is option 4.

Skepticism about the laws quickly follows because inhabitants of w0(mass, charge) could not possibly know how alien potencies, like schmass, would affect the distribution of w0-potencies, namely, mass and charge, so they could not possibly come to know all w0-potency-distributions, the best systematization of which determines the laws at w0. The problem generalizes and makes the actual laws unknowable too. In essence, the problem is this: at worlds with alien potencies, familiar potencies, like mass and charge, might behave very differently. We, at the actual world, cannot know how alien potencies will affect the distribution of, for example, mass and charge, so if the actual laws concerning mass and charge are supposed to systematize their distributions in the presence of alien potencies, we cannot know the laws. It is plausible, however, that we possess all sorts of knowledge about natural laws, or are at least in theory capable of acquiring such knowledge, so we are justified in rejecting any metaphysical view that would imply otherwise.

Alternatively, Demarest's definition of a w-potency distribution as ‘a possible distribution of only potencies appearing in w’ might be interpreted to mean a distribution of all the potencies at some possible world, w*, where the only potencies found at w* are potencies that are also found at w. On this reading, the distribution of mass at a world containing schmass would not be a w0-potency distribution, where w0(mass, charge). But distributions of mass at worlds with just mass, for example, as well as other possible distributions of mass and charge at worlds with no other potencies besides, would count as w0-potency distributions. On this interpretation, Demarest goes for option 2. Accordingly, when Demarest writes, regarding the impoverished world: ‘The laws of this world will systematize . . . all worlds that contain mass’, she must be read as speaking elliptically for ‘all worlds that contain only mass’. If this were the intended interpretation, I would suggest the following modification to the definition of the potency-BSA:

However, just as the distribution of mass, for all we know, might be radically unfamiliar at worlds where schmarge is instantiated, for all we know, the distribution of mass might be radically unfamiliar at worlds absent, say, charge. We inhabit a world where both mass and charge are instantiated (as well as other potencies), and in our world mass is distributed as it is, and we can make certain inferences about the possible distributions of mass. What we cannot know, I suggest, is how the absence of charge at a world would affect the behavior of masses, and this imposes a restriction on the range of possible mass distributions that we are able to know.

This concern is driven, in part, by reflection on the apparent fine-tuned-ness of the universe. It is often suggested that had certain fundamental physical constants been even slightly different, a radically different universe would have resulted; one without any carbon-based life or even any coalesced matter, perhaps:

But if minor tweaks to physical constants would result in such a radically different universe, it seems plausible that a big change—the omission of a ubiquitous fundamental potency, such as charge—might result in a world that is utterly unrecognizable. These considerations might reasonably inspire a distinct lack of confidence in our ability to know much at all about what such worlds would be like, including with respect to, say, how mass is distributed. It is better, then, not to allow those likely unknowable possible distributions of mass in such radically different worlds to be relevant to the actual laws.

It might be responded that given the success science has enjoyed when it comes to isolating potencies from each other, we, at the actual world, can be confident in our ability to make inferences about the possible behavior of, say, massive bodies in the absence of charges. But aside from the physical implausibility of the idea that we might completely isolate mass from charge, we cannot ever make it the case that mass is instantiated in a world where charge is uninstantiated and that we are there to observe the results. The skeptical concern is not that masses might behave oddly when isolated under lab conditions from the effects of charge at a world in which charge is nonetheless instantiated. The worry is that masses might behave oddly when instantiated in a world at which charge is nowhere instantiated—call this an S-type hypothesis. No lab can create these conditions; we are all world-bound.

There may be a temptation to dismiss S-type hypotheses as no more problematic than run-of-the-mill external world skepticism. However, S-type hypotheses are of a very different kind from run-of-the-mill skeptical hypotheses. A typical run-of-the-mill skeptical attack on knowledge argues that since I cannot know I am not a brain-in-a-vat (BIV), I cannot know all sorts of things about the actual world, such as that I have hands, because having hands is inconsistent with being a BIV. S-type hypotheses, by contrast, do not threaten our knowledge of the actual world; they threaten our modal knowledge. The fact that I cannot rule out the hypothesis that mass is distributed very strangely in worlds absent charge limits what I can know about other possible worlds. Furthermore, reflection on the apparent fine-tuned-ness of the universe provides S-type hypotheses with at least some prima facie plausibility not enjoyed by, say, the run-of-the-mill skeptical hypothesis that I am a BIV.

Given these differences, responses to run-of-the-mill skepticism should not be expected to be effective against the skeptical threat posed by S-type hypotheses. Consider, for example, a typical externalist response to run-of-the-mill skepticism (e.g., Nozick Reference Nozick1981), according to which a belief counts as knowledge just in case it is true and it tracks the truth at nearby worlds. Assume that I inhabit the actual, non-BIV-world and that I have a true belief that I have hands. This belief counts as knowledge because in nearby worlds in which I am handless (perhaps due to some unfortunate accident) I do not believe that I have hands and in nearby worlds where I do have hands I believe that I do. Sure, my belief would fail to track the truth at the BIV-world, but knowledge does not require truth-tracking at such distant worlds, on this account. This type of response justifies the dismissal of run-of-the-mill skeptical hypotheses by showing them to be compatible with much of my knowledge as well as by emphasizing the fact that run-of-the-mill skeptical hypotheses themselves enjoy no prima facie plausibility for being so distant.

No such response is available to the threat posed by S-type hypotheses. It is consistent with my having hands that, say, mass is distributed very strangely in worlds absent charge. Thus, it does nothing to quell the skeptical threat of S-type hypotheses to show that everyday knowledge of the actual world is consistent with our inability to rule them out. Furthermore, and as mentioned above, S-type hypotheses enjoy at least some prima facie plausibility once we reflect on the fine-tuned-ness of the universe. Unfortunately, I lack the space here to survey all possible responses to run-of-the-mill skepticism. But plausibly the point will extend to other responses given the very different kind of threat posed by S-type hypotheses compared with that posed by run-of-the-mill skeptical hypotheses and given the fact that the former, but not the latter, enjoy at least some prima facie plausibility. I thus take these considerations to show that S-type hypotheses should not be immediately dismissed as on a par with run-of-the-mill skepticism.

On the other hand, it might be argued that the skepticism ushered in by S-type hypotheses, if accepted, proves too much; for ought we, at the actual world, not also think that we cannot know how mass would behave at a world absent, say, Bill Clinton (BC)? The obvious response is that we have lots of evidence to suggest that the distribution of mass is completely independent of BC, and hence it seems reasonable to infer that the distribution of mass would be unaffected by his absence. Of course, we cannot rule out the logical possibility that BC's existence plays some key role in the law concerning mass, but this hypothesis deserves being taken no more seriously than Russell's teapot.

So why not employ a similar answer when charge is substituted for BC? (Of course, to do so would undermine my argument above.) Well, I note first there is a weak sense in which BC is relevant to the law concerning mass. Insofar as BC is composed of massive particles, he is relevant to the overall cross-world distribution of mass and hence the mass-law. But BC's negligible contribution to the distribution of mass is plausibly far from pivotal to the robustly best system of which the law concerning mass is an axiom. Whereas worlds absent BC would differ negligibly from the actual world, worlds absent any instances of charge whatsoever would be radically different from actuality. It seems at least plausible that in such a radically different world the distribution of mass would be significantly affected, and it is this prima facie plausibility that is lacking in the cases of hypotheses about BC and Russell's teapot. Charge is a ubiquitous, fundamental, potency. The concern is really that the possible distributions of ubiquitous fundamental potencies might be more tightly entwined than we could ever know. It would be a quite different matter to claim that any individual whatsoever might have some crucial, yet unobservable, impact on the possible evolution of the universe. BC is not a ubiquitous fundamental potency and so cannot be substituted for ‘charge’ in, for example, the hypothesis that we cannot know how mass would behave in a world absent charge without significantly altering the claim.

Thus, I suggest taking option 3, which avoids the skeptical problems by rendering only those worlds instantiating all and only the potencies instantiated at w relevant to w’s laws. But if for all that has been said you remain unconvinced, I offer one final consideration in favor of this option. Either S-type hypotheses pose no skeptical threat, for whatever reason, so we can embrace option 2: worlds with only those potencies found at w are relevant to w’s laws. Or S-type hypotheses are a threat and to ensure the epistemic accessibility of the laws we should go for option 3: worlds with all and only those potencies found at w are relevant to w’s laws. In a Pascal's Wager-type move, I suggest that unless we can be completely certain that S-type hypotheses pose absolutely no threat, we should go for option 3. This is because we stand to lose relatively little, perhaps even nothing, by choosing option 3 over option 2—maybe the laws of a world, w, will be slightly less informative than they might have been because they will systematize fewer possibilities. However, if we go for option 2 and it turns out that S-type hypotheses are problematic in the way described, we lose all epistemic access to the laws (which would plausibly count as infinitely bad in the context of an analysis of laws!). We cannot be absolutely certain that S-type hypotheses pose no threat whatsoever; we can only have perhaps a relatively high degree of confidence that they are unthreatening, and therefore option 3 is best.

In the next section I suggest a revision to the potency-BSA that guarantees to avoid skepticism about the laws and that, as I shall argue in 4.1, can satisfy the desiderata set out in section 2.

4.1 Satisfying the Desiderata

The laws at w, according to the revised potency-BSA, are a function of the modal profiles of all and only the potencies instantiated at w. In this subsection I shall say more about how to understand the term ‘modal profile’ in the course of gesturing at how the revised potency-BSA might be thought to satisfy the desiderata outlined in section 2.

I will define the modal profile of a potency, P, as the range of properties with which P is possibly coinstantiated by a property-bearer. In possible worlds talk, the modal profile of P will determine, for any property X, if there is a world, w, at which some individual, x, instantiates P and X. Since the definition does not specify that X is fundamental or sparse—X could stand for a conjunctive property—it captures the idea that the modal profile of P has to do with possible combinations of properties with which P is coinstantiated by a property-bearer. Furthermore, since the definition does not rule out that the Xs with which P is possibly coinstantiated are extrinsic properties—they might be relational—it captures that P’s modal profile determines how instances of P might possibly be distributed in space and time. A particular brick, for example, might coinstantiate toughness and redness, but the brick might also instantiate such extrinsic properties as being in a wheelbarrow or forming part of the foundations of a house. In other words, the property toughness is possibly coinstantiated with the extrinsic property forming the foundations of a house. The modal profile of the property toughness allows for such possibilities. Similarly, the potency electric charge, in virtue if its modal profile, is possibly coinstantiated with the property of partially constituting an atom of carbon—electrons, for example, instantiate electric charge and can also instantiate the extrinsic property of partially constituting an atom of carbon.

The laws at w are thus a function of the modal profiles of all and only those potencies instantiated at w, according to the revised potency-BSA. The laws are efficient summaries of the facts about possible distributions of those potency instances, where the possible distributions of potencies at w are determined by those potencies’ modal profiles. I propose an explanation of what it is for a potency to figure in some distribution in terms of the properties, including extrinsic properties, with which it is coinstantiated. For a given world, w, we thus have a hierarchical grounding structure at the base of which we find the potencies, with their irreducible modal profiles. These modal profiles then ground the possible distributions of the w-potency instances, which in turn ground the laws because the w-laws are summaries of the possible distributions of the w-potency instances that best balance the virtues of informativeness and simplicity.

I can now articulate more precisely how the revised potency-BSA satisfies C. Given a world of potencies, fully capable of ‘pushing and pulling’ things around, or determining their own distributions, in accordance with their modal profiles, there would seem to be no need for additional governing laws. The revised potency-BSA satisfies C because it says that the laws at w are the axioms of the system that best balances the virtues of informativeness and simplicity in its effort to convey all of the information about the distributions of the w-potencies in all possible worlds at which all and only w-potencies are instantiated. The potencies themselves might be thought to do some ‘pushing and pulling’ because their modal profiles metaphysically determine their possible distributions, but the potencies are not laws; the laws are features of a description of the possible distributions of those potencies that best balances the virtues of informativeness and simplicity.

The tougher task faced by any account of Humean laws in an unHumean world is that of satisfying I. Recall that according to dispositional essentialism, dispositional essences have an irreducible modal character that suffices to ground the laws independently of swathes of the fundamental mosaic; desideratum I is not satisfied by dispositional essentialism.

In order to satisfy desideratum I, the revised potency-BSA must understand the laws not as codifying the essences of particular potencies considered in isolation, as dispositional essentialism would have it, but as codifying the possible distributions of all potency instances. As we have seen, it is the potencies’ modal profiles that carry implications for their possible distributions because a potency's modal profile determines the range of properties with which it is possibly coinstantiated, including the distributions in which it can (metaphysically possibly) feature. Thus, talking in terms of modal profiles, as opposed to dispositional essences, facilitates discussion of the present account of laws according to which the w-laws are parts of an efficient integrated description of the possible arrangements of the w-potencies.

Crucially, the possible distributions of w-potencies across worlds instantiating all and only the w-potencies will have to do with the w-potencies considered collectively. The distribution of w-potencies across possible worlds will be determined by the various possible interactions between potency instances. The thought can be illustrated with a macroscopic example. Consider a vase encased in formaldehyde. Among the possible distributions of the stuff in a world, w1, that included vases and formaldehyde, there might be very few possibilities in which a vase encased in formaldehyde at one time, t_earlier , is then shattered at a later time, t_later , but in which there is no time between t_earlier and t_later at which the vase is not encased in formaldehyde. Put more simply, the point is that the possible interaction between the vase and the formaldehyde restricts how those things could possibly be distributed in space and time. In very few possibilities does an unbroken vase become broken in a time span in which it is encased in formaldehyde. Plausibly, fundamental potencies, charge, mass, etc., will exhibit analogous interactions. The w-laws of the revised potency-BSA thus respect desideratum I by summarizing all the information about possible configurations of w-potency instances in a manner that accounts for the various possible interactions between the w-potencies. To capture this information best, we need to ‘zoom out’, so to speak, to understand how the various potencies at a world, with their modal profiles, can possibly interact. No potency, or indeed cluster of potencies, considered in isolation from the entire distribution of potencies at a world could suffice to ground the laws on this view; hence I is satisfied.

I have said that potency instances will interact in various ways determined by their modal profiles. One way in which potencies might interact is by masking each other. The modal profile of the potency charge is such that distinct instances of charge can exert a force on each other. But this ability to exert a force conferred on an instance of charge, e, might be masked if extrinsic factors conspire to make it the case that e never manifests this ability. This is supposed to be analogous to the way in which wrapping a vase in bubble wrap masks its disposition to break. It would seem to follow that there is at least one possible world at which all and only those potencies instantiated at the actual world are instantiated and at which the instances of charge have their ability to exert a force on other instances of charge in accordance with Coulomb's law consistently masked. At this world, it so happens that distinct instances of charge never instantiate the property of exerting a force on each other in accordance with Coulomb's law because something always gets in the way, so to speak. Why, then, should Coulomb's law be an axiom of the best systematization of the possible distributions of all and only the potencies at the actual world? The answer comes, I suggest, from reflection upon the ceteris paribus nature of laws. It is implicit in the formulation of Coulomb's law (and other laws) that intervening factors are absent. All Coulomb's law says explicitly is that separated charges exert a force on each other proportional to the magnitude of their charge and inversely proportional to the square of the distance between them. What is left implicit is that this is only the case in the absence of, say, a nearby black hole or indeed of anything else that may negate the tendency of charged individuals to interact in accordance with Coulomb's law. What Coulomb's law tells us, on the current conception, is that in the absence of intervening factors, that is, ceteris paribus, charged bodies will interact in this and that way. Coulomb's law so conceived seems like a good candidate for entering into a strong, simple systematization of the possible distribution of all and only the potencies in the actual world, and hence it seems to be a good candidate for a law even given its ceteris paribus nature. Indeed, it really should count as a benefit of the present account that it accommodates the ceteris paribus nature of natural laws.

The laws, on this account, form parts of an integrated description of possible potency arrangements; desiderata C and I are satisfied. No potency instance considered in isolation can suffice to ground any law because the laws at a world, w, are the axioms of the best systematization of the possible interactions between the totality of potency instances at w. Possible arrangements of all and only the potencies at w, which are systematized as part of the revised potency-BSA, depend not on potency instances considered in isolation, but rather on the potency instances at w considered collectively.

5.1 Revised Potency-BSA Chances

The revised potency-BSA can account for nontrivial chances in much the same way as that suggested by Lewis: by showing them to follow from general probabilistic laws. However, as I will show, the revised potency-BSA blocks the credence=0 side of the contradiction because it is consistent with the chances of a world, w, that the entire history of w diverges dramatically from what we would expect given those chances.

Consider again the distribution of tritium decay events throughout the actual world, @. Now, if we were to systematize all actual tritium decay events, we might find that close to 50 percent of tritium atoms decay within 12.3 years of coming into existence. Indeed, the traditional BSA might offer this sort of fact as part of an analysis of the probabilistic law according to which the half-life of tritium is 12.3 years—the candidate law will increase the fit of a system. But as has been shown, this probabilistic law assigns nonzero chances to futures that are such that the actual present chances would be different; the bug bites.

According to the revised potency-BSA, however, it is not enough just to systematize @. The laws of @ systematize tritium decay events across all worlds at which all and only those potencies instantiated at @ are instantiated. If, and only if, according to the best systematization of potency distributions across all relevant worlds, ‘tritium has a half-life of 12.3 years’ is an axiom, then this fact will analyze relevant objective chances at @.

The bug does not bite this account. The @-law according to which tritium has a half-life of 12.3 years is consistent with an @-future in which vastly more tritium atoms than have ever existed previously come into being and all decay in well under 12.3 years—call this a recalcitrant future. It would not suffice to undermine the actual probabilistic law if a recalcitrant future were realized in @. This is because, according to the revised potency-BSA, the probabilistic law, which says that tritium has a half-life of 12.3 years, is grounded in a relevant range of possible worlds and their entire histories. Accordingly, while in @ it may turn out that most tritium atoms decay in well under 12.3 years, it can still be true that ‘tritium has a half-life of 12.3 years’ is an axiom of the best systematization of the potency distributions across all relevant worlds and hence a law at @. Since it does not follow that one's rational credence in a recalcitrant future coming to pass conditional on the relevant probabilistic law must be zero, the credence=0 side of the contradiction is blocked. It can be consistently maintained that one's credence in a recalcitrant future ought to be > 0.

At this point, the reader might wonder about the criterion of fit. The revised potency-BSA presents the following picture: all possible worlds are split up into equivalence classes under the relation ‘contains all and only the same potencies as’. Hence, to each world, w, there corresponds one such equivalence class, the w-class. The laws of w are then the axioms of the best systematization of potency distributions across all worlds in the w-class. Until now I have said that the best such system is the one that strikes the optimal strength/simplicity trade-off. However, with the introduction of probabilistic laws, fit must be maximized too. Furthermore, just as strength and simplicity of competing systems are evaluated at the interworld level, so too fit should be evaluated at the interworld level.

If fit were evaluated on a world-by-world basis, different systems would be best according to different w-class worlds; hence w-class worlds would differ with respect to their laws and chances, and the bug would still bite. Assuming that fit is to be evaluated on a world-by-world basis, consider two worlds in a given w-class, w1 and w2, and assume that w1 and w2 have different chances because different systems fit best in each case. Furthermore, assume that some initial segments of the histories of w1 and w2, H_w1 and H_w2 , match perfectly and that w2 contains finitely many chance events according to the laws of w1. Now let F be the proposition specifying the history of w2 after initial segment H_w2 . As there are only finitely many chance events occurring in F, the chance of F according to the laws of w1 is > 0. Accordingly, a subject in w1 whose evidence includes the w1-laws and hence the w1-chances ought to have a > 0 credence in F. But if F were to come to pass, the w1-chances would be different because, by hypothesis, w2, whose entire history is given by H_w2 +F (where H_w1 and H_w2 match perfectly), has different chances from w1. It also follows that the agent in w1 who knows the w1 chances should have 0 credence in F. The bug bites again. The only way out is to evaluate fit not on a world-by-world basis but at the interworld level such that all w-class worlds agree with respect to their laws and hence with respect to their chances.

How, then, is the fit of a system to be evaluated at the interworld level? Sure, the law ‘tritium has a half-life of 12.3 years’ may fit the history of the actual world, @, well, but there are many worlds in the @-class for which this law will be a very poor fit indeed. There may well be worlds in which all tritium atoms decay within a nanosecond and others in which no tritium atom decays in under a million years and everything else in between and more extreme. The hope must be that a system including the law ‘tritium has a half-life of 12.3 years’ fits the overall distribution of tritium decay events across all @-class worlds better than any competing system. Thus, it seems that some weighting function over possible worlds is required. This is a problem faced by any account of chance in terms of possible worlds. What the revised potency-BSA does, then, is shift the problem of chances undermining themselves onto the problem of devising a weighting function over possible worlds. Assuming that the prospects of solving the latter problem are brighter than the prospects of solving the former, this constitutes progress, but I leave further treatment of this issue for another time.