In this section, we will show that a “direct solution” to our puzzle by providing an underpinning relation at least commits one to a heavy-duty metaphysics. So, let us look at possible candidates for underpinning nomic explanations:
(i) Causation: The paradigm explanation-underpinner is causation. However, a causal connection does not seem to be suitable in the case of nomic explanations, since although the antecedent- or initial-conditions of a law may well be causes, a law itself seems to be of the wrong type of entity to figure as a relatum in a causal relation. As Marc Lange aptly notes ‘[a] law is a ‘because’ but not a cause’ (Lange (2009b: p. 288); see also Emery (2019: sec. 4) and Maudlin (2007: pp. 155–156)).Footnote 10 Notice, that this does not rely on restricting causation to event-causation. For instance, on plausible models of general causation such as presented in Woodward and Hitchcock (2003), laws are not eligible candidates for causes either: a law is not the kind of ‘thing’ that can be intervened on (cf. Hicks (forthcoming)).
Moreover, even setting the category-mistake-worry aside, thinking of laws as causing their explanandum facts, states or events seems to leave one with an unhappy choice: Either the (total) antecedent- or initial-condition of a law is never a complete cause, or there is some systematic causal overdetermination going on: although the (total) antecedent- or initial-condition is a full cause of the explanandum fact, state or event, the corresponding law always contributes some additional causal ‘influence’.
Finally, causation seems to be inadequate if we want to make room for synchronic non-causal nomic explanations by so-called laws of coexistence such as the law for the simple pendulum, Ohm’s law and Boyle’s law (see Hempel 1965: p. 352).Footnote 11
(ii) Laws (of nature): The underpinning relation we are looking is an explanatory relation that establishes an invariant connection between (facts about) laws and their instances. So, maybe the underpinning relation is itself a law. Moreover, even if it is claimed that nomic explanations are underpinned by causation or a non-causal explanatory relation like grounding (see (iii) below), it seems plausible that a corresponding law is working in the background (see Paul and Hall (2013: pp. 7–9) for claiming that causation is law-governed and see Kment (2014), Schaffer (2017a) and Wilsch (2015) for the role of laws in grounding).
However, if the underpinning is provided by a law, in order for a law l to (partially) explain one of its instances i, there must be a further ‘higher-order’ law l* (involved in the) underpinning (of) this explanation. Worse, a regress seems to be lurking by reapplying our two principles: by explanatory laws, l* itself has an explanatory capacity. Since l* apparently connects l and i, also l* seems to (partially) explain i. However, that explanation in turn needs to be backed according to explanatory realism and thus (given the assumption under scrutiny that nomic explanations are underpinned by laws) a further law l** is needed and so on (see Bird (2005) for a similar regress-objection against Armstrong’s theory of laws).
So, the prima facie innocuous assumptions of explanatory laws and explanatory realism seem to lead to the postulation of additional ‘higher-order’ laws and even to an endless cascading of such laws.Footnote 12 Is this regress benign or vicious? We do not have a general story about what makes a regress vicious. This regress, however, seems to be troubling for at least three reasons: First, postulating an infinity of different laws violates a parsimony constraint on fundamental metaphysics: since explanatory laws concerns fundamental laws, we end up with an infinite array of fundamental laws involved in the explanation of a single instance. Second, arguably one of the main reasons for introducing metaphysically robust laws is their role in explaining their instances. However, as the regress shows, in order to do exactly this job, further laws need to be introduced. And third, usually fundamental laws are held to be discovered by fundamental physics. However, it is highly unlikely that one day physicists come up with an infinite array of such ‘higher-order’ laws.
Can this regress be avoided?Footnote 13 A first option would be to restrict explanatory laws to first-order laws only. However, such a general restriction is not desirable, since plausibly at least some higher-order laws have explanatory power (see, e.g., Lange 2009a: sec. 3.5 for arguing that symmetries are meta-laws that explain conservation laws). If instead explanatory laws is restricted only to first-order and some higher-order laws, in order to avoid adhocery, a reason would be needed why only some higher-order laws have an explanatory capacity other than just stopping the above described regress. A second option to avoid the regress might be to postulate a ‘generic’ higher-order law like that it is a law that laws explain their instances. The idea is that the postulation of such an additional law would not create an endless regress of laws, since it can do all the ‘higher-order-explaining’: that a particular law l (partially) explains one of its instances i, is underpinned by the law that laws explain their instances. That the law that laws explain their instances explains why l (partially) explains i, is again underpinned by the law that laws explain their instances and so on. For this to work, however, at a minimum the law that laws explain their instances must be applicable to itself. However, a self-governing-law is quite unlike all the laws we know from science. We cannot discuss that option further here, but such an ‘exotic’ self-governing-law seems similarly problematic as the postulation of a cascading of different laws.
(iii) Grounding: It has been argued that laws explain by grounding their instances (see Emery 2019; cf. Hildebrand 2020). Assuming that grounding is a constitutive type of explanation and that, in the case of full grounding at least, the grounded is ‘nothing over and above’ its ground (see, e.g., Fine 2001; Schaffer 2009), a grounding-account of nomic explanation is hardly acceptable for Humeans, DTA-theorists or dispositionalists.Footnote 14
There are two challenges for grounding-theorists about nomic explanations: First, several theorists (e.g., Armstrong 1983; Carroll 1994; Maudlin 2007; Tooley 1977) have argued that the laws are to a high degree independent from their particular instances and even that a law could have existed as an uninstantiated law without having any positive instances. Assume that, say, we want to know ‘[w]hy was the event of applying a net force of 1 N to the rock [of mass 1 kg, starting at rest] at t1 followed by the rock traveling at a speed of 1 m/s at t2 [1 second later]?’ (Emery 2019: p. 1541). Remember, according to the second reading (b) of explanatory laws, that it is a law that f = ma (see Emery 2019) fully explains this fact. Now, on standard conceptions of grounding, if the fact that it is a law that f = ma were to (fully) ground the above succession of events, the former would have to metaphysically necessitate the latter. However, that contradicts the independence of laws according to which that it is a law that f = ma could have obtained without this particular instance or even without having any single positive instance at all. In fact, any plausible account of nomological or natural necessity should render it nomologically possible that (at least) one single of a law’s actual positive instances is missing.Footnote 15 That its link to necessitation threatens to make grounding inappropriate for underpinning nomic explanations is also revealed if we take seriously the possibility of fundamental indeterministic laws: the instances of an indeterministic law are not nomologically (and a fortiori not metaphysically) necessitated by and thus are not grounded in (or supervenient upon) their total nomic antecedents; yet also indeterministic laws arguably explain their positive instances.
A second challenge arises in connection with the first reading (a) of explanatory laws.Footnote 16 Consider a case where a dynamical law is supposed to be a partial explainer and so (fully) explains (later) states only together with its antecedent- or initial-conditions. Claiming that the law’s explanatory contribution is underpinned by grounding raises a question: Is the initial- or antecedent-state also a (partial) ground of the later states?Footnote 17 If the answer is ‘yes’, that seems to be in conflict with the widespread assumption that grounding is synchronic in the sense that, in contrast to causation, grounding does not relate distinct events at different timesFootnote 18; and assuming that the universe has no first moment, that may also compromise grounding’s well-foundedness (see Schaffer 2016a: p. 95). If the answer is ‘no’, one seems to be committed to claiming that only the explanatory contribution of the (dynamical) law is backed by grounding whereas the explanatory contribution of the antecedent- or initial-state is backed, say, by causation. So, we have a case of a single full explanation that is somehow a result of two partial explanations that are ex hypothesi each underpinned by a different relation (the first by grounding, the second by causation). But then, a story is needed how causation and grounding ‘interact’ in backing this single full explanation. However, as long as it is assumed that there are significant ‘structural’ differences between causation and grounding, it is questionable whether there is a coherent such story. For instance, the former is supposed to relate completely distinct events or states and to be diachronic whereas the latter is supposed to be constitutive and synchronic (see Bernstein 2016; and Schaffer 2016a: sec. 4.5 for these and further structural differences between grounding and causation). It is hard to see how such disparate relations could combine in one single (full) explanation, and to our knowledge no one has yet addressed how they could.Footnote 19 Worse, on the plausible assumption that something is a partial ground only in case it forms a full ground together with other factors, this alleged ‘mixed’ underpinning is even incoherent: The only thing the law could ‘team up with’ to fully ground the later state is the antecedent- or initial state which ex hypothesi is not a ground but a cause.Footnote 20
In response to these worries, it might be suggested to cap the connections between the notion of ground and the notions of constitution and ‘nothing-over-and-above-ness’ (see Audi 2012). It might be also be denied that (full) grounds metaphysically or nomologically necessitate what they ground (see Emery 2019: sec. 7, and Skiles 2015), that grounding is synchronic, that it cannot relate entirely distinct events, and maybe even that grounding is well-founded (see Bliss 2013). Admittedly, every single deviation from the received view about grounding that is necessary to get a grounding-account of nomic explanation off the ground has been argued for. However, the whole package of deviations seems to leave us with a rather weak notion of ground that threatens to blur the distinction between grounding and causation (see Bernstein (2016); and Schaffer (2016a: sec. 4.5) for relevant distinguishing features; cf. also the discussion in Hildebrand (2020: ch. 3.3)).Footnote 21
(iv) A sui generis governing relation: Finally, a sui generis governing relation might be posited to provide the relevant underpinning (see Hildebrand 2020). A primitive governing relation seems to make the respective account of laws less parsimonious. Proponents of certain forms of non-Humeanism about laws might not be much moved by parsimony-considerations, since they think additional primitive law(maker)s such as sui generis laws (see Maudlin 2007) or instantiations of a higher-order necessitation relation (see Armstrong 1983) need to be posited to explain facts about their instances anyways. However, if governing is a primitive determinative relation it is a further cost over and above the primitive non-Humean law(maker): Non-Humeans then not only posit primitive metaphysical entities to explain the pattern of occurrent facts. They also add, as a further posit, a primitive explanatory nexus to connect their primitive law(maker)s to facts about their instances.Footnote 22
Moreover, postulating a primitive governing relation does not seem to be an option for proponents of any law-account that in some way makes the laws depended on facts about their instances. Since that might not be obvious, let us briefly explain: Although in the relevant literature there is little consensus about what the governing relation is, there is a broad agreement that, in order to govern, the laws must be independent from the facts about their instances in at least two ways: First, the laws must not reduce to or supervene on facts about their instances (see Armstrong 1983: p. 106; Maudlin 2007: p. 175; Hildebrand 2019: p. 176) and, second, it needs to be metaphysically possible that a law remains a law in all of its models (see Loewer 1996: p. 115; Maudlin 2007: p. 67).Footnote 23
The first non-supervenience-condition obviously rules out Humean reductionism. Moreover, although non-supervenience is characteristic for most non-Humean accounts of laws, it is incompatible with dispositionalism about laws which is committed to the claim that ‘[t]he laws of the world supervene on the totality of the instantiations of potencies’ (see Bird 2007: p. 82).Footnote 24 Mumford (2004) takes the fact that a dispositionalist metaphysics violates non-supervenience about laws to be a reason to eliminate laws altogether and to adopt a ‘lawless’ version of dispositionalism. Bird (2007: ch. 9), by contrast, defends non-governing, supervenient dispositionalist laws.Footnote 25
The second independence-condition additionally rules out Armstrong’s Aristotelian version of the DTA-account. An empty absolute space devoid of any particles seems to be a model of Newton’s law of gravitation.Footnote 26 However, since in the entire history of such a world neither mass-, distance-, nor force-universals are instantiated anywhere, the required universals for the Armstrongian necessitation-relation to relate are lacking. Hence, on Armstrong’s Aristotelian view there cannot be a Newtonian empty absolute space and thus Armstrongian laws violate the second independence-requirement on governing laws.
So, it seems that only radical law-accounts such as law-primitivism (see Maudlin 2007) or a Platonist version of the DTA-account (see Tooley 1977; Hildebrand 2020) or of dispositionalism (see Tugby 2013) that cap the laws’ dependence on facts about their particular instances allow for the laws’ independence required for a primitive governing relation.
Finally, it is worth mentioning that such a sui-generis-governing-account seems to face a similar challenge as arises for the grounding-view (iii): According to the first reading (a) of explanatory laws, a (dynamical) law is a partial explainer and only jointly with its antecedent-state fully explains later states. Even if it is granted that laws stand in a sui generis governing relation to the states they explain, the (sub-nomic) initial- or antecedent-states plausibly do not. If instead, say, the antecedent-states explain later states by causing them, the question arises how a primitive sui generis governing relation ‘interacts’ with causation in backing one single full explanation. Again, structural differences between governing and causation make it difficult so see how they could ‘interact’. For instance, causation is usually construed as diachronic relating earlier to later events or states; governing cannot be, since it relates the laws that are widely regarded to be atemporal.
The above discussion of this selection of four candidates for underpinning nomic explanations – namely causation, laws, grounding, and sui generis governing – shows that the tweaking or augmenting of the world’s determinative structure necessary to provide such an underpinning requires adopting radical (non-naturalist) metaphysical positions. Since for many metaphysicians of laws, including Humean reductionists and proponents of various naturalistic versions of anti-reductionism, such radical metaphysics is no option, it is worth examining the ramifications of denying the conjunction of explanatory laws and explanatory realism.