Humean laws and explanation

Abstract

A common objection to Humeanism about natural laws is that, given Humeanism, laws cannot help explain their instances, since, given the best Humean account of laws, facts about laws are explained by facts about their instances rather than vice versa. After rejecting a recent influential reply to this objection that appeals to the distinction between scientific and metaphysical explanation, I will argue that the objection fails by failing to distinguish between two types of facts, only one of which Humeans should regard as laws. I will then conclude by rebutting a variant of this objection that appeals to a principle of metaphysical explanation recently put forward by Kit Fine.

1 Introduction

Humeanism about natural laws can be formulated in terms of the notion of a broadly explanatory expression. Broadly explanatory expressions include nomic expressions, such as ‘is a natural law’ and ‘it is a natural law that’, causal expressions, such as ‘causes’ and ‘because’ (on its causal reading), natural modality expressions, such as ‘it is physically necessary that’ and ‘it is physically likely that’, and narrowly explanatory expressions, such as ‘explains’ and ‘because’ (on its explanatory reading). Humeanism about natural laws is the view that the property expressed by ‘is a natural law’ and the operator expressed by ‘it is a natural law that’ can be expressed using only expressions that are not broadly explanatory.¹ Traditionally, Humeans about natural laws have also held the more general thesis of Humeanism about all broadly explanatory notions, according to which the properties and operators expressed by any broadly explanatory expression can be expressed using only expressions that are not broadly explanatory.² One important motivation for both Humeanism about natural laws and the stronger thesis of Humeanism about all broadly explanatory notions is ideological parsimony: everything else being equal, since Humeans avoid being committed to the fundamental broadly explanatory ideology anti-Humeans are committed to, Humeans are able to endorse a more ideologically parsimonious total theory than anti-Humeans. Humeans have traditionally also claimed that, since fundamental broadly explanatory notions are obscure or epistemically problematic, Humeans avoid the obscure or epistemically problematic notions anti-Humeans traffic in.

The best Humean account of natural laws is widely agreed to be David Lewis’s best systems account (BSA). According to BSA, a natural law is a contingent regularity that is expressed by a theorem in every best system, where a best system is an eligible true deductive system that achieves the best combination of simplicity and strength.³ BSA may also be taken as holding that ‘It is a law that $\varphi$’ expresses the same state of affairs as ‘The state of affairs of it being the case that $\varphi$ is a law’, and that ‘It is a fundamental law that $\varphi$’ expresses the same state of affairs as ‘The state of affairs of it being the case that $\varphi$ is a fundamental law’.⁴ Given these identifications, BSA also provides an account of how the operators expressed by ‘it is a law that’ and ‘it is a fundamental law that’ can be expressed without using any broadly explanatory expressions.⁵

As well as being the best Humean account of natural laws, it is also widely agreed that BSA entails that whether a particular regularity is a natural law is determined by, and hence explained by, all, or nearly all, of the particular matters of fact that make up reality. Loewer, for example, writes that, given BSA:⁶

What makes a proposition a law at a world w is the “vast mosaic of particular matters of fact at w”. There is no part of reality that can be isolated that makes a general proposition lawful or accidental. (Loewer 1996, p. 108)

This consequence of BSA, however, appears to conflict with the fact that laws can help explain their instances. Tim Maudlin puts the objection as follows:

If one is a Humean, then the Humean Mosaic itself appears to admit of no further explanation. Since it is the ontological bedrock in terms of which all other existent things are to be explicated, none of these further things can really account for the structure of the Mosaic itself. This complaint has been long voiced, commonly as an objection to any Humean account of laws. If the laws are nothing but generic features of the Humean Mosaic, then there is a sense in which one cannot appeal to those very laws to explain the particular features of the Mosaic itself: the laws are what they are in virtue of the Mosaic rather than vice versa. (Author’s emphasis) (Maudlin 2007, p. 172)

Lange sums up the objection by writing:

[I]f the Humean mosaic is responsible for making certain facts qualify as laws, then the facts about what the laws are cannot be responsible for features of the mosaic. (Lange 2013, p. 256)

Humeans about natural laws, then, face the following objection from explanation: Humeanism about laws is false since, given the best Humean account of laws, laws are explained by their instances rather than vice versa.

To state this objection more precisely, let ‘$\underline{\phi }$’ symbolise ‘the state of affairs of it being the case that $\phi$’, and say that f partly explains (or helps to explain) g iff f, together with possibly other facts $f_1,\ldots f_n$, jointly explain g.⁷ Suppose that a is F, and that it is a law that all Fs are Gs. Given this, if laws help to explain their instances, then (1) is true.

• 1. The law that all Fs are Gs, together with $\underline{a \hbox { is } F}$, jointly explains $\underline{a \hbox { is } G}$.

Given BSA, whether $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is determined by what the best systems are, which is in turn determined by what things there are and how those things are. As a result, given BSA, whether $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is determined by $\underline{a \hbox { is } G}$, together with many other particular matters of fact, plus plausibly a totality fact (where a totality fact is a fact expressed by a sentence of the form ‘$\forall x(x=a_1 \vee x=a_2 \vee \ldots )$’).⁸ As a result, given BSA, it is plausible that the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is jointly explained by $\underline{a \hbox { is } G}$, together with many other particular matters of fact and a totality fact. Hence $\underline{a \hbox { is } G}$ partly explains the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law. Given $\underline{a \hbox { is } G}$ partly explains the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, however, it follows from (A) that this fact fails to partly explain $\underline{a \hbox { is } G}$.

• A. If f partly explains g, then g does not partly explain f.

Hence, the law that all Fs are Gs does not partly explain $\underline{a \hbox { is } G}$, which conflicts with (1). Hence, given BSA, laws fail to partly explain their instances. Since laws do partly explain their instances, and since BSA is the best Humean account of natural laws, it follows that Humeanism about natural laws is false.⁹

In this paper I will argue that the objection from explanation against Humeanism fails. In Sect. 2, I will first reject a recent influential response to the objection that appeals to the distinction between scientific and metaphysical explanation. In Sect. 3, I will then put forward what I take to be the correct response to the objection before discussing a variant of the objection in Sect. 4 and arguing that it also fails. As well as defending Humeanism from the objection from explanation, these two sections will also provide an answer to the important question of what explanatory relations Humeans should think nomic facts, regularities, and their instances stand in to each other. I will then finish with a brief conclusion in Sect. 5.

2 Loewer’s response

Barry Loewer has defended Humeanism from the objection from explanation by claiming that there are two ways in which one fact can explain another fact: a fact can explain another fact by scientifically explaining it, or it can explain it by metaphysically explaining it. He characterises the difference between these two types of explanation as follows:

The relevant kind of metaphysical explanation is one in which a type of fact—say mental facts—is shown to be grounded in or constituted by some other kind of fact—say neurological fact. Metaphysical explanation need not involve laws and the explanandum and explanans must be co-temporal (if the explanans is a temporal fact or property). Scientific explanation of a particular event or fact need not show that it is grounded in a more fundamental event or fact but rather, typically, shows why the event occurred in terms of prior events and laws. (Loewer 2012, p. 131)

According to Loewer, f can both partly explain g and be partly explained by g, provided the way in which f partly explains g and the way in which g partly explains f are different. In particular, while he thinks (A-s) and (A-m) are both valid, he thinks (A) is invalid.

• A-s. If f partly scientifically explains g, then g does not partly scientifically explain f.

• A-m. If f partly metaphysically explains g, then g does not partly metaphysically explain f.

Loewer thinks that laws partly scientifically explain (but don’t metaphysically explain) their instances, while they are partly metaphysically explained by (but not scientifically explained by) their instances. Since this is consistent with (A-s) and (A-m), he thinks there is no conflict between Humeanism about laws and the fact that laws can partly explain their instances.¹⁰ Unfortunately, there are three problems with Loewer’s response, each of which plausibly shows that it is unsatisfactory. I will discuss each of these in turn.

2.1 Asymmetry

The first problem with Loewer’s response is that the fact (granting that it is a fact) that there are these different types of explanation does not, by itself, provide any reason to think (A) is invalid, and hence does not provide any reason to think that the application of (A) in the argument from explanation fails. For comparison, the fact that there are two types of proper parthood—proper parthood where one of the relata has a smaller volume than the other, and proper parthood where this is not the case—does not provide any reason to think proper parthood is not asymmetric. Similarly, the fact that we can draw a distinction between two types of explanation does not by itself provide any reason to think that explanation fails to be asymmetric. Moreover, no further reason to reject (A) is provided by Loewer’s particular way of drawing this distinction. This is illustrated by the fact that, if we assume that all explanation is either scientific or metaphysical, and we slightly simplify his characterisation so that it holds that f partly metaphysically explains g only if f and g are co-temporal or both non-temporal, and that f partly scientifically explains g only if f is temporally prior to g, then his characterisation will entail (A) rather than refute it!¹¹

Given the high intuitive plausibility of (A), what is needed to shed real doubt on (A) are independently plausible examples of facts f and g, where f partly scientifically explains g while g partly metaphysically explains f. Elizabeth Miller has recently suggested that the following is such an example:

Consider a universe consisting of thirteen particles and fluctuating between two states, $S_1$, in which exactly seven of those particles are blue, and $S_2$, in which exactly seven are green. Suppose that, in both cases, every particle is either blue or green, but there are no further facts about which individual particles will be blue or green, except for this: whenever the universe is in state $S_1$, then at least one of a and b is among the seven blue particles. It seems we might offer a scientific explanation of the fact that a is blue in part by citing the fact that the universe is in state $S_1$. But, plausibly, the fact that a is blue still could help [metaphysically explain] the fact that the universe is in state $S_1$; what it is for the universe to be in state $S_1$ in this instance is for it to contain exactly seven blue particles, with a among them. (Miller 2014, Sect. 4)

Miller’s example, however, is not a convincing case of a failure of (A). While the fact that the universe is in state $S_1$ epistemically supports the fact that a is blue, it is not clear why we should think that it explains why a is blue, anymore than the fact that it looks like someone is hungry explains the fact that someone is hungry, as opposed to merely epistemically supporting that fact.

It is also important to note that, even if Miller’s example was a case where (A) convincingly fails, this would not provide much of a reason to think that the application of (A) in the objection from explanation also fails. Given its plausibility and utility, if there are pathological cases where (A) fails, then it is plausible that a restricted version of (A) which vindicates most of our applications of (A) will still be valid. Miller’s example at most provides reason to restrict (A) so that it does not apply to probabilistic explanation, which is the kind of explanation involved in her example, but instead only applies to deterministic explanation, which is the kind of explanation involved in the objection from explanation. Unless we can find an example that undermines the application of (A) in the objection from explanation, Loewer’s response to the objection from explanation therefore appears unsuccessful.¹²

2.2 Lange’s transitivity principle

The second problem with Loewer’s response is that, as Marc Lange has recently pointed out, it is possible to modify the objection from explanation so that it does not rely on (A), but instead relies on the transitivity principle (T*) that explicitly recognises the distinction between scientific and metaphysical explanation.¹³

• T*. If f partly metaphysically explains g, and g partly scientifically explains h, then f partly scientifically explains h.

Lange argues in favour of (T*) by providing several plausible examples of scientific and philosophical arguments that appear to rely on (T*). Two of Lange’s examples are the following:

[S]uppose that a given balloon expands because of various laws and the fact that the pressure of the gas inside the balloon is greater that the atmospheric pressure outside of the balloon. Then since the fact that the internal pressure is greater than the external pressure is grounded in the value of the internal pressure and the value of the external pressure, it follows from [(T*)] that the internal and external pressures help to scientifically explain why the balloon expands...The internal pressure, in turn, is grounded in the forces exerted by various gas molecules as they collide with the balloon’s interior walls. By [(T*)], then, those forces help to scientifically explain why the balloon expands. (Lange 2013, p. 257)

[A] coin’s chance of landing heads explains its actual relative frequency of landing heads, so if the chance were grounded in the actual relative frequency, then [as (Hájek 1996, p. 79) argues] the actual relative frequency would have to explain itself, which it cannot do. (Lange 2013, p. 257)

The objection from explanation can be reformulated using (T*) as follows: Given Humeanism, laws are partly metaphysically explained by their instances. If laws also partly scientifically explain their instances, then, by (T*), laws partly scientifically explain themselves. Since nothing can partly scientifically explain itself, it follows that laws cannot partly scientifically explain their instances given Humeanism about laws. Since laws can partly scientifically explain their instances, it follows that Humeanism about laws is false.¹⁴

Michael Townsen Hicks and Peter van Elswyk have proposed counterexamples to (T*), as has Miller. Hicks and van Elswyk discuss the following case involving an electron that is part of a lion:

The position of electron e partly metaphysically explains the position of lion L. The position of L scientifically explains the number of prey animals in region R. But the position of electron e does not explain the number of prey animals in region R. For if the electron were elsewhere, L would still be warding prey animals out of R. (Hicks and van Elswyk 2015, pp. 437–438)

This case, however, is hardly a convincing counterexample to (T*), since the counterfactual principle (CD) Hicks and van Elswyk rely on is not valid.

• CD. If f partly scientifically explains g then, had f not obtained, g would not have obtained.

(CD) can fail, for example, in a case where a man is simultaneously shot by three bullets that jointly kill him. In such a case, the fact that bullet 1 is fired partly scientifically explains the fact that the man dies, since this fact, together with the fact that bullet 2 is fired, the fact that bullet 3 is fired, and some more facts jointly scientifically explain the fact that the man dies. However, it is still the case, we may suppose, that, had the bullet 1 not been fired, the man would still have died due to the firing of the other two bullets.

The situation in the example Hicks and van Elswyk describe seems relevantly similar. The fact that e has a certain position, together with facts about the other properties of e, and facts describing the properties of all the other particle parts of L and all the other lions in region R and how they are related to each other, may scientifically explain the number of prey animals in region R, even though, had e been elsewhere, the number of prey animals in R would have been the same. If so, then the location of e does partly explain the number of prey animals in R, and the example fails to be a counterexample to (T*).¹⁵

Miller discusses the following case involving James the snail.¹⁶ Suppose p and q are both atomic parts of James the snail, l is a location property, and B is a biological property. Suppose that the fact that p has l at $t_1$, together with facts about the location of James’s other atomic parts at $t_1$ metaphysically explains the fact that James has B at $t_1$, which in turn scientifically explains the fact that q has E at $t_2$. Suppose also that B is a member of a plurality of mutually incompatible biological properties such that, for any B* that is a member of this plurality, p having l at $t_1$ is compatible with James having B* at $t_1$. Finally, suppose that James having B at $t_1$ is compatible with p not having l at $t_1$. Miller claims that, in this example, since p having l at $t_1$ is compatible with James having a number of different incompatible biological properties, and since James having B at $t_1$ is compatible with p failing to have l at $t_1$, p having l at $t_1$ does not scientifically explain q having E at $t_2$, and hence (T*) fails in the example.

Miller’s argument that (T*) fails in her example relies on the validity of what might be called Miller’s principle (MP).

• MP. If f metaphysically explains g, and g scientifically explains h, f does not scientifically explain h when

  • (a) g is among a plurality of mutually incompatible states of affairs, all of whom are more general, or at a higher level, than f, and all of whom are compatible with f obtaining, and

  • (b) g is compatible with f not obtaining.

It is not clear, however, why we should think that (MP) is valid.

One reason to doubt (MP) is that, if it were valid, it would not only entail that the James the snail example is a case where (T*) fails, it would also entail that (T*) fails in Lange’s balloon example above, which was meant to illustrate the plausibility of (T*). In Lange’s example, facts about the various forces exerted by various gas molecules when they collide with the balloon’s interior walls metaphysically explain what internal pressure the balloon has, which in turn partly scientifically explains the fact that the balloon expands. Each fact about these various forces is individually compatible with the balloon having a different internal pressure than the one it actually has, and hence is compatible with a plurality of more general facts including the fact describing the balloon’s actual internal pressure. Since the fact describing the balloon’s actual internal pressure is also compatible with the non-obtaining of any one of the facts about the various forces, it follows from (MP) that these facts do not scientifically explain the fact that the balloon expands. However, in Lange’s example, these facts plausibly do scientifically explain the fact that the balloon expands.

Another principle Miller might appeal to in order to argue that (T*) fails both in the James the snail case and in Lange’s balloon case is (Sp), where f is more specific than g iff f necessitates g but is not necessitated by g.¹⁷

• Sp. If f is more specific than g, and g explains h, then f does not explain h.

The rationale for (Sp) is that, if g explains h, and f is more specific than g, then f will be too specific to explain h since it will contain explanatorily irrelevant information. A plausible instance of (Sp) is the following. Suppose Suzy breaks a window at $t_2$ by throwing a rock at it at $t_1$. The conjunction of the fact that Suzy throws a rock at $t_1$ with a number of other facts then explains the fact that the window breaks at $t_2$. If we conjoin this conjunctive fact with another fact, say the fact that Suzy once ate pancakes, thereby obtaining a more specific fact, then the more specific fact will have some explanatorily irrelevant information, namely information about Suzy having eaten pancakes. As a result, the more specific fact will not explain the breaking of the window.¹⁸

If (Sp) is valid then it can be used to show that (T*) fails in both the James the snail example and Lange’s balloon example. In the James the snail example, for instance, if the location of p at $t_1$ partly explains q having E at $t_2$, then it must be that the location of p at $t_1$ conjoined with a number of other microscopic facts explains q having E at $t_2$. Since this conjunction necessitates the fact that James is in B at $t_1$ but is not necessitated by it, however, and since the fact that James is in B at $t_1$ explains the fact that q is in E at $t_2$, it follows from (Sp) that this conjunction does not explain the fact that q is in E at $t_2$. Hence, the location of p at $t_1$ does not partly explain the fact that q is in E at $t_2$, which entails that (T*) fails. A similar argument shows that, given (Sp), (T*) fails in Lange’s balloon example as well.

If (Sp) is valid, then, (T*) fails in both the James the snail example and Lange’s balloon example, and Lange’s modified objection from explanation is greatly undermined. There are reasons, however, to suspect that it is (Sp) that fails to be valid rather than (T*). First, the fact that (Sp) entails that the facts about the forces in Lange’s example do not explain the expansion of the balloon provides some reason to reject (Sp), since, as noted above, the claim that they do explain this fact is prima facie plausible. Second, and more importantly, (Sp) seems to have clear counterexamples. For example, (Sp) not only entails that the fact that Suzy threw a rock at $t_1$ and once had pancakes fails to help explain the fact that the window broke at $t_2$, (Sp) also entails that the fact that Suzy threw the rock at $t_1$ does not help explain this fact either. The reason for this is the following. If the fact that Suzy threw a rock at $t_1$ did help explain this fact, then the conjunction of this fact about Suzy throwing the rock at $t_1$ plus some other facts should explain this fact. However, given (Sp), this conjunction doesn’t explain this fact, since there is a less specific fact that does explain it, namely the purely existential fact that someone having such and such properties throws a rock at $t_1$. Hence, given (Sp), while the fact that there is something that throws a rock at $t_1$ helps to explain the fact that the window broke at $t_2$, the fact that Suzy threw a rock at $t_1$ does not. A Humean might bite the bullet at this point and claim that the fact involving Suzy does not strictly explain the fact that the window broke. However, it would clearly be better if Humeans could respond to the objection from explanation without being forced to make such a commitment.¹⁹

I will discuss one further kind of proposed counterexample which Hicks and van Elswyk put forward, and which involves a kind of causation they call “immanent causation”. Suppose there is immanent causation, where immanent causation is causation that “proceeds from an object not by way of the object’s parts, but from the object as a whole” (Hicks and van Elswyk 2015, p. 439). Given there is such causation, we could have a case where microscopic facts about a macroscopic object metaphysically explains a macroscopic fact about that object, which is then causally responsible for another fact h that is not caused by the microscopic facts, and hence is arguably not scientifically explained by them. Hicks and van Elswyk do not give any specific example of such a case, but such an example might be the following. Suppose that, whenever something is a perfect solid sphere of 1 m radius the fact that it is a solid sphere of 1 m radius is causally responsible for each of its atomic parts going into state E, and suppose that this is always a case of immanent causation. Suppose x is a solid sphere of 1 m radius. Then, according to Hicks and van Elswyk, while the microscopic facts describing the locations of the atomic parts of x metaphysically explain the fact that x is a solid sphere of 1 m radius, which in turn scientifically explains each fact describing an atomic part of x being in state E, these latter facts are not scientifically explained by the former microscopic facts due to the lack of a causal connection between them. If this is the case, then (T*) fails.

As a counterexample to (T*), the above case has several problems. First, the case relies on the claim that immanent causation is possible, but it would be better if Humeanism could be defended without having to rely on such a controversial thesis. Secondly, the counterexample relies on a principle concerning scientific explanation such as (CSE), where ‘nomic connection’ is Hicks and van Elswyk’s name for the relation that backs explanation in the case where a natural law helps to explain one of its instances.

• CSE. If f scientifically explains g, then g is either causally responsible for g, nomically connected to g, or a combination of these.

It is not clear, however, whether the distinction between scientific explanation and metaphysical explanation needs to be drawn so that (CSE), rather than for example (CSE*), is true, where metaphysical grounding is a relation that backs metaphysical explanation.

• CSE*. If f scientifically explains g, then g is either causally responsible for g, nomically connected to g, or a combination of these, or a combination of one or both of these together with metaphysical grounding.

Finally, even if (T*) fails due to the possibility of immanent causation and a principle like (CSE), Lange’s modified version of the objection from explanation can arguably be easily fixed to avoid this problem by replacing (T*) with (T), which avoids the distinction between scientific explanation and metaphysical explanation altogether.

• T. If f partly explains g, and g partly explains h, then f partly explains h.

(T) is supported by Lange’s examples just as much as (T*).²⁰ If (T*) faces problems due to where the precise division between scientific and metaphysical explanation should be drawn, Lange can therefore forsake the distinction and use (T) instead in formulating his version of the objection from explanation. Without more convincing counterexamples to (T*) (or to (T)), then, Loewer’s response to the objection from explanation again appears to be inadequate.²¹

2.3 Metaphysical laws

The final problem with Loewer’s response is that it does not apply to Humeans who wish to be Humeans about metaphysical laws as well as natural laws. Three possible examples of metaphysical laws are (P), (S) and (L).

• P. For any xs, there is a mereological fusion of the xs.

• S. For any xs, if there are finitely many xs, then there is a set whose members are all and only the xs.

• L. For any x, if x has location l, then $\{x\}$ has location l.

According to Humeanism about metaphysical laws, the property expressed by ‘is a metaphysical law’ and the operator expressed by ‘it is a metaphysical law that’ can be expressed using only expressions that are not broadly explanatory. Humeanism about metaphysical laws is entailed by Humeanism about all broadly explanatory notions and is motivated by the same considerations that motivate Humeanism in general: namely, ideological parsimony and the avoidance of obscure or epistemically problematic primitives. It is therefore natural for Humeans about natural laws to also endorse Humeanism about metaphysical laws.

If there are metaphysical laws, such as (P), (S) and (L), a Humean about metaphysical laws needs to be able to explain how a fact like the fact ascribing lawhood to the fact expressed by (L) can be expressed using only expressions that are not broadly explanatory. The natural way of attempting to do this is to extend Lewis’s BSA so that it applies to metaphysical laws as well as natural laws. For example, a Humean might claim that the key difference between metaphysical laws and natural laws is that the former are necessary while the latter are contingent. They might then claim that the fact that p is a metaphysical law is the fact that p is a necessary regularity that is expressed by a theorem in every best system. Similarly, they might claim that the fact that p is a fundamental metaphysical law is the fact that p is a necessary regularity that is expressed by an axiom in every best system.

Given a Humean endorses a best systems account of metaphysical laws along the above lines, she is going to face the same objection from explanation with respect to metaphysical laws as she faces with respect to natural laws. Given there are metaphysical laws, such laws can presumably help explain their instances just as much as natural laws can. For example, given (L) expresses a metaphysical law, this law, together with the fact that Dan has location l, presumably explains the fact that $\{\text {Dan}\}$ has location l. A best systems type account of metaphysical laws, however, just like the best systems account of natural laws, plausibly entails that what the metaphysical laws are is partly explained by the instances of those laws. Hence, we have the same conflict as we have in the case of natural laws: metaphysical laws partly explain their instances, but, given Humeanism, they appear unable to do this. In this case, however, Loewer’s response to this conflict appears clearly inadequate, since metaphysical laws presumably help metaphysically explain their instances rather than merely help scientifically explain them.

3 A better response

I argued in Sect. 2 that Loewer’s response to the objection from explanation against Humeanism, which involves distinguishing between scientific and metaphysical explanation, and rejecting (A), (T*) and (T), is unsuccessful. In this section, I will argue that, fortunately for Humeans, there is a more straightforward response that is successful. Instead of distinguishing between two kinds of explanation, this better response merely distinguishes between regularities, such as $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, and facts that ascribe lawhood to regularities, such as the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law. The former kind of facts are distinct from the latter kind of facts since, at least if the former are contingent, the former can obtain without the latter obtaining. With the distinction between these two kinds of facts firmly in mind, Humeans can respond to the objection from explanation as follows.

Given BSA, whether $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is determined by what the best systems are, which in turn is determined by $\underline{a \hbox { is } G}$, together with a multitude of other particular matters of fact and a totality fact. As noted in Sect. 1, it plausibly follows from this that $\underline{a \hbox { is } G}$ partly explains the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law. However, as pointed out above, the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is distinct from the fact $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$. Hence, it does not follow from it being the case that $\underline{a \hbox { is } G}$ partly explains the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law that $\underline{a \hbox { is } G}$ partly explains $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$. Hence there is no conflict with (A) in accepting that $\underline{a \hbox { is } G}$ partly explains the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, while also holding that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ partly explains $\underline{a \hbox { is } G}$. Since Humeans think that it is the regularity $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ that is the law that all Fs are Gs, rather than the second order fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, there is therefore no problem with Humeans holding that the law that all Fs are Gs partly explains $\underline{a \hbox { is } G}$.

On this response to the objection from explanation, Humeans should accept the following two claims. First, laws are certain regularities, such as $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, and these regularities partly explain their instances but are not partly explained by their instances. Second, higher order facts that ascribe lawhood to other facts, such as the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, aren’t themselves laws, don’t partly explain the instances of their associated laws, but are rather partly explained by those instances. Given these two claims, there is no incompatibility between laws partly explaining their instances, their instances partly explaining higher order facts ascribing lawhood to laws, and the principle of asymmetry (A). As a result, the conflict alleged by the objection from explanation is resolved.

The claim that facts ascribing lawhood to laws don’t partly explain the instances of those laws might at first seem counterintuitive, since (2) might at first seem plausible.

• 2. The fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, together with $\underline{a \hbox { is } F}$, explains $\underline{a \hbox { is } G}$.

While the above response provides a simple resolution to the apparent conflict between laws explaining their instances, the instances of laws determining what the laws are, and (A), the fact that it requires rejecting (2) might be thought to provide a good reason to reject the response. A Humean who endorses the response, however, can make the following two points.

First, there is a good reason to think that, provided they endorse BSA, any Humean, no matter how they respond to the objection from explanation, should reject (2). Given BSA, the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is expressed by a theorem in every eligible true deductive system that achieves the best combination of strength and simplicity. In short, given BSA, the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law is the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is expressed by a theorem in every best axiomatization of all the particular matters of fact. The latter fact, however, does not seem to be of the right kind to be able to help to explain any particular matter of fact, since facts of the form ‘the fact that p is expressed by a theorem in every best axiomatization of facts of type T’ plausibly cannot help to explain any fact of type T. The claim that this fact cannot help to explain any particular matters of fact can be bolstered, given (A), by the similarly plausible claim that this fact is explained by particular matters of fact. However, even without (A), and hence even without this extra support, the claim remains highly plausible. In other words, even if p can help to explain q, even though q helps to explain p, the above case appears not to be an instance of this. Hence, given BSA is the best Humean theory of laws, Humeans should arguably reject (2), even if they also reject principles like (A) and (T).

Second, Humeans can give the following plausible argument that they incur no significant cost in rejecting (2).²² Humeans should admit that there is a traditionally important and still influential concept of lawhood on which (2) is true. This is the concept of lawhood on which lawhood requires a divine lawmaker. On this concept of lawhood, the fact that it is a law that $\varphi$ is the fact that God commands that $\varphi$. Since God is omnipotent, if God commands that all Fs are Gs, then the fact that God makes this commandment explains why all Fs are Gs, and hence explains why a is G given a is F. Hence, on the divine lawmaker concept of lawhood, the fact that it is a law that all Fs are Gs partly explains the instances of the law that all Fs are Gs. Hence, given the identification of the fact that it is a law that all Fs are Gs with the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law, it follows that, on the divine lawmaker concept, the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law does partly explain $\underline{a \hbox { is } G}$, just as (2) claims.²³

Humeans, however, do not aim at giving an analysis of the divine lawmaker concept of lawhood. Instead, they aim to give an analysis of the concept of lawhood that those who wish to have a concept of law not involving God either do, or should, adopt. Hence, the fact that (2) is true on the divine lawmaker concept does not entail that Humeans should not reject (2). Moreover, the fact that the divine lawmaker concept remains influential, that (2) is true on the divine lawmaker concept, and that people who officially reject the divine lawmaker concept aren’t always clear about this rejection or about what concept they are employing in its place, provides an explanation for why (2) might seem true, even to those who officially wish not to employ the divine lawmaking concept.

Humeans can also claim that those who don’t employ the divine lawmaker concept of lawhood either do employ, or should employ, a concept of lawhood according to which laws are those things that play all of the most important roles laws play in science, such as supporting counterfactuals, constraining physical possibility, and helping to explain particular matters of fact. Humeans can then claim that the things that play these roles are the things that are laws according to BSA.²⁴ Since being such that the fact ascribing lawhood to one partly explains one’s instances is not an important role laws play in science, Humeans can therefore claim that they incur no significant cost in rejecting (2).²⁵

While Humeans cannot allow facts about which facts are laws to partly explain the instances of those laws, it is important to note that they can easily account for the fact that scientists are interested in such facts: Scientists are interested in knowing such facts as the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law because they want to know what the laws are, and they want to know what the laws are because laws are explanatorily powerful. Moreover, although Humeans cannot allow facts about which facts are laws to partly explain the instances of laws, they can allow such higher order facts to play another important explanatory role. In particular, they can claim that, while the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law cannot partly explain $\underline{a \hbox { is } G}$, it can partly explain why $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ partly explains $\underline{a \hbox { is } G}$. In defence of this claim, they can claim that the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law necessitates that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ explains $\underline{a \hbox { is } G}$ when combined with $\underline{a \hbox { is } F}$. They can also point out that, if $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ was not a law, then, if it obtained at all, it would only be an accidental regularity, and hence would not be able to combine with $\underline{a \hbox { is } F}$ to explain $\underline{a \hbox { is } G}$. They might then claim that this provides a good reason to think that it is only because $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ can combine with $\underline{a \hbox { is } F}$ to explain $\underline{a \hbox { is } G}$, and so conclude that there is good reason to think that the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law partly explains why $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ partly explains $\underline{a \hbox { is } G}$.

Given Humeans hold that facts that ascribe lawhood to laws can explain the ability of those laws to partly explain their instances, they can give a further reason why (2) might seem true, even though it is false. (2) might seem true since we might fail to distinguish between a fact explaining another fact and a fact explaining why a related lower order fact explains a further fact.²⁶ That is, (2) might at first seem true because we fail to distinguish it from the true (3).

• 3. The fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ is a law partly explains the fact that ($\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, together with $\underline{a \hbox { is } F}$, explains $\underline{a \hbox { is } G}$).

Since it is easy to confuse (3) with (2), or to falsely infer (2) from (3), the fact that (3) is true provides a good explanation for why (2) might at first seem true, even to someone who transparently disavows the divine lawmaker concept of lawhood.

To sum up: Humeans have a straightforward response to the objection from explanation. While it is true that, given Humeanism, facts about what regularities are laws cannot help to explain the instances of those laws, this does not prevent the laws themselves from explaining those instances. Humeans, then, can allow laws to help explain their instances.²⁷

4 Fine’s principle

Some philosophers might think that a variant of the objection from explanation poses a greater threat to Humeanism than the version considered in the previous three sections. Instead of appealing to features of BSA, as the original version does, the variant objection appeals to the principle that regularities are metaphysically explained by their instances, which is a principle that has recently been endorsed by Kit Fine. It is, in fact, this variant of the objection that Miller, as well as Hicks and van Elswyk, in effect address, rather than the original one.

The principle endorsed by Kit Fine is (FP), where ‘T($a_1,\,a_2,\ldots$)’ expresses the totality state of affairs that obtains iff $a_1,\,a_2,\ldots$ are all the things there are.²⁸

• FP. If $a_1,\,a_2,\ldots$ are all the things there are, then $\underline{\varphi (a_1)}, \underline{\varphi (a_2)},\ldots$ and $\underline{\hbox {T}(a_1, a_2,\ldots )}$ jointly metaphysically explain $\underline{\forall x \varphi (x)}$.

The variant objection is the following. Suppose a is F and that it is a law that all Fs are Gs. Then (4) follows from (FP).

• 4. If $a_1,\,a_2,\ldots$ are all the things there are, then $\underline{\lnot Fa_1 \vee Ga_1}, \underline{\lnot Fa_2 \vee Ga_2},\ldots$ and $\underline{\hbox {T}(a_1, a_2, \ldots )}$ jointly metaphysically explain $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$.

It follows from a being F, and it being a law that all Fs are Gs, that a is G. Since $\underline{a \hbox { is } G}$ metaphysically explains $\underline{\lnot Fa \vee Ga}$, it therefore follows from (4) and the transitivity of metaphysical explanation that $\underline{a \hbox { is } G}$ metaphysically explains $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$. Hence, by (A), $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ doesn’t partly explain $\underline{a \hbox { is } G}$. Hence, laws cannot partly explain their instances given Humeanism about laws, and hence Humeanism about laws should be rejected.

One problem with the variant objection is that, if it works, then it also raises problems for anti-Humean accounts of laws. According to typical anti-Humean accounts, laws aren’t facts of the form $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, but are instead facts of the form $\underline{\hbox {It is a law that all } F\hbox {s are } G\hbox {s}}$, where this fact might get identified with a fact like $\underline{\hbox {Being } G\hbox { is causally necessitated by being } F}$.²⁹ Anti-Humeans also typically claim that laws like $\underline{\hbox {It is a law that all } F\hbox {s are } G\hbox {s}}$ help to explain both the regularity $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ and the instance $\underline{a \hbox { is } G}$, given $\underline{a \hbox { is } F}$. It is natural for them to go further and hold that It is a law that $\underline{\varphi }$ directly explains $\underline{\varphi }$, where f directly explains g iff (a) f explains g, and (b) there is no fact h such that f explains h, and h explains g. Given this, it is natural for anti-Humeans to hold that $\underline{\hbox {It is a law that all } F\hbox {s are } G\hbox {s}}$ and $\underline{Fa}$ jointly explain $\underline{Ga}$ by virtue of the fact that $\underline{\hbox {It is a law that all } F\hbox {s are } G\hbox {s}}$ explains $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, and the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ and $\underline{Fa}$ jointly explain $\underline{Ga}$. However, this natural claim for an anti-Humean to make conflicts with the combination of (FP) and (A). The reason for this is that it follows from (FP) that $\underline{a \hbox { is } G}$ partly explains $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$, which is impossible given (A) and the fact that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ partly explains $\underline{a \hbox { is } G}$. If (FP) is valid, then, both Humeans and anti-Humeans have difficulty allowing laws to partly explain their instances.³⁰

The other problem with the variant objection is that it is unclear why we should believe (FP), especially given Fine (and also Hicks and van Elswyk who also endorse a version of the principle) do not give any argument for it. In particular, it is not clear why we should think $\underline{a \hbox { is } G}$ partly explains $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ as (FP) dictates, as opposed to thinking that $\underline{\hbox {All } F\hbox {s are } G\hbox {s}}$ and $\underline{a \hbox { is } F}$ jointly explain $\underline{a \hbox { is } G}$. Given (A), we need to choose between these options. However, prima facie, the first option is not more plausible than the second option.

One reason Fine might give for endorsing (FP) is that it allows a simpler account of the principles that govern metaphysical explanation. It is plausible to think that purely accidental regularities are explained by their instances rather than the other way round. For example, if there are only two people, Jack and Jill, that own exactly 217 pairs of shoes and they each have 20 cents in their pocket, then the accidental regularity expressed by ‘Everyone who owns exactly 217 shoes has 20 cents in their pocket’ is plausibly explained in terms of particular matters of fact about Jack and Jill, rather than vice versa. (FP), then, plausibly works for accidental regularities, and so simplicity considerations might be thought to suggest that it should work for all regularities. The small gain in simplicity that would be provided by such a uniform treatment, however, comes at a terrible cost in terms of explanatory parsimony, and hence in terms of overall simplicity. If laws can partly explain their instances then a vast number of particular matters of fact can plausibly be explained in terms of a much smaller number of particular matters of fact, together with a small number of laws. If (FP) is true, on the other hand, and laws cannot partly explain their instances, then a vast number of particular matters of fact will instead have to be foundational, where a fact is foundational iff it is not explained by any other fact (and is not explanatorily trivial).³¹ (FP) therefore plausibly entails there being a much greater number of foundational facts than there would be if (FP) was false, and this great increase in explanatory bruteness greatly outweighs the small increase in simplicity in the principles governing explanation that (FP) provides. Therefore overall simplicity considerations provide no support for (FP). As a result of this, and the fact that it lacks any intuitively compelling support, Humeans and anti-Humeans should both reject (FP).

Given Humeans and anti-Humeans reject (FP), what account of what explains universal facts should they endorse in its place? In the case of Humeanism, an attractive alternative to (FP) is the following. If a universal fact is a law then it is either foundational (if it is a fundamental law), or it is metaphysically explained by other laws (and perhaps by some other facts that are expressed by axioms in every best system). If a universal fact is not a law, on the other hand, then it either metaphysically explained by its instances in the way (FP) claims, or it is metaphysically explained by a combination of laws, facts expressed by axioms in every best system, and instances. Anti-Humeans can similarly claim that a universal fact $\underline{\forall x \varphi (x)}$ is metaphysically explained by the fact that it is a law that $\forall x\varphi (x)$, given it is a law that $\forall x\varphi (x)$, and is otherwise metaphysically explained by its instances as (FP) dictates or by a mixture of particular matters of fact and laws.³²

Humeans, therefore, can respond to the variant objection from explanation by arguing that both Humeans and anti-Humeans should reject Fine’s principle (FP) and instead endorse a more sophisticated account of what explains universal facts which distinguishes between those that are laws and those that are mere regularities.

5 Conclusion

I have argued that Humeans about laws can successfully respond to both the objection from explanation that relies on Lewis’s best systems account of laws, and the variant of this objection that relies on Kit Fine’s principle that holds that regularities are metaphysically explained by their instances. In order to respond to these objections, Humeans do not need to reject plausible principles governing explanation, such as the principles that explanation is asymmetric and transitive. Instead they need to get clear about what account of the explanatory structure of reality they should endorse. The account they should endorse is one on which the foundational explanatory facts consist of a relatively small number of fundamental laws and particular matters of fact. These explanatorily foundational facts can then be held to explain all the other facts, including all the other particular matters of fact and the second order facts that ascribe lawhood to laws. Universal facts, on this picture, are either foundational, if the they are fundamental laws, or they are explained either by laws, or by particular matters of fact, or jointly by both of these kinds of facts. Endorsing this picture allows Humeans to respond to both the original objection from explanation and its variant. They can respond to the original objection from explanation by claiming that, while laws partly explain their instances, and their instances partly explain facts ascribing lawhood to laws, no explanatory circle results, since the latter facts do not partly explain either of the former facts. And they can respond to the variant objection from explanation by rejecting Fine’s principle (FP) as being too simplistic and claiming that it is only accidental universal facts that are partly explained by their instances.