Laws of Nature, Explanation, and Semantic Circularity

Laws of Nature, Explanation, and Semantic Circularity Abstract Humeans and anti-Humeans agree that laws of nature should explain scientifically particular matters of fact. One objection to Humean accounts of laws contends that Humean laws cannot explain particular matters of fact because their explanations are harmfully circular. This article distinguishes between metaphysical and semantic characterizations of the circularity and argues for a new semantic version of the circularity objection. The new formulation suggests that Humean explanations are harmfully circular because the content of the sentences being explained is part of the content of the sentences doing the explaining. I describe the nature of partial content and demonstrate how this account of partial content renders Humean explanations ineffective while sparing anti-Humean explanations from the same fate. 1 Introduction 2 Standard Formulations of the Circularity Charge 3 Humean Responses 4 Semantic Characterizations of the Circularity Worry 4.1 Hempel and Oppenheim’s semantic circularity concern 4.2 A new version of the semantic circularity charge 4.3 Partial content as a guide to circularity 5 Humean Responses to the Semantic Circularity Charge 5.1 Smuggling in metaphysics through the back door? 5.2 Do anti-Humean laws fare any better? 5.3 The over-generalization concern 6 Conclusion Appendix 1 Introduction The world, it seems, is composed of many individual events or property-instances: a flame’s igniting in Greece in 200 B.C.E., a football’s travelling in a projectile motion in Belize in 1950, and so on. The totality of these property-instantiations across space and time is commonly referred to as the ‘Humean mosaic’. Humeans take scientific laws to derive from the particular matters of fact comprising the Humean mosaic. While not all Humeans adopt the same account of the laws, many accept or defend the best systems account (Lewis [1973], [1999]; Beebee [2000]; Schrenk [2006]; Cohen and Callender [2009]; Loewer [2012]; Bhogal and Perry [2017]). The best systems account is a regularity account of laws; it takes laws to be ‘certain true propositions and equations’ that capture regularities holding in the mosaic. The laws of the best systems account are the generalizations that are entailed by the ideally best scientific theory, where the ideally best scientific theory is the one that best balances simplicity and informativeness (Loewer [2012], p. 119). For instance, if our simplest and most powerful scientific theory maintains that across the span of the universe, all qualitatively identical fermions occupy different quantum states, then a statement of the form (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a law. In fact, the former is taken to be a law called the Pauli exclusion principle. Anti-Humeans, on the other hand, deny that laws are generalizations derived from particular matters of fact. Anti-Humean views take many forms. Tim Maudlin ([2007]) advocates taking laws as primitive. David Armstrong ([1983]) argues that laws are brute necessitation relations holding among universals. Marc Lange ([2009]) considers laws to be dependent on certain primitive counterfactuals—counterfactuals that do not depend on the Humean mosaic. Alexander Bird ([2005], [2007]) takes laws to arise from dispositional properties.1 Uniting views under the umbrella of anti-Humeanism is the idea that the laws are not summaries of the events occurring in the Humean mosaic. Anti-Humeans typically take laws to be powerful entities that have the ability to govern, direct, or guide the progression of events. One objection to Humean accounts of laws is that we cannot successfully use them in scientific explanations (Hempel and Oppenheim [1948]; Feigl [1981]; Armstrong [1983]; Maudlin [2007]; Loewer [2012]; Lange [2013]; Paul [2013]; Hicks and van Elswyk [2014]; Miller [2015]; Marshall [2015]) because doing so would be to use parts of the Humean mosaic to explain themselves. We will refer to this as the ‘circularity charge’. The structure of this article is first to introduce the existing versions of this charge against the Humean and to show that the Humean can avoid these versions of the circularity charge because these versions rely on metaphysical assumptions the Humean can plausibly reject. I then offer a new semantic version of the circularity charge that does not rely on strong metaphysical assumptions. Finally, I discuss possible responses the Humean can make to this semantic circularity charge and argue that these responses are unsuccessful. 2 Standard Formulations of the Circularity Charge Following other discussions of the circularity charge, this article concerns explanations that employ deterministic laws of nature in order to explain particular events. I will call these ‘nomological explanations’. These are explanations that appeal to exceptionless laws in their explanans.2 I will not presuppose that such explanations have a specific structure. To clarify, I will leave open whether the Humean or anti-Humean appealing to laws in explaining the occurrence of particular events can adopt a covering law model of explanation, a unificationist view of explanation that appeals to laws, or appeals to causal explanations that make use of specifically causal laws—or some other view of explanation entirely. Covering law explanations are those, like the deductive-nomological (D-N) account, that maintain scientific explanations involve laws of nature and initial conditions that inductively entail statements of the phenomenon they purport to explain. While a covering law model of explanation may seem (at first glance) to be one of the most appropriate to nomological explanations, there are a couple of reasons I do not assume proponents of nomological explanations are making use of it: First, the Humean and anti-Humean need not assume that laws (and initial conditions) deductively entail which events occur. We may take laws to explain facts concerning which events occur, but deny that this constitutes a deductive entailment. Second, the Humean and anti-Humean may take laws to feature only in some scientific explanations. Other scientific explanations may primarily use information about causal processes or causal history to explain particular matters of fact. Humeans and anti-Humeans alike appeal to laws in providing scientific explanations, but they are subject to attack from one another. Humeans fail to see how the anti-Humeans’ mysterious laws entail universal generalizations about particular matters of fact, and anti-Humeans maintain that Humeans can only provide circular explanations of particular matters of fact.3 The circularity charge concerns us here. We can informally state the circularity objection as follows: since Humean laws derive from particular matters of fact, they cannot be used as the explanans in a scientific explanation where the same particular matters of fact are the ones to be explained. This is a casual formulation of the circularity worry, and it needs specification. What does it mean for the laws to ‘derive from’ the same matters of fact they purport to partially explain? This section considers two ways to spell out the circularity objection that rely on assumptions concerning the nature of metaphysical relations of truthmaking and ‘in virtue of’. David Armstrong ([1983]) and Tim Maudlin ([2007]) raise versions of the circularity charge invoking these notions. Armstrong uses this example to illustrate his point: If ‘all Fs are Gs’ were a law, we would use it to explain ‘all observed Fs are Gs’. But [all Fs are Gs] is a conjunctive state of affairs, which we can also pick out as [all observed Fs are Gs and all unobserved Fs are Gs].4 And ‘as a result trying to explain why all observed Fs are Gs by postulating that all Fs are Gs is a case of trying to explain something by appealing to a state of affairs part of which is the thing to be explained’ (Armstrong [1983], p. 40). Armstrong frames the worry in terms of states of affairs. States of affairs are complexes of objects and properties, and we should think of our physical universe as being composed of states of affairs. States of affairs stand in parthood relations. On Armstrong’s account, [all observed Fs are Gs] is part of [all Fs are Gs] because the latter is the conjunctive state of affairs [all observed Fs are Gs and all unobserved Fs are Gs] and conjunctive states of affairs have the conjunct states as their parts. He maintains that one cannot explain why a state of affairs obtains by appealing to another state of affairs where the former is a proper part of the latter. Maudlin ([2007], p. 172) puts the concern differently: ‘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’. This version of the circularity objection invokes the ‘in virtue of’ relation, which Maudlin does not elaborate upon. One way we can understand this relation is in terms of grounding; although, it is not clear that Maudlin himself has the grounding relation in mind when advancing this version of the circularity charge.5 Grounding is a relation that holds between facts, sentences, or objects.6 It is typically considered to be irreflexive, asymmetric, and transitive. Grounding relations are supposed to capture metaphysical explanation: if P grounds Q, then P metaphysically explains Q. To state Maudlin’s concern in terms of grounding: If the parts of the Humean mosaic ground the existence of the laws, then the existence of the laws cannot in turn scientifically explain parts of the mosaic. If the parts of the mosaic taken together fully ground the law and the law explains parts of the mosaic, then the mosaic, in some sense, explains its parts. We will now explore some reasons why we should not develop the circularity charge in these ways. 3 Humean Responses The two versions of the worry above are forceful only if we adopt metaphysical accounts of states of affairs or grounding relations. A Humean can avoid the circularity worry by denying that she must appeal to such entities in the first place. In response to Armstrong, the Humean can reject that she is invested in states of affairs explaining other states of affairs: the Humean is concerned with how sentences—namely, certain universal generalizations—explain other sentences about the mosaic. So even if Armstrong is correct about how the two states of affairs are mereologically related, the Humean can deny that this impacts her account of how laws explain statements of particular matters of fact. Armstrong can reply that the Humean should still take states of affairs seriously as they are supposed to be truthmakers for law-statements. Yet, the Humean is not forced to take seriously a metaphysical truthmaking relation either. There are other ways to account for the truth of sentences that posit a less metaphysically robust apparatus than a truthmaking relation. Similarly, in response to the formulation of the worry in terms of grounding, the Humean can reject metaphysical grounding relations as part of her ontology. Grounding is supposed to capture metaphysical explanation, and it is open to the Humean to deny that this notion of metaphysical explanation is coherent or useful for her purposes.7 The Humean need not be wholly averse to grounding and related metaphysical notions in order to take this option. She can claim that grounding is a metaphysically useful notion, but that the Humean does not appeal to it here. One classic way to account for the relation between the laws and the Humean mosaic is by appeal to supervenience. When the laws supervene on the Humean mosaic, facts involving the mosaic necessitate which laws obtain. There can be no change in the laws of nature without change in the mosaic. Supervenience is a purely modal relation that does not strive to capture metaphysical explanation. This circularity charge does not arise if one takes the Humean mosaic to be the supervenience base instead of the grounding base for laws of nature.8 Another option for avoiding these circularity charges is to distinguish between the kinds of explanation at work in the problematic examples. Barry Loewer ([2012]) asserts that even if the Humean appeals to grounding relations, she can respond to the charge by claiming there are two notions of explanation in play: metaphysical explanation and scientific explanation. While the Humean mosaic metaphysically explains or grounds the laws, the laws scientifically explain the parts of the mosaic. Since these are two distinct kinds of explanations, it does not follow that the Humean mosaic either metaphysically or scientifically explains itself. This response relies on the existence of a distinction between metaphysical and scientific explanation, and it must ensure that explanations of particular matters of fact in terms of laws count as scientific and not metaphysical explanations. It is not clear there is sufficient motivation for upholding this distinction; nevertheless, we will give Humeans the benefit of doubt and assume explanations can be distinguished in this manner.9 There are now two different ways the Humean can respond to these versions of the circularity charge: (i) the Humean can reject the applicability of the metaphysical accounts of truthmaking and ground needed to press the circularity charge, and (ii) even if she endorses truthmaking or ground in this context, the Humean can try to avoid the circularity by distinguishing between the kinds of explanation in play. 4 Semantic Characterizations of the Circularity Charge 4.1 Hempel and Oppenheim’s semantic circularity concern I advocate a semantic version of the circularity objection that does not rest on heavyweight metaphysical assumptions. This version of the circularity objection descends from a version Carl Hempel and Paul Oppenheim ([1948]) raise and dismiss when discussing the D-N model of explanation. Here I argue that the semantic circularity objection is stronger than Hempel and Oppenheim take it to be. However, I also maintain that we need a different understanding of semantic circularity than the one Hempel and Oppenheim appeal to in order to advance the objection against the Humean. Hempel and Oppenheim accept a Humean account of laws in which laws are generalizations expressing regularities. As Hempel and Oppenheim’s D-N explanations involve Humean laws, the circularity worry they raise is quite similar to the ones discussed in the context of the best systems account. So, while I do not assume that either the Humean or anti-Humean accepts the D-N model as the correct account of scientific explanation, I will show that Hempel and Oppenheim’s semantic circularity worry can be raised against Humean explanations in general. The D-N model of explanation maintains that we have a scientific explanation when statements of the laws of nature, along with particular matters of fact, logically entail the explanandum in question. D-N explanations require two provisos to be in place: the statements in the explanans must be true and the law must be ‘essential’ to the explanation (Hempel [1942]). The explanation of the distinct quantum states of two fermions considered in the previous section also counts as an instance of a D-N explanation. Hempel and Oppenhiem ([1948], p. 162) write the following: It has […] been argued that in a sound explanation the content of the explanandum is contained in that of the explanans. That is correct since the explanandum is a logical consequence of the explanans; but this peculiarity does not make scientific explanation trivially circular since the general laws occurring in the explanans go far beyond the content of the specific explanandum. They suggest that such explanations may be circular because the content of the explanandum is ‘contained’ in the explanans. They characterize this containment in terms of logical consequence. They would maintain that in the case of the explanation concerning fermions above, the content of the sentence ‘a and b are in different quantum states’ is contained in the content of ‘a and b are distinct fermions and all distinct fermions are in different quantum states’, because the former is a logical consequence of the latter. Hempel and Oppenheim deny that content-containment is a problematic feature of explanation; in fact, they take it to be a virtue, but they are aware of the concern that one could consider such explanations ‘trivially circular’ in light of this containment. They pre-empt this reaction by stating that the content of an explanans’ containing the content of the explanandum is unproblematic because the general laws have much more content than just the explanandum. But why should the fact that the explanans have much more content than the explanandum help assuage this circularity worry? When a fact is used to explain itself, adding in more facts to the explanans does not detract from the circularity. For example, it would be harmfully circular to explain a particular fact such as ‘Mercury is in retrograde’ by appealing to the fact that Mercury is in retrograde. Even if we add a lot of information to the explanans—facts about the motion of other planets or the behaviour of Mercury at other times, for instance—that, in conjunction with the fact that Mercury is in retrograde, still does not explain ‘Mercury in retrograde’. It is the other information that helps explain that Mercury is in retrograde, not this other information in conjunction with the fact that Mercury is in retrograde, and the latter does not constitute a better explanation of the fact. Hempel and Oppenheim have not explained why this circularity is harmless. Hempel and Oppenheim have not successfully responded to this charge of circularity, but for the charge to be forceful we must understand the notion of containment appealed to above. What does it mean for the explanans to have the explanandum as part of its content? Hempel and Oppenheim do not adequately address this. They maintain that A has B as part of its content whenever B is a logical consequence of A. I think that while sentences contain some of their logical consequences as content-parts, there are many pairs of sentences where the contents are wholly unrelated, even though one is the logical consequence of the other. B’s being a logical consequence of A does not entail that the content of B is part of the content of A. ‘George is in the philosophy department’ does not have ‘It is raining or it is not raining’ as part of its content, for example, even though the latter is a logical consequence of the former. The contents of these sentences do not stand in a parthood relation because the sentences have entirely distinct subject matters; the first sentence is about George and the latter is about rain. Before the Humean is forced to take this version of the circularity charge seriously, we need to provide a satisfactory account of containment and content-parthood. 4.2 A new version of the semantic circularity charge This section provides a characterization of containment that differs from Hempel and Oppenheim’s and explains why it is a mark of a poor explanation that the content of the explanans contains the content of the explanandum. I address the second task and then the first. By advancing explanations where the explanandum is part of the content of the explanans, the Humean violates a non-circularity condition for scientific explanations, which is put forth in the following principle CON as follows: CON: If the content of a sentence E is part of the content of a set of sentences, then an explanation of E in terms of Γ is unsuccessful. If the explanans contains the explanandum as a part of its content, the purported explanation fails. This failure amounts to a type of circularity because the explanandum is being used, possibly together with additional content, to explain itself. Contra Hempel and Oppenheim, this kind of circularity is not negligible. CON does not presuppose a particular account of scientific explanation. Instead CON is supposed to be a generally desirable feature of scientific explanations. When a set of sentences explains another, the explaining sentences should provide us with an understanding of why a particular phenomenon occurs. If the content of the sentence being explained is already contained in the sentences doing the explaining, then we are appealing to the occurrence of the phenomenon to give us a deeper understanding of why that same phenomenon occurs. We should not appeal to a phenomenon in order to explain its own occurrence. In the next few paragraphs, I will try to articulate why violations of CON are problematic. Quintessentially successful explanations accord with CON, while quintessentially circular explanations violate CON. For example, we considered an attempt to explain that Mercury is in retrograde by appeal to the sentence ‘Mercury is in retrograde’ in conjunction with other sentences concerning celestial motion in the previous section. We can now diagnose why this explanation fails by appealing to CON above: the explanandum is contained as part of the explanans. Many explanations—such as typical causal and reasons-based explanations—do not violate CON. Following Salmon, let us take a causal explanation of some particular outcome to ‘consist of citing (some portion of) the causal processes and interactions leading up to that outcome’ (Woodward [1989], p. 358). Consider a causal explanation like the following: The lightning striking the barn—perhaps along with certain background conditions such as the barn’s being dry, the presence of oxygen, and so on—explains why the barn is now on fire. Here the statements comprising the explanans taken together (‘lightning strikes the barn and the barn is dry and…’) do not contain as part of their content that the barn is on fire.10 Standard reasons-based explanations behave similarly: Lola ate the cake because she desired cake and Maria offered her some. The statement of Lola’s desire and Maria’s offer do not contain as part of their content the fact that Lola ate the cake. We should expect explanations involving scientific laws to likewise obey CON. One source of motivation for CON follows from two constraints on explanation: (i) explanations should not contain irrelevant information, and (ii) explanations should not violate irreflexivity, that is, a sentence cannot explain itself or help explain itself. Constraint (ii) is a familiar structural constraint on explanations.11 As for Constraint (i), we can restate it as follows: If a set of sentences, Γ, fully explains another sentence, E, then Γ is wholly relevant to E, namely, all of the content of the sentences of Γ is relevant to explaining E. If any part of the content of Γ were irrelevant to E, then we would not need to appeal to it in explaining E. We would instead appeal to Γ-, the set of sentences of Γ that do not involve the irrelevant content, to provide a good explanation of E. Now, in explanations where the content of the explanandum is part of the content of the explanans, a sentence capturing the content of explanandum E will be relevant to explaining itself. In other words, it will help explain itself, which violates irreflexivity. We can restate the point more clearly as follows: If Γ fully explains E and Γ contains nothing irrelevant to E, then sentence B containing some of the content of Γ (and nothing outside of the content of Γ) helps explain E. If B = E, then we will violate irreflexivity; a sentence cannot help explain itself. CON is appealing as a condition for acceptable scientific explanations, but we need a more rigorous conception of a content-part in order to grasp exactly what CON amounts to and to understand where it gains its force. In recent literature (van Fraassen [1969]; Angell [1989]; Gemes [1993], [1994]; Correia [2004]; Fine [2012a], [2013], [2017], [unpublished]; Yablo [2014]), philosophers have developed and refined the notion of partial content for a wide range of purposes. These notions of partial content are well suited to our purposes because they aim to capture a finer-grained relation than logical consequence; as we will see, one sentence’s entailing the other will not suffice for the content of the latter being part of the content of the former on these views. Among other applications, philosophers have used the notion of partial content in order to capture notions of verisimilitude and account for differences in confirmation.12 I maintain that we can also use the notion of a content-part in order to rigorously characterize explanatory circularity. Here I spell out the notion of a content-part and show how it reveals the circularity in Humean explanations. 4.3 Partial content as a guide to circularity We can provide an account of partial content using different semantic pictures as backdrops. Most recently, Stephen Yablo ([2014]) and Kit Fine ([2012a], [2012b]) characterize partial content using the background of a possible-worlds semantics and truthmaker semantics, respectively. Possible-worlds semantics maintains that we can understand the semantic content of sentences in terms of sets of possible worlds (and sets of sets of possible worlds), while truthmaker semantics advocates using sets of parts of possible worlds to best capture the semantic content of sentences. The truthmakers here are not understood in Armstrong’s sense, but that will be explained below. The primary difference between the two approaches is that possible-worlds semantics determines whether a statement is true or false at a possible world, whereas truthmaker semantics ‘tells us what it is in the world that makes the statement true if it is true or what it is in the world that makes it false if it is false’ (Fine [2012a], p. 235). Here I follow Fine’s characterization of partial content framed in terms of truthmaker semantics. I have attached an appendix where I develop the concern using Yablo’s possible-worlds semantics. While accepting truthmaker semantics is not essential for formulating a rigorous conception of partial content, it provides a straightforward depiction of how scientific explanations either satisfy or violate circularity in terms of CON.13 In order to understand the account of partial-content, we need to grasp this semantic picture. Truthmaker semantics posits a state-space, a parthood relation over the state-space, and verification and falsification relations holding between states in the state-space and sentences. The state-space plays the role for the truthmaker semanticist that the pluriverse of possible worlds plays for the advocate of possible-worlds semantics. States pick out proper parts (or in some cases, improper parts) of possible worlds, but usually no stand is taken on what exactly these states or situations are supposed to be. Truthmaker semantics remains neutral as to whether we should think of states as concrete or abstract, and if the latter, what the nature of the abstract entities in question is. This neutrality is similar to the attitude possible-worlds semanticists take to the nature of possible worlds. We can think of the appeal to possible worlds and possible states as a modelling tool, allowing us to account for the content of our sentences. As Humeans can differ as to whether possible worlds are concrete or abstract, they can also differ on the concreteness or abstractness of possible states. For that reason, I will not assume that states are either abstract or concrete here; rather, I will show how Humean explanations violate CON on either a concrete or abstract characterization of states. States can contain other states as parts where the kind of parthood is similar to mereological parthood. Of course, if states turn out to be purely abstract entities, it may be inappropriate to consider this relation to be strictly one of mereological parthood; in this case we can take the relation to be merely ‘part-like’, in the sense that this kind of parthood obeys some of the same abstract mereological axioms as concrete parthood. Here are some examples of states containing other states: the state [a bowl of marbles is in the Philosophy Department] contains the state [a marble is in the philosophy department] as a part. Any part of a possible world containing a bowl of marbles must have a smaller part containing a single marble. The state [Colorado is mountainous and the sun is shining in Nevada] is a fusion of [Colorado is mountainous] and [the sun is shining in Nevada]. Truthmaker semantics accounts for the semantic content of sentences by appealing to how states or situations ‘exactly verify’ or ‘falsify’ the sentences in question. The content of a statement is picked out by the possible states that exactly verify it. Fine ([2017]) maintains that states exactly verify or falsity statements when ‘the state is wholly relevant’ to the truth or falsity of the statement. Exact verification is, in part, a modal notion. For a state to exactly verify a sentence, it must entail the truth of the sentence. But a state’s entailing the truth of a sentence is not sufficient for it to exactly verify the sentence. This is because states can entail the truth of totally unrelated sentences. For instance, the state [there is a donkey in the Eiffel Tower] necessitates the truth of ‘either it is raining or it is not raining’, but the content of that sentence is unrelated to the donkey’s presence in the Eiffel Tower. This is why we cannot just look at which states entail a sentence in order to determine that sentence’s content. While we cannot provide a reductive definition of what it is for a state to be ‘wholly relevant’ to the truth of a sentence, there are constraints on exact verification that deliver a good grasp of how it works. For instance, the sentence ‘the sun was shining last Tuesday’ will be exactly verified by the state [the sun was shining last Tuesday]; it will not be exactly verified by [the sun was shining every day last week]. The latter state is not an exact verifier of ‘the sun was shining last Tuesday’ because it includes extra information pertaining to the weather on days other than Tuesday. It is inconsequential to the truth or falsity of ‘the sun was shining last Tuesday’ whether the sun was shining on these other days. But [the sun was shining last Tuesday] does not contain any extra information, which allows it to be an exact verifier for the sentence. This is the sense in which an exact verifier is wholly relevant to the truth or falsity of the sentence in question. An atomic sentence, P, is exactly verified by any possible state, [p], whose obtaining is wholly relevant to the truth of P. Following Fine ([2012a]), here are clauses demonstrating how states exactly verify truth-functional conjunctions, disjunctions, and negations: State [s] exactly verifies A &B if and only if [s] is the fusion of a state [s1] that exactly verifies A and a state [s2] that exactly verifies B. State [s] exactly verifies A ∨B if and only if [s] exactly verifies A or [s] exactly verifies B. State [s] exactly verifies ¬A if and only if [s] falsifies A. Here are a few examples of sentences and the states that exactly verify them: ‘The barn is on fire and there is a thunderstorm’ is exactly verified by the fusion of [the barn is on fire] with [there is a thunderstorm]. The state [water is in Lake Tahoe] exactly verifies ‘Water is in Lake Tahoe or chlorine is in Lake Tahoe’. Negations of the form ¬P are exactly verified by the state that exactly falsifies P. For instance, ‘it is not the case that the grass is yellow’ will be exactly verified by the state that exactly falsifies ‘the grass is yellow’, which is plausibly the state [the grass is green]. We can now provide a definition of content-part as follows (Fine [unpublished], p. 11): C is part of the content of A if and only if: Every possible state that exactly verifies A contains a possible state that exactly verifies C. Every possible state that exactly verifies C is contained in a possible state that exactly verifies A. The first clause is satisfied if every exact verifier of A has an exact verifier of C as a part, and the second clause is satisfied if every exact verifier of C is part of at least one exact verifier of A. When both clauses are satisfied, C is part of the content of A. For instance, ‘the barn is on fire and there is a thunderstorm’ has ‘the barn is on fire’ as part of its content. The first clause is satisfied because every state that verifies ‘a barn is on fire and there is a thunderstorm’ is a fusion of a state that verifies ‘the barn is on fire’ and a state that verifies ‘there is a thunderstorm’. Such states contain an exact verifier of ‘the barn is on fire’ as a proper part. The second clause is satisfied because every state that verifies ‘a barn is on fire’ will be a proper part of at least one possible state that is a verifier of the conjunction. On this account of partial content, whenever C is part of A, A deductively entails C; but not every sentence entailed by A will be part of the content of A. Given this notion of a content-part, along with the principle of non-circularity, we can explicate the circularity charge against the Humean. The Humean wants to explain why particular matters of fact hold by appeal to the laws and antecedent conditions. To redeploy our example from the previous section, the Pauli exclusion principle and the fact that a and b are distinct fermions are supposed to explain the fact that a and b are in distinct quantum states. Here, the explanans is the conjunction of the Pauli exclusion principle and the antecedent condition that a and b are distinct fermions. We can rewrite this example of a Humean explanation as follows: ExplanansH: (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states], and a and b are distinct fermions. ExplanandumH: a and b are in different quantum states. The content of ‘a and b are in different quantum states’ is part of the content of the explanans, which is the conjunction of the law ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ and ‘a and b are distinct fermions’. To see why, let us consider which states are supposed to exactly verify universal generalizations. The most natural candidate for an exact verifier of a universal generalization will include the fusion of the states verifying all the instances of the generalization.14 The trouble is in determining what it takes to verify an instance of a universal generalization. I will consider two proposals. First, take a simpler universal generalization than the one featured in the explanans above: ‘ (∀x)[Fx ⊃Gx]’. One of its instances is ‘Fa ⊃Ga’. Since ‘ ⊃’ is supposed to be a material conditional, we can rewrite the instance as ‘ ¬Fa ∨Ga’. The exact verifier of this disjunction will be the states that verify either or both of its disjuncts. So, on this characterization, the exact verifier of ‘ (∀x)[Fx ⊃Gx]’ is the state that is the fusion of all the states that verify the disjunctive instances. According to a second approach, we deny that verifiers for ‘ ¬Fa’ count as part of the exact verifier of ‘ (∀x)[Fx ⊃Gx]’ because this would allow irrelevant verifiers, all the non-Fs, to be part of the verifier of the universal generalization. The proponent of the second approach emphasizes that the generalization gives us features that all the Fs have and that the behaviour of the non-Fs should be irrelevant. Here I take the instances that verify ‘ (∀x)[Fx ⊃Gx]’ to be the states of each F also being G. The fusion of states that verify statements of the form ‘Fx&Gx’ verify the universal generalization.15 For instance, if there are three things that have feature F: a, b, and c, then an exact verifier for ‘ (∀x)[Fx ⊃Gx]’ would be the fusion of the states that verify ‘Fa&Ga’, ‘Fb&Gb’, and ‘Fc&Gc’, or the fusion of the states [Fa], [Fb], [Fc], [Ga], [Gb], and [Gc]. We are now in a position to show how the content of the explanandum is contained in the content of the explanans for the Humean. Recall the explanans is a conjunction of a universal generalization and the claim that a and b are distinct fermions. The latter conjunct is straightforwardly verified by the state [a and b are distinct fermions]. On the first characterization of verifying a universal generalization, the state that verifies the first conjunct of the explanans, ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’, is the fusion of states that exactly verify instances of the following disjunction: (∀x)(∀y)[ ¬(x and y are distinct fermions) ∨(x and y occupy different quantum states)]. This is just the original universally quantified material conditional rewritten as a universally quantified disjunction. So for a and b in particular, an exact verifier of ‘[ ¬(a and b are distinct fermions) ∨(a and b are in distinct quantum states)]’ is part of the verifier for the universal generalization. But the first disjunct is false in this case since a and b are distinct fermions. Recall that [a and b are distinct fermions] is also part of the verifier for the explanans. Thus, the exact verifier of the entire explanans must include a verifier for the right disjunct. And the exact verifier of the right disjunct is [a and b are in different quantum states]. This state is the same one that serves as the exact verifier for the explanandum. The explanandum is part of the content of the explanans: Every exact verifier for the explanans includes a verifier for the explanandum. And every verifier for the explanandum is included in a verifier for the explanans.16 Humean explanations violate CON under the second approach to verifying universal generalizations as well. Here the Pauli exclusion principle is exactly verified by the conjunction of its positive instances. The fusion of states verifying the instances of ‘x and y are distinct fermions and x and y are in different quantum states’ counts as part of the exact verifier of the explanans. So the fusion of [a and b are distinct fermions] and [a and b are in different quantum states] is part of any exact verifier for the explanans. And this state contains [a and b are in different quantum states] as a part. So the explanans contains the explanandum as a part on this characterization as well. This follows generally: Since the Humean takes laws to be universal generalizations and explains particular matters of fact by appealing to the laws along with antecedent conditions, the Humean’s explanations will be harmfully circular in the way CON specifies.17 While this version of the circularity objection depicts a semantic circularity, it differs from Hempel and Oppenheim’s objection to D-N explanations. Here, not every logical consequence of a sentence counts as part of its content. The content of ‘It is either raining or it is not raining’ is not part of the content of ‘George is in the philosophy department’ because exact verifiers of the latter do not contain exact verifiers of the former as parts. An exact verifier for the latter sentence is the state [George is in the philosophy department]. And exact verifiers of this form do not contain exact verifiers for ‘it is either raining or it is not raining’. ‘It is either raining or it is not raining’ has as exact verifiers states that verify each of its disjuncts, that is, the state [it is raining] or the state [it is not raining]. Thus, not every scientific explanation that proceeds via logical consequence will exhibit semantic circularity. The notion of content part in use here is stricter than the one Hempel and Oppenheim employ. Unlike Maudlin's and Armstrong’s charges of circularity, the Humean cannot avoid this semantic circularity by avoiding certain metaphysical relations or distinguishing between scientific and metaphysical explanation. There is only scientific explanation at work here. The claim that the explanans of Humean explanations contain their explananda as part of their content rests on only semantic notions. 5 Humean Responses to the Semantic Circularity Charge This section explores four ways the Humean can respond to the circularity charge above. The responses, however, do not dismiss the threat of circularity. 5.1 Smuggling in metaphysics through the back door? The Humean may respond that this characterization of circularity illicitly appeals to metaphysical assumptions in the background and thus fares no better than Armstrong or Maudlin’s characterizations. There are two ways to develop this concern. First, perhaps the exact verification relation too closely resembles Armstrong’s truthmaking relation. Armstrong takes states of affairs involving the mosaic to be truthmakers for Humean laws, and here the truthmaker semanticist takes the states concerning the mosaic to exactly verify the Humean laws. Given the similarity of these claims, one may worry that exact verification is a metaphysically heavyweight relation in disguise. While truthmaker semantics invokes states and a verification relation between states and sentences, we have remained neutral as to the nature of the states. This differs from Armstrong’s truthmaking relation in that his relation obtains between concrete states of affairs and sentences. Furthermore, truthmaker semantics is used to specify the content of sentences, not the constituents of the physical world that settle or necessitate the truth of sentences. An account of the content of sentences is valuable for metaphysicians and non-metaphysicians alike. And unlike the grounding relation, the exact verification relation is not posited as a metaphysically primitive relation, nor is it intended to be a metaphysically explanatory relation. Exact verification is a relation appearing in our semantic theory, not in our metaphysical theories. The second way the Humean may allege that I have smuggled illicit metaphysical assumptions into the circularity charge is in the notion of parthood that covers the truthmaker semanticist’s state-space. Recall that states can contain other states as parts and that there are fusions of states on this account. This parthood relation over the state-space is crucial for making sense of exact verification. It would be troubling if the account forced Humeans to automatically accept a metaphysics of parthood as metaphysical theories of parthood are controversial. However, this theory of truthmaker semantics does no such thing. It requires that the states in the state-space stand in parthood relations, but this does not commit one to a particular metaphysics of parthood holding among objects in general, or for there to be metaphysically fundamental parthood relations in the world. As remarked above, if the states and the state-space are characterized abstractly, this may not even be the same kind of parthood relation that one finds among concrete objects. Philosophers often take mereological theories to hold among only concrete objects (although Fine [2010] is an exception), in which case the states only stand in ‘part-like’ relations to one another, relations that bear structural similarities to mereological parthood relations. Accepting relations that have some structural similarities to the mereological parthood relation does not force the Humean to adopt any robust mereological theory. Thus, we have no reason to believe that the account of partial content used to formulate this concern relies on any metaphysical assumptions that would be controversial for the Humean. 5.2 Do anti-Humean laws fare any better? 5.2.1 Primitivist accounts of laws and those involving necessitation relations If the circularity charge poses a distinctive threat to the Humean, we must ensure that anti-Humean explanations do not suffer from the same circularity. Let’s consider a few anti-Humean views. Anti-Humeans who take statements of the laws to be verified by states involving necessitation relations among universals or primitive modal entities offer non-circular explanations as well. For the anti-Humean who takes laws to be necessitation relations among universals, the exact verifiers of the explanans ‘The Pauli exclusion principle is a law and a and b are distinct fermions’ will be the state [F-ness and G-ness stand in the nomic necessitation relation] (for the relevant universals F and G underlying this law) along with the state [a and b are distinct fermions]. The fusion of these states do not contain as a part [a and b are in different quantum states]. So there is no circularity. Suppose the anti-Humean primitivist about laws offers an explanation of why fermions a and b are in distinct quantum states. Such an anti-Humean can appeal to the fact that the Pauli exclusion principle is a law in her explanans. I rewrite this sentence as ‘LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]’. This sentence together with ‘a and b are distinct fermions’ constitute the explanans of an anti-Humean explanation for a and b’s being in distinct quantum states. ExplanansAH-P: LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]], and a and b are distinct fermions. ExplanandumAH-P: a and b are in different quantum states. Although universal quantifiers appear in it, the sentence ‘LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]’ is not a universal generalization, and the state [a and b are in different quantum states] does not appear in an exact verifier for the explanans. The primitivist takes the fact that the Pauli exclusion principle is a law of nature to be a brute fact in the world; thus, states involving the behaviour of individual fermions do not verify the statement. A primitive state such as [LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]] serves as the exact verifier for the first conjunct of the explanans. And the state [LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]] along with the state [a and b are distinct fermions] comprise the exact verifier for the explanans. An anti-Humean who takes laws to be necessitation relations between universals will be able to avoid the circularity charge as well. Let’s consider the explanation of ‘Ga’ by ‘Fa’ and the law ‘all Fs are Gs’. Here, the anti-Humean can claim that it is the sentence ‘F-ness necessitates G-ness’ or ‘N(F, G)’, conjoined with ‘Fa’ will explain ‘Ga’. The verifier for ‘N(F, G)’ will involve universals standing in a higher-order relation to one another, and the state [Ga] need not be involved. We should note that while these versions of anti-Humeanism can avoid the circularity charge, their verifiers for the statement of the law are still somewhat obscure. It is still questionable whether we are justified in positing primitive law-states or primitive necessitation relations between universals and what they are like. The potential obscurity of these verifiers highlights a different explanatory burden for many anti-Humeans: the inference problem. As Lewis ([1983]) and van Fraassen ([1989]) noted, it is mysterious how anti-Humean laws, along with initial conditions, are supposed to entail particular matters of fact if they do not do so via a universal generalization.18 Nevertheless, this is a distinct issue from the one at hand. What matters in this context is that these versions of anti-Humeanism do not fall victim to the semantic circularity afflicting the Humean. 5.2.2 Counterfactual and dispositionalist accounts of laws It is more difficult to determine whether anti-Humean accounts that employ primitive counterfactuals and dispositional properties violate CON. There are two sources of this difficulty: (i) accounts of partial content have not been fully worked out for counterfactual conditionals, and (ii) counterfactual and dispositional anti-Humean accounts of laws can take different forms. Let’s consider a counterfactualist account of laws first. Marc Lange’s ([2009]) account analyses a law of the form ‘It is a law that all Fs are Gs’, and concludes that such laws are a set of counterfactuals of the form ‘p □→∀x[Fx⊃Gx]’ for relevant counterfactual antecedents, p.19 Relevant antecedents include ones that specify certain features of the possible worlds in question, such as initial conditions of the universe that are slightly different from the ones of our actual world. First, we should note that if the counterfactuals in question do not (along with the initial conditions) entail the fact that the explanandum holds, then the explanans of the explanation will not contain the explanandum as a part. However, if one of the law-constituting counterfactuals includes the conditional in which p picks out the initial conditions of the actual world, then it seems as though an explanation using that conditional may entail the explanandum and exhibit the same kind of content-containment as the Humean’s. This will be the case if the counterfactual contains the corresponding material conditional as part of its verifier. Even in cases where p does not pick out the initial conditions of the actual world, the anti-Humean will face a problem if the verifier for the counterfactual ‘p □→∀x[Fx⊃Gx]’ contains a verifier of the consequent as a part. In this case, the counterfactual will involve verifiers for ‘∀x[Fx⊃Gx]’. Thus, whether accounts of laws based on counterfactuals survive the circularity charge will depend on which counterfactuals count as law-like and what the verifiers for counterfactuals should be. For similar reasons, it is difficult to see where dispositional accounts of laws stand on violating CON. If we analyse dispositional properties counterfactually, then the dispositionalist will have to answer similar questions as the ones facing the proponent of counterfactual-based accounts of laws. Say x has disposition P if and only if: if x had stimulus conditions F, then x would have manifestation conditions G (Px if and only if Fx □→Gx). Here our explanation of ‘Ga’ is ‘(Fa □→Ga)&Fa’. If we understand the content of this counterfactual as entailing and as having as a part the material conditional Fa⊃Ga, then the dispositionalist’s account will face the same circularity as above. However, if instead of analysing dispositions in terms of such counterfactuals, we appeal to primitive potencies of objects, then an explanation appealing to laws whose verifiers are picked out by dispositional properties will not, presumably, contain the explanandum as part of its content. This is because there is no reason to think that the verifier for ‘Pa’, if P is a primitive power or potency, has as a part the verifier for ‘Ga’. 5.2.3 Can the Humean avoid the circularity charge by changing the content of her explanation? Unlike the Humean explanations in Section 4, the anti-Humean explanations that avoid the circularity charge (the accounts considered in Section 5.2.1) do not directly involve a universal generalization like ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ in the explanans. Some anti-Humeans will invoke a statement that a necessitation relation holds among universals. Others, like the primitivist, use ‘(∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a law’ in the explanans of their explanations. Should we consider modifying the Humean’s explanation to imitate the primitivist’s? Perhaps it is the fact that the universal generalization in question is a law, which is explanatory for the Humean. The Humean still faces circularity under this modification. For the proponent of the best systems account, ‘The Pauli exclusion principle is a law’ amounts to the conjunctive claim that a certain regularity obtains and that it belongs in the best system: ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] and (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a simple and powerful generalization in our best system’. And the exact verifier for this state will be the fusion of states that exactly verify the first and second conjuncts of this claim. Since the first conjunct just is the universal generalization, the verifiers for the universal generalization still factor into the exact verifier of the explanans, and the circularity charge proceeds as before. Alternatively, the Humean could maintain that only the second conjunct belongs in her explanation: L: ( ∀x)( ∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a simple and powerful generalization in our best system. The problem here is that L will plausibly contain the original universal generalization ‘( ∀x)( ∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ as part of its content.20 First, any verifier for L will include a verifier for the universal generalization. This is because part of what it is to be a generalization of the best system is for the generalization to obtain. Presumably, a verifier for a sentence that predicates a feature onto X (here the feature of ‘being part of the best system’) includes a verifier for X existing or obtaining in the first place. Second, it is plausible that every verifier for the universal generalization is part of a possible verifier for L. Take the states verifying the universal generalization (states that verify the instances, possibly together with a totality state). Here it seems like there will be a possible state verifying L that will contain the verifiers for the universal generalization.21 In this case, the Humean will be led back to the original circularity charge. I have demonstrated that anti-Humean explanations need not violate CON, but this is not to say that every anti-Humean account can avoid the circularity charge. As we witnessed above, it is not clear whether anti-Humean laws invoking counterfactuals can avoid the charge. Moreover, if an anti-Humean insists that explanations involving laws must have the universal generalizations as part of the explanans, her explanation will be circular as well. Nevertheless, this provides greater impetus to insist that it is really the anti-Humean laws—instead of the universal generalizations they are supposed to entail—that carry the explanatory weight. 5.3 The over-generalization concern The Humean may take issue with the principle that the explanans of a scientific explanation should not contain the explanandum as a content-part. CON looks initially plausible, but one could claim that it fails as a criterion for scientific explanation because too many cases of purported explanations violate it; in other words, perhaps the argument against the Humean over-generalizes. In addition to the Humean’s explanations involving universal generalizations violating CON, there are many other cases of deductively valid arguments where the content of the conclusion is contained in the content of the premises. Are these arguments unable to reflect genuine scientific explanations as well? For instance, some explanations may proceed by way of disjunctive syllogism. Consider the following: Explanans: The ducks are either in the pond or on the shore, and the ducks are not in the pond. Explanandum: The ducks are on the shore. If taken to explain why the ducks are on the shore, this violates CON because the content of the explanandum is part of the content of the explanans, according to our definition of a content-part. Perhaps having to discard cases like this from counting as scientific explanations is too high a price to pay for CON. I do not think that CON is threatened by such examples. CON tells us that arguments like these do not themselves constitute scientific explanations. Nevertheless, the example above may point us in the direction of a scientific explanation; in this case, it is a sentence that explains the disjunction that does the real explanatory work. This accounts for why the example above looks like it constitutes an explanation of why the ducks are on the shore. The disjunction initially looks like it can explain the ducks’ location, but this is only because there is a fact that explains the disjunction; the fact that it’s a principle that ducks will reside wherever they have access to the most food is what is capable of partially explaining the location of the ducks. The treatment of this case is similar to the anti-Humean’s treatment of the law-like explanations from the previous section. While it looks like the fact that all Fs are Gs and the fact that a is F explain that a is G, what actually explains, a is G is what motivates the universal generalization in the first place. The universal generalization itself does not do the explanatory work. CON provides a constraint on a genuine scientific explanation: a genuine scientific explanation must not presuppose the content of what is to be explained in the conjunction of the facts doing the explanatory work. These cases demonstrate that in order for scientific explanations to avoid triviality, we should look for the fact that explains the relevant universal generalizations (or, in this case, the relevant disjunction) in the first place. There are still escape routes for the Humean we have not explored: In order to escape the circularity, perhaps the Humean can reject CON as an adequate criterion for circularity, or develop an alternative account of the content of universal generalizations—or develop an alternative account of partial content altogether. It is not clear what such an account of partial content will look like; but even if one attempt is successful, it will yield a surprising consequence for Humean theories of laws of nature. It will show that if Humean laws are explanatory, the Humean must make substantive claims about the nature of semantic content in order to ensure this result. 6 Conclusion This article offered an account of the circularity of scientific explanations invoking Humean laws. First, I discussed existing versions of the circularity charge against the Humean and acknowledged that the Humean can coherently reject the metaphysical assumptions needed to spell out these versions of the worry. I then argued that the Humean succumbs to a semantic circularity charge. The semantic circularity charge discussed relies on a notion of partial-content that does not invoke controversial metaphysical assumptions. Finally, I considered responses on behalf of the Humean to the semantic circularity charge and determined that they are unsuccessful. Appendix: Partial Content and Possible-World Semantics In this article, I used Fine’s theory of partial content to argue that the content of the explanandum of a Humean explanation is contained in the content of the explanans. Fine’s account of partial content uses truthmaker semantics. Here I provide Yablo’s ([2014]) alternative characterization of partial content, which uses possible-worlds semantics instead, and I aim to show that it yields the same results for Humean explanations. Yablo provides an account of partial content in the framework of possible-worlds semantics. Yablo’s account of partial content invokes the notion of a ‘subject matter’.22 He maintains that the content of sentence B is part of the content of another sentence, A, just in case the inference from A to B is both truth-preserving and ‘aboutness-preserving’. And an inference from A to B is aboutness-preserving if and only if A’s subject matter includes B’s, and A’s subject anti-matter includes B’s. As for notation, single quotes—‘S’, ‘T’, and so on—continue to pick out sentences. Angle brackets—⟨S⟩, ⟨T⟩, and so on—designate propositions, which are sets of possible worlds. Possible worlds are picked out as w, w′, and so on. Yablo ([2014], p. 45) maintains that B is part of the content of A if and only if: A entails B. A’s subject matter includes B’s subject matter. A’s subject anti-matter includes B’s subject anti-matter. To understand this notion of partial content, we need to know what a sentence’s subject matter (and subject anti-matter) is, and what it is for some subject matter (and anti-matter) to include another. According to Yablo, A’s subject matter is the relation one possible world, w, bears to another, w′, if and only if A has a truthmaker in w that it also has in w′. Yablo maintains that we can think of the subject matter as a set of potential truthmakers. Moreover, A’s subject antimatter is the set of potential falsemakers. A truthmaker for sentence A in w is a proposition obtaining in w that necessitates A’s truth.23 Not all propositions necessitating A will count as truthmakers for A. A’s truthmakers are the propositions that A holds ‘because of’, or ‘in virtue of’.24 Yablo does not mean to pick out any heavy-duty metaphysical notions by ‘because’ or ‘in virtue of’ here. He thinks we can determine which proposition the sentence holds ‘in virtue of’ by finding the proposition that exhibits the best balance of ‘naturalness’ and ‘proportionality’.25 A falsemaker for sentence ‘S’ in w is a truthmaker for S’s negation. Yablo ([2014], Appendix) discusses a few different characterizations of truthmaking. Below I present one characterization, which Yablo calls the ‘recursive view of truthmaking’: Atomic sentences: Distinct atomic sentences will have different kinds of truthmakers on Yablo’s apparatus. For instance, a good truthmaker for the sentence ‘Spot has less than six legs’ will be the proposition ⟨Spot has four legs⟩. A truthmaker for ‘The flower is red’ will be ⟨The flower is scarlet⟩. Conjunctions: The truthmaker for a conjunction like P&Q will be the union of propositions that are truthmakers for P and Q. Disjunctions: The truthmaker for disjunction P ∨Q will be the proposition that is the truthmaker for P or the one that is the truthmaker for Q (both can be truthmakers for P ∨Q). A truthmaker for negation ‘ ¬P’ will be a falsemaker for P. Such a proposition will entail that P is false. For instance, a truthmaker for ‘The sun isn’t shining in Casablanca’ is ⟨It is raining in Casablanca⟩. Universal generalization: The truthmaker for universal generalization ‘( ∀x)Fx’ will plausibly be the propositions ⟨Fa⟩, ⟨Fb⟩, and so on, together with a totality fact.26 Yablo’s truthmakers resemble Fine’s truthmakers in that many sentences have the same kinds of truthmakers: a truthmaker for a disjunction will be the truthmaker for one of its disjuncts on both Fine’s and Yablo’s accounts, for example. The main distinction between Yablo’s and Fine’s accounts is that for Fine, truthmakers are states, which we can think of as parts of possible worlds; while for Yablo, truthmakers are propositions, or sets of possible worlds. We can now understand subject-matter inclusion as follows: A’s subject matter includes B’s when the truthmaker for B cannot change without the truthmaker(s) for A changing. And the falsemaker for B cannot change without a corresponding change in the falsemaker for A. In the case of an explanation of the form ‘ (∀x)(Fx ⊃Gx)&Fa’ explains ‘Ga’ (where the universal generalization contains a material conditional), Yablo’s apparatus will have the same results as the truthmaker semantics account. The explanandum is part of the content of the explanans. First, the explanans entails the explanandum, so the inference is truth-preserving (it will satisfy Condition 1, above). The inference from ‘(∀x)(Fx ⊃Gx)&Fa’ to ‘Ga’ is also aboutness preserving. The subject matter of explanans ‘(∀x)(Fx ⊃Gx)&Fa’ will include the subject matter of ‘Ga’. Here the truthmakers for the first conjunct of explanans ‘(∀x)(Fx ⊃Gx)&Fa’ will be the truthmakers for the instances of the generalization ‘¬Fa ∨Ga’, ‘¬Fb ∨Gb’, and so on (in addition to a totality proposition), and the truthmaker for ‘¬Fa’. So the truthmaker must include a truthmaker for ‘¬Fa ∨Ga’. This truthmaker cannot be the truthmaker for the first disjunct ‘¬Fa’ because such a truthmaker will be a falsemaker for ‘Fa’, and we need a truthmaker for ‘Fa’ to be the truthmaker for the second conjunct of the explanans. So the truthmaker of the conjunction will be ⟨Ga⟩. Furthermore, A’s subject anti-matter includes B’s subject anti-matter when each falsemaker for B is a falsemaker for A as well. Here, a falsemaker for ‘Ga’ (a truthmaker for ‘¬Ga’) should also serve as a falsemaker for the explanans. This is because a falsemaker for ‘Ga’ will make it such that either ‘Fa’ must be false or it is not the case that all Fs are Gs.27 Footnotes 1 Although not every account of laws invoking dispositions falls wholly under anti-Humeanism. For instance, Demarest ([2017]) also discusses a view that is ‘anti-Humean in its ontology’ because it accepts fundamental powers or dispositions, but ‘Humean in its laws’ because it retains a best systems account of laws. 2 In the discussion of Humean accounts below, discussion will be limited to laws that are exceptionless regularities taking the form of universal generalizations. As a result, this discussion will not incorporate laws that do not hold universally, nor will it incorporate probabilistic laws. We need a separate discussion of whether Humean explanations invoking probabilistic laws of nature are circular. Thanks to an anonymous referee here. 3 See (Lewis [1983]; van Fraassen [1989]) for this charge against the anti-Humean. 4 In this article, states of affairs, as well as states in truthmaker semantics (introduced below), will be picked out using square brackets ‘[ ]’ in order to distinguish them from sentences, which are picked out using single quotation marks. 5 There are other ways to develop Maudlin’s concern that avoid grounding and the ‘in virtue of’ relation altogether. One way to understand the semantic formulations of the circularity charge discussed in Section 4 is as an attempt to develop Maudlin’s circularity charge without using metaphysically robust notions, like that of grounding. 6 The relata of the grounding relation are contested. For sentential-operator accounts of ground, see (Fine [2012b]). See (Rosen [2010]) for a fact-based account, and see (Schaffer [2009]) for an object-based account. I will speak in terms of fact-grounding, but none of the points raised will rest on that choice. 7 See (Miller [2015]) for a discussion of rejecting grounding in this context. For general scepticism about grounding, see (Hofweber [2009]; Daly [2012]; Wilson [2014]; Koslicki [2015]). 8 This option may not be open to someone who thinks the laws are ‘nothing over and above’ the mosaic. Sometimes one fact or state can supervene on another, while the former still exists over and above the latter. For instance, the fact that {Socrates} exists supervenes on the existence of Socrates, but the existence of the set still exists as something over and above Socrates. Thanks to an anonymous reviewer here. 9 For further discussion of the distinction between metaphysical and scientific explanation, see (Lange [2013]) for criticism of Loewer’s approach. See (Hicks and van Elswyk [2014]) for a response to Lange, and (Marshall [2015]) for a response to Hicks and van Elswyk. 10 Here I take causal explanations to consist of the citing of causal processes and interactions leading to the event in question. However, maybe some causal explanations cite a causal law in the background. If causal laws are understood as mere generalizations over the mosaic, then they will be subject to violations of CON. 11Ruben ([1990], p. 204) discusses irreflexivity in scientific explanation. Irreflexivity constraints for metaphysical explanation (in terms of grounding) can be found in (Rosen [2010]; Raven [2013]). While irreflexivity constraints in scientific explanations are not explicitly discussed very often, explanatory asymmetry is more commonly addressed (Salmon [1970], p. 71; Kitcher [1989]; Strevens [2008], Chapter 1). If we take explanations to be asymmetric (in other words, if A explains or helps to help explain D, then D cannot help explain A), then the irreflexivity of explanation will follow: if A helps explain A, then asymmetry is violated). 12 See (Yablo [2014], pp. 95–112) for a discussion of paradoxes of confirmation and content. 13 One way Yablo’s and Fine’s accounts differ is in their treatment of certain necessarily true sentences, such as ‘2 + 2 = 4’ and ‘3 + 3 = 6’. For Yablo, the content of a sentence will be the set of possible worlds in which it is true, along with the ‘subject matter’ of the sentence. Intuitively, a subject matter is a division across possible worlds that picks out different ways in which the sentence can be true or false. For instance, the sentence ‘Bob the table is blue’ can be true in different ways: it can be true because Bob the table is turquoise, navy, royal blue, and so on. On Yablo’s apparatus, it is difficult to distinguish the content of mathematical claims like ‘2 + 2 = 4’ and ‘3 + 3 = 6’. Both sentences are true in all possible worlds, and it does not seem as if these two sentences can be true in different ways. As explained below, for Fine, the content of a necessarily true sentence will be picked out by the states that verify it in each possible world. Different states will verify ‘2 + 2 = 4’ and ‘3 + 3 = 6’, namely, the states of [2 + 2 = 4] and [3 + 3 = 6]. If one has the intuition that these two mathematical sentences don't have the same content, an account based on truthmaker semantics will secure this result more straightforwardly than one based on possible-worlds semantics. While I am using Fine's account of partial content to develop the circularity objection here, I do not intend to take a stand on this issue. 14 However, it is notoriously difficult to fully account for the verification conditions of universal generalizations, and perhaps the fusion of states verifying the instances of a generalization is not sufficient for exact verification of the generalization: we may also need something similar to a totality state, a state that ensures that there are no other Fs that have been left out of the verifiers for the instances (for a discussion of totality states, see Yablo [2014], pp. 45–54; Fine [2012b]). We can include a totality state as part of the exact verifier of the universal generalization in addition to the states that verify the instances of the generalization. There are different ways to try to characterize this kind of state, but whether and how we incorporate totality states does not impact the points made here. 15 This picture accords well with a rejection of the standard universal quantifier of first-order logic in favour of restricted quantifiers. To illustrate how such an account would handle this example, we can use a restricted quantifier that ranges over ravens, and ‘All ravens are black’ would then be written as ‘∀x : x is a raven [x is black]’. Thanks to Cian Dorr for discussion of this point. 16 I think it’s plausible that the verifiers for universal generalizations are the same as the verifiers for their equivalent negative existentials, and I take both to include verifiers for the instances. But were the Humean to deny this, she may have a response to the circularity worry I raise. An alternative would be to take laws to have the same verifiers as negative existential sentences instead. For instance, if the verifier for the law ‘all Fs are Gs’ is the verifier for ‘ ¬(∃x)[Fx& ¬Gx]’ and this sentence is not verified by the instances of the universal generalization ‘ (∀x)[Fx ⊃Gx]’, then the Humean may avoid the circularity charge depending upon what she takes the verifiers for negative existentials to be. Three questions arise for someone inclined to take this route: (i) What reasons do we have to deny that ‘ ¬(∃x)[Fx& ¬Gx]’ includes verifiers for ‘ (∀x)[Fx ⊃Gx]’? (ii) Which states exactly verify negative existentials? (iii) Do we have independent motivation for taking laws to have the same verifiers as these negative existentials instead of universal generalizations? Thanks to two anonymous reviewers for helpful discussion here. 17 This discussion is restricted to semantics involving possible worlds and states, and excludes semantics involving impossible worlds and states. Fine ([unpublished]) wants to extend the space of possible states to impossible states as well. Admitting the (controversial) existence of impossible worlds and states allows the Humean to avoid violating CON in some contexts because there may be some verifiers for (∀x)[Fx ⊃Gx]&Fa that are not verifiers for ‘Ga’, namely, impossible states that are the fusions of verifiers for Fa and ¬Fa. However, invoking impossible worlds and states will not help to avoid every violation of CON; it will not help if we adopt the second understanding of universal generalizations where the universal quantifier is restricted. Thanks to Kit Fine and Dominic Alford-Duguid for discussion here. 18 Thanks to an anonymous reviewer and Siegfried Jaag for valuable input here. 19 See (Lange [2009], Chapter 1). 20 Not every Humean will be happy to appeal to L in their explanations. Marshall ([2015], p. 3158) states: [...] there is a good reason to think that, provided they endorse BSA [the best systems account], any Humean, no matter how they respond to the objection from explanation, should reject [the fact that All Fs are Gs is a law, together with a is F, explains a is G] Given BSA, the fact that ‘All Fs are Gs is a law’ is the fact that ‘All Fs are Gs’ 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. 21 It is not entirely clear that L will contain the universal generalization as a part. The second condition for parthood states that every verifier for C must be contained in a possible verifier for A. But for the Humean, whether every possible verifier for the universal generalization will be contained as part of a possible verifier of L is controversial. Consider a possible world with one fermion. In such a world, the falsifier for ‘x and y are distinct fermions’ and a totality state will presumably be the verifier for the universal generalization. But is there any possible state where this verifier will be included in the verifier for L? That is unclear and will depend on whether the Humean thinks there can be a law like L in worlds that are sparsely populated in certain respects—for instance, in worlds with a single fermion. Nevertheless, it would be odd if the Humean were required to rely on intuitions about such controversial cases to avoid the content containment. 22 Yablo’s account of subject matters is similar, but not identical, to Lewis’s ([1988]). For Lewis, subject matters of sentences are partitions across the space of possible worlds, picked out by equivalence relations. Yablo thinks it is more appropriate to appeal to similarity relations instead of equivalence relations to understand subject matters. 23 Yablo’s truthmakers (sets of possible worlds) are distinct from both Armstrong’s truthmakers (states of affairs) and Fine’s truthmakers (states or parts of possible worlds). 24 See (Yablo [unpublished]) for a discussion of this. 25 To explain naturalness and proportionality, Yablo ([unpublished]) writes: Suppose that ⟨T⟩ and ⟨T′⟩ both imply X, and are to that extent candidates for the role of truthmaker. Naturalness: ⟨T⟩ is more natural if it obtains in a more compact, principled set of worlds. ⟨I am a man⟩ is preferred to ⟨I am a man or a mouse⟩ as truthmaker for ‘I am a man or a mouse’. ⟨T⟩ is also more natural if goings-on in a more compact, well-defined region determine whether it obtains. ⟨That chair is empty⟩ is preferred to ⟨No one is ten feet tall⟩ as truthmaker for ‘No one in that chair is ten feet tall’. Proportionality: ⟨T⟩ is more proportional to ‘X’ if it involves fewer irrelevant extras in whose absence it would still imply ‘X’. Proportionality favours ⟨Sparky weighs 16 pounds⟩ over ⟨ is a black and white dog weighing 16 pounds⟩ as truthmaker for ‘Sparky weighs under 20 pounds’. 26 Yablo’s ([2014], Appendix) account of truthmaking for propositional logic differs from the recursive view presented here. Yablo will take the truthmaker for sentence ‘X’ to be a minimal model of ‘X’. A falsemaker for ‘X’ will be a minimal model of ‘ ¬X’, which is a minimal counter-model of ‘X’. This is different from the recursive view in certain respects. On the recursive view, P ∨P&Q will plausibly have both ⟨P⟩ and ⟨Q⟩ as truthmakers. But if we are taking truthmakers to be minimal models, the truthmaker for P ∨P&Q will only be ⟨ P⟩. I do not think that this difference will impact the discussion of explanations invoking universal generalizations below. 27 As in the discussion of truthmaker semantics above, we will need to revisit the circularity charge if we admit the existence of impossible worlds in addition to possible worlds. Yablo ([2014], Appendix) himself provides a characterization of impossible worlds. Acknowledgments I am indebted to Martín Abreu Zavaleta, Dominic Alford-Duguid, Harjit Bhogal, Cian Dorr, Katrina Elliott, Kit Fine, Vera Flocke, Ian Grubb, Yu Guo, Ronald Houts, Siegfried Jaag, Zee Perry, Betty Shumener, Ang Tong, Tobias Wilsch, Jessica Wilson, and very patient anonymous reviewers for many helpful comments. Thanks also to audiences at New York University, the University of Köln, the University of Toronto, and the University of Pittsburgh. References Angell R. B. [ 1989 ]: ‘Deducibility, Entailment, and Analytic Containment’, in Norman J. , Sylvan R. (eds), Reason and Argument , Berlin : Kluwer Academic , pp. 1191 – 43 . Armstrong D. M. [ 1983 ]: What Is a Law of Nature ? Cambridge : Cambridge University Press . Beebee H. [ 2000 ]: ‘ The Non-governing Conception of Laws of Nature ’, Philosophy and Phenomenological Research , 61 , pp. 571 – 94 . Google Scholar CrossRef Search ADS Bhogal H. , Perry Z. R. [2017]: ‘ What the Humean Should Say about Entanglement ’, Noûs , 51, pp. 74–94. Bird A. [ 2005 ]: ‘ The Dispositionalist Conception of Laws ’, Foundations of Science , 10 , pp. 353 – 70 . Google Scholar CrossRef Search ADS Bird A. [ 2007 ]: Nature's Metaphysics: Laws and Properties , Oxford: Oxford University Press. Cohen J. , Callender C. [ 2009 ]: ‘ A Better Best System Account of Lawhood ’, Philosophical Studies , 145 , pp. 1 – 34 . Google Scholar CrossRef Search ADS Correia F. [ 2004 ]: ‘ Semantics for Analytic Containment ’, Studia Logica , 77 , pp. 87 – 104 . Google Scholar CrossRef Search ADS Daly C. [ 2012 ]: ‘Scepticism about Grounding’, in Correia F. , Schnieder B. (eds), Metaphysical Grounding: Understanding the Structure of Reality , Cambridge : Cambridge University Press , pp. 81 – 100 . Google Scholar CrossRef Search ADS Demarest H. [2017]: ‘Powerful Properties, Powerless Laws’, in Jacobs J. (ed.), Causal Powers, Oxford: Oxford University Press, pp. 38–55. Feigl H. [ 1981 ]: ‘Operationism and Scientific Method’, in Cohen R. (ed.), Inquiries and Provocations , Dordrecht, Holland : D. Reidel , pp. 1711 – 91 . Google Scholar CrossRef Search ADS Fine K. [ 2010 ]: ‘ Towards a Theory of Part ’, Journal of Philosophy , 107 , pp. 559 – 89 . Google Scholar CrossRef Search ADS Fine K. [ 2012a ]: ‘ Counterfactuals without Possible Worlds ’, Journal of Philosophy , 109 , pp. 221 – 46 . Google Scholar CrossRef Search ADS Fine K. [ 2012b ]: ‘Guide to Ground’, in Correia F. , Schnieder B. (eds), Metaphysical Grounding: Understanding the Structure of Reality , Cambridge : Cambridge University Press , pp. 37 – 80 . Google Scholar CrossRef Search ADS Fine K. [ 2013 ]: ‘ Truth-Maker Semantics for Intuitionistic Logic ’, Journal of Philosophical Logic , 43 , pp. 1 – 29 . Fine K. [2017]: ‘Truthmaker Semantics’, in B. Hale, C. Wright and A. Miller (eds), A Companion to the Philosophy of Language , Chichester, UK: Wiley, pp. 556–77. Fine K. [unpublished]: ‘Constructing the Impossible’. Gemes K. [ 1993 ]: ‘ Hypothetico-deductivism, Content, and the Natural Axiomatization of Theories ’, Philosophy of Science , 60 , pp. 477 – 87 . Google Scholar CrossRef Search ADS Gemes K. [ 1994 ]: ‘ Explanation, Unification, and Content ’, Noûs , 28 , pp. 225 – 40 . Google Scholar CrossRef Search ADS Hempel C. G. [ 1942 ]: ‘ The Function of General Laws in History ’, Journal of Philosophy , 39 , pp. 35 – 48 . Google Scholar CrossRef Search ADS Hempel C. G. , Oppenheim P. [ 1948 ]: ‘ Studies in the Logic of Explanation ’, Philosophy of Science , 15 , pp. 135 – 75 . Google Scholar CrossRef Search ADS Hicks M. T. , van Elswyk P. [ 2014 ]: ‘ Humean Laws and Circular Explanation ’, Philosophical Studies , 172 , pp. 1 – 11 . Hofweber T. [ 2009 ]: ‘Ambitious, yet Modest, Metaphysics’, in Chalmers D. , Manley D. , Wasserman R. (eds), Metametaphysics: New Essays on the Foundations of Ontology , Oxford : Oxford University Press , pp. 2602 – 89 . Kitcher P. [ 1989 ]: ‘Explanatory Unification and the Causal Structure of the World’, in Kitcher P. , Salmon W. (eds), Scientific Explanation , Minneapolis, MN : University of Minnesota Press , pp. 410 – 505 . Koslicki K. [ 2015 ]: ‘The Coarse-Grainedness of Grounding’, in Bennett K. , Zimmerman D. (eds), Oxford Studies in Metaphysics , Volume 9 , Oxford : Oxford University Press , pp. 3063 – 44 . Lange M. [ 2009 ]: Laws and Lawmakers , Oxford : Oxford University Press . Lange M. [ 2013 ]: ‘ Grounding, Scientific Explanation, and Humean Laws ’, Philosophical Studies , 164 , pp. 255 – 61 . Google Scholar CrossRef Search ADS Lewis D. K. [ 1973 ]: Counterfactuals , Oxford : Blackwell Publishers . Lewis D. K. [ 1983 ]: ‘ New Work for a Theory of Universals ’, Australasian Journal of Philosophy , 61 , pp. 343 – 70 . Google Scholar CrossRef Search ADS Lewis D. K. [ 1988 ]: ‘ Relevant Implication ’, Theoria , 54 , pp. 161 – 74 . Google Scholar CrossRef Search ADS Lewis D. K. [ 1999 ]: Papers in Metaphysics and Epistemology , Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Loewer B. [ 2012 ]: ‘ Two Accounts of Laws and Time ’, Philosophical Studies , 160 , pp. 115 – 37 . Google Scholar CrossRef Search ADS Marshall D. [ 2015 ]: ‘ Humean Laws and Explanation ’, Philosophical Studies , 172 , pp. 3145 – 65 . Google Scholar CrossRef Search ADS Maudlin T. [ 2007 ]: The Metaphysics within Physics , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS Miller E. [ 2015 ]: ‘ Humean Scientific Explanation ’, Philosophical Studies , 172 , pp. 1 – 22 . Google Scholar CrossRef Search ADS Paul L. A. [ 2013 ]: ‘ Categorical Priority and Categorical Collapse ’, Aristotelian Society Supplementary Volume , 87 , pp. 89 – 113 . Google Scholar CrossRef Search ADS Raven M. [ 2013 ]: ‘ Is Ground a Strict Partial Order? ’, American Philosophical Quarterly , 50 , pp. 191 – 9 . Rosen G. [ 2010 ]: ‘Metaphysical Dependence: Grounding and Reduction’, in Hale B. , Hoffman A. (eds), Modality: Metaphysics, Logic, and Epistemology , Oxford : Oxford University Press , pp. 1091 – 36 . Ruben D. [ 1990 ]: Explaining Explanation , London : Routledge . Google Scholar CrossRef Search ADS Salmon W. C. [ 1970 ]: ‘Statistical Explanation’, in Colodny R. (ed.), The Nature and Function of Scientific Theories , Pittsburgh, PA : University of Pittsburgh Press , pp. 173 – 231 . Schaffer J. [ 2009 ]: ‘On What Grounds What’, in Chalmers D. , Manley D. , Wasserman R. (eds), Metametaphysics: New Essays on the Foundations of Ontology , Oxford : Oxford University Press , pp. 3473 – 83 . Schrenk M. [ 2006 ]: ‘A Theory for Special Science Laws’, in Bohse H. , Walter S. (eds), Selected Papers Contributed to the Sections of GAP.6 , Paterbom: Mentis. Strevens M. [ 2008 ]: Depth: An Account of Scientific Explanation , Cambridge, MA : Harvard University Press . van Fraassen B. [ 1969 ]: ‘ Facts and Tautological Entailments ’, Journal of Philosophy , 66 , pp. 477 – 87 . Google Scholar CrossRef Search ADS van Fraassen B. [ 1989 ]: Laws and Symmetry , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS Wilson J. M. [ 2014 ]: ‘ No Work for a Theory of Grounding ’, Inquiry , 57 , pp. 535 – 79 . Google Scholar CrossRef Search ADS Woodward J. [ 1989 ]: ‘The Causal Mechanical Model of Explanation’, in Kitcher P. , Salmon W. (eds), Scientific Explanation , Minnesota, MN: University of Minnesota Press, pp. 3573 – 83 . Yablo S. [ 2014 ]: Aboutness , Princeton, NJ : Princeton University Press . Yablo S. [unpublished]: ‘Aboutness Theory’. © The Author 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. 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Laws of Nature, Explanation, and Semantic Circularity

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Abstract

Abstract Humeans and anti-Humeans agree that laws of nature should explain scientifically particular matters of fact. One objection to Humean accounts of laws contends that Humean laws cannot explain particular matters of fact because their explanations are harmfully circular. This article distinguishes between metaphysical and semantic characterizations of the circularity and argues for a new semantic version of the circularity objection. The new formulation suggests that Humean explanations are harmfully circular because the content of the sentences being explained is part of the content of the sentences doing the explaining. I describe the nature of partial content and demonstrate how this account of partial content renders Humean explanations ineffective while sparing anti-Humean explanations from the same fate. 1 Introduction 2 Standard Formulations of the Circularity Charge 3 Humean Responses 4 Semantic Characterizations of the Circularity Worry 4.1 Hempel and Oppenheim’s semantic circularity concern 4.2 A new version of the semantic circularity charge 4.3 Partial content as a guide to circularity 5 Humean Responses to the Semantic Circularity Charge 5.1 Smuggling in metaphysics through the back door? 5.2 Do anti-Humean laws fare any better? 5.3 The over-generalization concern 6 Conclusion Appendix 1 Introduction The world, it seems, is composed of many individual events or property-instances: a flame’s igniting in Greece in 200 B.C.E., a football’s travelling in a projectile motion in Belize in 1950, and so on. The totality of these property-instantiations across space and time is commonly referred to as the ‘Humean mosaic’. Humeans take scientific laws to derive from the particular matters of fact comprising the Humean mosaic. While not all Humeans adopt the same account of the laws, many accept or defend the best systems account (Lewis [1973], [1999]; Beebee [2000]; Schrenk [2006]; Cohen and Callender [2009]; Loewer [2012]; Bhogal and Perry [2017]). The best systems account is a regularity account of laws; it takes laws to be ‘certain true propositions and equations’ that capture regularities holding in the mosaic. The laws of the best systems account are the generalizations that are entailed by the ideally best scientific theory, where the ideally best scientific theory is the one that best balances simplicity and informativeness (Loewer [2012], p. 119). For instance, if our simplest and most powerful scientific theory maintains that across the span of the universe, all qualitatively identical fermions occupy different quantum states, then a statement of the form (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a law. In fact, the former is taken to be a law called the Pauli exclusion principle. Anti-Humeans, on the other hand, deny that laws are generalizations derived from particular matters of fact. Anti-Humean views take many forms. Tim Maudlin ([2007]) advocates taking laws as primitive. David Armstrong ([1983]) argues that laws are brute necessitation relations holding among universals. Marc Lange ([2009]) considers laws to be dependent on certain primitive counterfactuals—counterfactuals that do not depend on the Humean mosaic. Alexander Bird ([2005], [2007]) takes laws to arise from dispositional properties.1 Uniting views under the umbrella of anti-Humeanism is the idea that the laws are not summaries of the events occurring in the Humean mosaic. Anti-Humeans typically take laws to be powerful entities that have the ability to govern, direct, or guide the progression of events. One objection to Humean accounts of laws is that we cannot successfully use them in scientific explanations (Hempel and Oppenheim [1948]; Feigl [1981]; Armstrong [1983]; Maudlin [2007]; Loewer [2012]; Lange [2013]; Paul [2013]; Hicks and van Elswyk [2014]; Miller [2015]; Marshall [2015]) because doing so would be to use parts of the Humean mosaic to explain themselves. We will refer to this as the ‘circularity charge’. The structure of this article is first to introduce the existing versions of this charge against the Humean and to show that the Humean can avoid these versions of the circularity charge because these versions rely on metaphysical assumptions the Humean can plausibly reject. I then offer a new semantic version of the circularity charge that does not rely on strong metaphysical assumptions. Finally, I discuss possible responses the Humean can make to this semantic circularity charge and argue that these responses are unsuccessful. 2 Standard Formulations of the Circularity Charge Following other discussions of the circularity charge, this article concerns explanations that employ deterministic laws of nature in order to explain particular events. I will call these ‘nomological explanations’. These are explanations that appeal to exceptionless laws in their explanans.2 I will not presuppose that such explanations have a specific structure. To clarify, I will leave open whether the Humean or anti-Humean appealing to laws in explaining the occurrence of particular events can adopt a covering law model of explanation, a unificationist view of explanation that appeals to laws, or appeals to causal explanations that make use of specifically causal laws—or some other view of explanation entirely. Covering law explanations are those, like the deductive-nomological (D-N) account, that maintain scientific explanations involve laws of nature and initial conditions that inductively entail statements of the phenomenon they purport to explain. While a covering law model of explanation may seem (at first glance) to be one of the most appropriate to nomological explanations, there are a couple of reasons I do not assume proponents of nomological explanations are making use of it: First, the Humean and anti-Humean need not assume that laws (and initial conditions) deductively entail which events occur. We may take laws to explain facts concerning which events occur, but deny that this constitutes a deductive entailment. Second, the Humean and anti-Humean may take laws to feature only in some scientific explanations. Other scientific explanations may primarily use information about causal processes or causal history to explain particular matters of fact. Humeans and anti-Humeans alike appeal to laws in providing scientific explanations, but they are subject to attack from one another. Humeans fail to see how the anti-Humeans’ mysterious laws entail universal generalizations about particular matters of fact, and anti-Humeans maintain that Humeans can only provide circular explanations of particular matters of fact.3 The circularity charge concerns us here. We can informally state the circularity objection as follows: since Humean laws derive from particular matters of fact, they cannot be used as the explanans in a scientific explanation where the same particular matters of fact are the ones to be explained. This is a casual formulation of the circularity worry, and it needs specification. What does it mean for the laws to ‘derive from’ the same matters of fact they purport to partially explain? This section considers two ways to spell out the circularity objection that rely on assumptions concerning the nature of metaphysical relations of truthmaking and ‘in virtue of’. David Armstrong ([1983]) and Tim Maudlin ([2007]) raise versions of the circularity charge invoking these notions. Armstrong uses this example to illustrate his point: If ‘all Fs are Gs’ were a law, we would use it to explain ‘all observed Fs are Gs’. But [all Fs are Gs] is a conjunctive state of affairs, which we can also pick out as [all observed Fs are Gs and all unobserved Fs are Gs].4 And ‘as a result trying to explain why all observed Fs are Gs by postulating that all Fs are Gs is a case of trying to explain something by appealing to a state of affairs part of which is the thing to be explained’ (Armstrong [1983], p. 40). Armstrong frames the worry in terms of states of affairs. States of affairs are complexes of objects and properties, and we should think of our physical universe as being composed of states of affairs. States of affairs stand in parthood relations. On Armstrong’s account, [all observed Fs are Gs] is part of [all Fs are Gs] because the latter is the conjunctive state of affairs [all observed Fs are Gs and all unobserved Fs are Gs] and conjunctive states of affairs have the conjunct states as their parts. He maintains that one cannot explain why a state of affairs obtains by appealing to another state of affairs where the former is a proper part of the latter. Maudlin ([2007], p. 172) puts the concern differently: ‘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’. This version of the circularity objection invokes the ‘in virtue of’ relation, which Maudlin does not elaborate upon. One way we can understand this relation is in terms of grounding; although, it is not clear that Maudlin himself has the grounding relation in mind when advancing this version of the circularity charge.5 Grounding is a relation that holds between facts, sentences, or objects.6 It is typically considered to be irreflexive, asymmetric, and transitive. Grounding relations are supposed to capture metaphysical explanation: if P grounds Q, then P metaphysically explains Q. To state Maudlin’s concern in terms of grounding: If the parts of the Humean mosaic ground the existence of the laws, then the existence of the laws cannot in turn scientifically explain parts of the mosaic. If the parts of the mosaic taken together fully ground the law and the law explains parts of the mosaic, then the mosaic, in some sense, explains its parts. We will now explore some reasons why we should not develop the circularity charge in these ways. 3 Humean Responses The two versions of the worry above are forceful only if we adopt metaphysical accounts of states of affairs or grounding relations. A Humean can avoid the circularity worry by denying that she must appeal to such entities in the first place. In response to Armstrong, the Humean can reject that she is invested in states of affairs explaining other states of affairs: the Humean is concerned with how sentences—namely, certain universal generalizations—explain other sentences about the mosaic. So even if Armstrong is correct about how the two states of affairs are mereologically related, the Humean can deny that this impacts her account of how laws explain statements of particular matters of fact. Armstrong can reply that the Humean should still take states of affairs seriously as they are supposed to be truthmakers for law-statements. Yet, the Humean is not forced to take seriously a metaphysical truthmaking relation either. There are other ways to account for the truth of sentences that posit a less metaphysically robust apparatus than a truthmaking relation. Similarly, in response to the formulation of the worry in terms of grounding, the Humean can reject metaphysical grounding relations as part of her ontology. Grounding is supposed to capture metaphysical explanation, and it is open to the Humean to deny that this notion of metaphysical explanation is coherent or useful for her purposes.7 The Humean need not be wholly averse to grounding and related metaphysical notions in order to take this option. She can claim that grounding is a metaphysically useful notion, but that the Humean does not appeal to it here. One classic way to account for the relation between the laws and the Humean mosaic is by appeal to supervenience. When the laws supervene on the Humean mosaic, facts involving the mosaic necessitate which laws obtain. There can be no change in the laws of nature without change in the mosaic. Supervenience is a purely modal relation that does not strive to capture metaphysical explanation. This circularity charge does not arise if one takes the Humean mosaic to be the supervenience base instead of the grounding base for laws of nature.8 Another option for avoiding these circularity charges is to distinguish between the kinds of explanation at work in the problematic examples. Barry Loewer ([2012]) asserts that even if the Humean appeals to grounding relations, she can respond to the charge by claiming there are two notions of explanation in play: metaphysical explanation and scientific explanation. While the Humean mosaic metaphysically explains or grounds the laws, the laws scientifically explain the parts of the mosaic. Since these are two distinct kinds of explanations, it does not follow that the Humean mosaic either metaphysically or scientifically explains itself. This response relies on the existence of a distinction between metaphysical and scientific explanation, and it must ensure that explanations of particular matters of fact in terms of laws count as scientific and not metaphysical explanations. It is not clear there is sufficient motivation for upholding this distinction; nevertheless, we will give Humeans the benefit of doubt and assume explanations can be distinguished in this manner.9 There are now two different ways the Humean can respond to these versions of the circularity charge: (i) the Humean can reject the applicability of the metaphysical accounts of truthmaking and ground needed to press the circularity charge, and (ii) even if she endorses truthmaking or ground in this context, the Humean can try to avoid the circularity by distinguishing between the kinds of explanation in play. 4 Semantic Characterizations of the Circularity Charge 4.1 Hempel and Oppenheim’s semantic circularity concern I advocate a semantic version of the circularity objection that does not rest on heavyweight metaphysical assumptions. This version of the circularity objection descends from a version Carl Hempel and Paul Oppenheim ([1948]) raise and dismiss when discussing the D-N model of explanation. Here I argue that the semantic circularity objection is stronger than Hempel and Oppenheim take it to be. However, I also maintain that we need a different understanding of semantic circularity than the one Hempel and Oppenheim appeal to in order to advance the objection against the Humean. Hempel and Oppenheim accept a Humean account of laws in which laws are generalizations expressing regularities. As Hempel and Oppenheim’s D-N explanations involve Humean laws, the circularity worry they raise is quite similar to the ones discussed in the context of the best systems account. So, while I do not assume that either the Humean or anti-Humean accepts the D-N model as the correct account of scientific explanation, I will show that Hempel and Oppenheim’s semantic circularity worry can be raised against Humean explanations in general. The D-N model of explanation maintains that we have a scientific explanation when statements of the laws of nature, along with particular matters of fact, logically entail the explanandum in question. D-N explanations require two provisos to be in place: the statements in the explanans must be true and the law must be ‘essential’ to the explanation (Hempel [1942]). The explanation of the distinct quantum states of two fermions considered in the previous section also counts as an instance of a D-N explanation. Hempel and Oppenhiem ([1948], p. 162) write the following: It has […] been argued that in a sound explanation the content of the explanandum is contained in that of the explanans. That is correct since the explanandum is a logical consequence of the explanans; but this peculiarity does not make scientific explanation trivially circular since the general laws occurring in the explanans go far beyond the content of the specific explanandum. They suggest that such explanations may be circular because the content of the explanandum is ‘contained’ in the explanans. They characterize this containment in terms of logical consequence. They would maintain that in the case of the explanation concerning fermions above, the content of the sentence ‘a and b are in different quantum states’ is contained in the content of ‘a and b are distinct fermions and all distinct fermions are in different quantum states’, because the former is a logical consequence of the latter. Hempel and Oppenheim deny that content-containment is a problematic feature of explanation; in fact, they take it to be a virtue, but they are aware of the concern that one could consider such explanations ‘trivially circular’ in light of this containment. They pre-empt this reaction by stating that the content of an explanans’ containing the content of the explanandum is unproblematic because the general laws have much more content than just the explanandum. But why should the fact that the explanans have much more content than the explanandum help assuage this circularity worry? When a fact is used to explain itself, adding in more facts to the explanans does not detract from the circularity. For example, it would be harmfully circular to explain a particular fact such as ‘Mercury is in retrograde’ by appealing to the fact that Mercury is in retrograde. Even if we add a lot of information to the explanans—facts about the motion of other planets or the behaviour of Mercury at other times, for instance—that, in conjunction with the fact that Mercury is in retrograde, still does not explain ‘Mercury in retrograde’. It is the other information that helps explain that Mercury is in retrograde, not this other information in conjunction with the fact that Mercury is in retrograde, and the latter does not constitute a better explanation of the fact. Hempel and Oppenheim have not explained why this circularity is harmless. Hempel and Oppenheim have not successfully responded to this charge of circularity, but for the charge to be forceful we must understand the notion of containment appealed to above. What does it mean for the explanans to have the explanandum as part of its content? Hempel and Oppenheim do not adequately address this. They maintain that A has B as part of its content whenever B is a logical consequence of A. I think that while sentences contain some of their logical consequences as content-parts, there are many pairs of sentences where the contents are wholly unrelated, even though one is the logical consequence of the other. B’s being a logical consequence of A does not entail that the content of B is part of the content of A. ‘George is in the philosophy department’ does not have ‘It is raining or it is not raining’ as part of its content, for example, even though the latter is a logical consequence of the former. The contents of these sentences do not stand in a parthood relation because the sentences have entirely distinct subject matters; the first sentence is about George and the latter is about rain. Before the Humean is forced to take this version of the circularity charge seriously, we need to provide a satisfactory account of containment and content-parthood. 4.2 A new version of the semantic circularity charge This section provides a characterization of containment that differs from Hempel and Oppenheim’s and explains why it is a mark of a poor explanation that the content of the explanans contains the content of the explanandum. I address the second task and then the first. By advancing explanations where the explanandum is part of the content of the explanans, the Humean violates a non-circularity condition for scientific explanations, which is put forth in the following principle CON as follows: CON: If the content of a sentence E is part of the content of a set of sentences, then an explanation of E in terms of Γ is unsuccessful. If the explanans contains the explanandum as a part of its content, the purported explanation fails. This failure amounts to a type of circularity because the explanandum is being used, possibly together with additional content, to explain itself. Contra Hempel and Oppenheim, this kind of circularity is not negligible. CON does not presuppose a particular account of scientific explanation. Instead CON is supposed to be a generally desirable feature of scientific explanations. When a set of sentences explains another, the explaining sentences should provide us with an understanding of why a particular phenomenon occurs. If the content of the sentence being explained is already contained in the sentences doing the explaining, then we are appealing to the occurrence of the phenomenon to give us a deeper understanding of why that same phenomenon occurs. We should not appeal to a phenomenon in order to explain its own occurrence. In the next few paragraphs, I will try to articulate why violations of CON are problematic. Quintessentially successful explanations accord with CON, while quintessentially circular explanations violate CON. For example, we considered an attempt to explain that Mercury is in retrograde by appeal to the sentence ‘Mercury is in retrograde’ in conjunction with other sentences concerning celestial motion in the previous section. We can now diagnose why this explanation fails by appealing to CON above: the explanandum is contained as part of the explanans. Many explanations—such as typical causal and reasons-based explanations—do not violate CON. Following Salmon, let us take a causal explanation of some particular outcome to ‘consist of citing (some portion of) the causal processes and interactions leading up to that outcome’ (Woodward [1989], p. 358). Consider a causal explanation like the following: The lightning striking the barn—perhaps along with certain background conditions such as the barn’s being dry, the presence of oxygen, and so on—explains why the barn is now on fire. Here the statements comprising the explanans taken together (‘lightning strikes the barn and the barn is dry and…’) do not contain as part of their content that the barn is on fire.10 Standard reasons-based explanations behave similarly: Lola ate the cake because she desired cake and Maria offered her some. The statement of Lola’s desire and Maria’s offer do not contain as part of their content the fact that Lola ate the cake. We should expect explanations involving scientific laws to likewise obey CON. One source of motivation for CON follows from two constraints on explanation: (i) explanations should not contain irrelevant information, and (ii) explanations should not violate irreflexivity, that is, a sentence cannot explain itself or help explain itself. Constraint (ii) is a familiar structural constraint on explanations.11 As for Constraint (i), we can restate it as follows: If a set of sentences, Γ, fully explains another sentence, E, then Γ is wholly relevant to E, namely, all of the content of the sentences of Γ is relevant to explaining E. If any part of the content of Γ were irrelevant to E, then we would not need to appeal to it in explaining E. We would instead appeal to Γ-, the set of sentences of Γ that do not involve the irrelevant content, to provide a good explanation of E. Now, in explanations where the content of the explanandum is part of the content of the explanans, a sentence capturing the content of explanandum E will be relevant to explaining itself. In other words, it will help explain itself, which violates irreflexivity. We can restate the point more clearly as follows: If Γ fully explains E and Γ contains nothing irrelevant to E, then sentence B containing some of the content of Γ (and nothing outside of the content of Γ) helps explain E. If B = E, then we will violate irreflexivity; a sentence cannot help explain itself. CON is appealing as a condition for acceptable scientific explanations, but we need a more rigorous conception of a content-part in order to grasp exactly what CON amounts to and to understand where it gains its force. In recent literature (van Fraassen [1969]; Angell [1989]; Gemes [1993], [1994]; Correia [2004]; Fine [2012a], [2013], [2017], [unpublished]; Yablo [2014]), philosophers have developed and refined the notion of partial content for a wide range of purposes. These notions of partial content are well suited to our purposes because they aim to capture a finer-grained relation than logical consequence; as we will see, one sentence’s entailing the other will not suffice for the content of the latter being part of the content of the former on these views. Among other applications, philosophers have used the notion of partial content in order to capture notions of verisimilitude and account for differences in confirmation.12 I maintain that we can also use the notion of a content-part in order to rigorously characterize explanatory circularity. Here I spell out the notion of a content-part and show how it reveals the circularity in Humean explanations. 4.3 Partial content as a guide to circularity We can provide an account of partial content using different semantic pictures as backdrops. Most recently, Stephen Yablo ([2014]) and Kit Fine ([2012a], [2012b]) characterize partial content using the background of a possible-worlds semantics and truthmaker semantics, respectively. Possible-worlds semantics maintains that we can understand the semantic content of sentences in terms of sets of possible worlds (and sets of sets of possible worlds), while truthmaker semantics advocates using sets of parts of possible worlds to best capture the semantic content of sentences. The truthmakers here are not understood in Armstrong’s sense, but that will be explained below. The primary difference between the two approaches is that possible-worlds semantics determines whether a statement is true or false at a possible world, whereas truthmaker semantics ‘tells us what it is in the world that makes the statement true if it is true or what it is in the world that makes it false if it is false’ (Fine [2012a], p. 235). Here I follow Fine’s characterization of partial content framed in terms of truthmaker semantics. I have attached an appendix where I develop the concern using Yablo’s possible-worlds semantics. While accepting truthmaker semantics is not essential for formulating a rigorous conception of partial content, it provides a straightforward depiction of how scientific explanations either satisfy or violate circularity in terms of CON.13 In order to understand the account of partial-content, we need to grasp this semantic picture. Truthmaker semantics posits a state-space, a parthood relation over the state-space, and verification and falsification relations holding between states in the state-space and sentences. The state-space plays the role for the truthmaker semanticist that the pluriverse of possible worlds plays for the advocate of possible-worlds semantics. States pick out proper parts (or in some cases, improper parts) of possible worlds, but usually no stand is taken on what exactly these states or situations are supposed to be. Truthmaker semantics remains neutral as to whether we should think of states as concrete or abstract, and if the latter, what the nature of the abstract entities in question is. This neutrality is similar to the attitude possible-worlds semanticists take to the nature of possible worlds. We can think of the appeal to possible worlds and possible states as a modelling tool, allowing us to account for the content of our sentences. As Humeans can differ as to whether possible worlds are concrete or abstract, they can also differ on the concreteness or abstractness of possible states. For that reason, I will not assume that states are either abstract or concrete here; rather, I will show how Humean explanations violate CON on either a concrete or abstract characterization of states. States can contain other states as parts where the kind of parthood is similar to mereological parthood. Of course, if states turn out to be purely abstract entities, it may be inappropriate to consider this relation to be strictly one of mereological parthood; in this case we can take the relation to be merely ‘part-like’, in the sense that this kind of parthood obeys some of the same abstract mereological axioms as concrete parthood. Here are some examples of states containing other states: the state [a bowl of marbles is in the Philosophy Department] contains the state [a marble is in the philosophy department] as a part. Any part of a possible world containing a bowl of marbles must have a smaller part containing a single marble. The state [Colorado is mountainous and the sun is shining in Nevada] is a fusion of [Colorado is mountainous] and [the sun is shining in Nevada]. Truthmaker semantics accounts for the semantic content of sentences by appealing to how states or situations ‘exactly verify’ or ‘falsify’ the sentences in question. The content of a statement is picked out by the possible states that exactly verify it. Fine ([2017]) maintains that states exactly verify or falsity statements when ‘the state is wholly relevant’ to the truth or falsity of the statement. Exact verification is, in part, a modal notion. For a state to exactly verify a sentence, it must entail the truth of the sentence. But a state’s entailing the truth of a sentence is not sufficient for it to exactly verify the sentence. This is because states can entail the truth of totally unrelated sentences. For instance, the state [there is a donkey in the Eiffel Tower] necessitates the truth of ‘either it is raining or it is not raining’, but the content of that sentence is unrelated to the donkey’s presence in the Eiffel Tower. This is why we cannot just look at which states entail a sentence in order to determine that sentence’s content. While we cannot provide a reductive definition of what it is for a state to be ‘wholly relevant’ to the truth of a sentence, there are constraints on exact verification that deliver a good grasp of how it works. For instance, the sentence ‘the sun was shining last Tuesday’ will be exactly verified by the state [the sun was shining last Tuesday]; it will not be exactly verified by [the sun was shining every day last week]. The latter state is not an exact verifier of ‘the sun was shining last Tuesday’ because it includes extra information pertaining to the weather on days other than Tuesday. It is inconsequential to the truth or falsity of ‘the sun was shining last Tuesday’ whether the sun was shining on these other days. But [the sun was shining last Tuesday] does not contain any extra information, which allows it to be an exact verifier for the sentence. This is the sense in which an exact verifier is wholly relevant to the truth or falsity of the sentence in question. An atomic sentence, P, is exactly verified by any possible state, [p], whose obtaining is wholly relevant to the truth of P. Following Fine ([2012a]), here are clauses demonstrating how states exactly verify truth-functional conjunctions, disjunctions, and negations: State [s] exactly verifies A &B if and only if [s] is the fusion of a state [s1] that exactly verifies A and a state [s2] that exactly verifies B. State [s] exactly verifies A ∨B if and only if [s] exactly verifies A or [s] exactly verifies B. State [s] exactly verifies ¬A if and only if [s] falsifies A. Here are a few examples of sentences and the states that exactly verify them: ‘The barn is on fire and there is a thunderstorm’ is exactly verified by the fusion of [the barn is on fire] with [there is a thunderstorm]. The state [water is in Lake Tahoe] exactly verifies ‘Water is in Lake Tahoe or chlorine is in Lake Tahoe’. Negations of the form ¬P are exactly verified by the state that exactly falsifies P. For instance, ‘it is not the case that the grass is yellow’ will be exactly verified by the state that exactly falsifies ‘the grass is yellow’, which is plausibly the state [the grass is green]. We can now provide a definition of content-part as follows (Fine [unpublished], p. 11): C is part of the content of A if and only if: Every possible state that exactly verifies A contains a possible state that exactly verifies C. Every possible state that exactly verifies C is contained in a possible state that exactly verifies A. The first clause is satisfied if every exact verifier of A has an exact verifier of C as a part, and the second clause is satisfied if every exact verifier of C is part of at least one exact verifier of A. When both clauses are satisfied, C is part of the content of A. For instance, ‘the barn is on fire and there is a thunderstorm’ has ‘the barn is on fire’ as part of its content. The first clause is satisfied because every state that verifies ‘a barn is on fire and there is a thunderstorm’ is a fusion of a state that verifies ‘the barn is on fire’ and a state that verifies ‘there is a thunderstorm’. Such states contain an exact verifier of ‘the barn is on fire’ as a proper part. The second clause is satisfied because every state that verifies ‘a barn is on fire’ will be a proper part of at least one possible state that is a verifier of the conjunction. On this account of partial content, whenever C is part of A, A deductively entails C; but not every sentence entailed by A will be part of the content of A. Given this notion of a content-part, along with the principle of non-circularity, we can explicate the circularity charge against the Humean. The Humean wants to explain why particular matters of fact hold by appeal to the laws and antecedent conditions. To redeploy our example from the previous section, the Pauli exclusion principle and the fact that a and b are distinct fermions are supposed to explain the fact that a and b are in distinct quantum states. Here, the explanans is the conjunction of the Pauli exclusion principle and the antecedent condition that a and b are distinct fermions. We can rewrite this example of a Humean explanation as follows: ExplanansH: (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states], and a and b are distinct fermions. ExplanandumH: a and b are in different quantum states. The content of ‘a and b are in different quantum states’ is part of the content of the explanans, which is the conjunction of the law ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ and ‘a and b are distinct fermions’. To see why, let us consider which states are supposed to exactly verify universal generalizations. The most natural candidate for an exact verifier of a universal generalization will include the fusion of the states verifying all the instances of the generalization.14 The trouble is in determining what it takes to verify an instance of a universal generalization. I will consider two proposals. First, take a simpler universal generalization than the one featured in the explanans above: ‘ (∀x)[Fx ⊃Gx]’. One of its instances is ‘Fa ⊃Ga’. Since ‘ ⊃’ is supposed to be a material conditional, we can rewrite the instance as ‘ ¬Fa ∨Ga’. The exact verifier of this disjunction will be the states that verify either or both of its disjuncts. So, on this characterization, the exact verifier of ‘ (∀x)[Fx ⊃Gx]’ is the state that is the fusion of all the states that verify the disjunctive instances. According to a second approach, we deny that verifiers for ‘ ¬Fa’ count as part of the exact verifier of ‘ (∀x)[Fx ⊃Gx]’ because this would allow irrelevant verifiers, all the non-Fs, to be part of the verifier of the universal generalization. The proponent of the second approach emphasizes that the generalization gives us features that all the Fs have and that the behaviour of the non-Fs should be irrelevant. Here I take the instances that verify ‘ (∀x)[Fx ⊃Gx]’ to be the states of each F also being G. The fusion of states that verify statements of the form ‘Fx&Gx’ verify the universal generalization.15 For instance, if there are three things that have feature F: a, b, and c, then an exact verifier for ‘ (∀x)[Fx ⊃Gx]’ would be the fusion of the states that verify ‘Fa&Ga’, ‘Fb&Gb’, and ‘Fc&Gc’, or the fusion of the states [Fa], [Fb], [Fc], [Ga], [Gb], and [Gc]. We are now in a position to show how the content of the explanandum is contained in the content of the explanans for the Humean. Recall the explanans is a conjunction of a universal generalization and the claim that a and b are distinct fermions. The latter conjunct is straightforwardly verified by the state [a and b are distinct fermions]. On the first characterization of verifying a universal generalization, the state that verifies the first conjunct of the explanans, ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’, is the fusion of states that exactly verify instances of the following disjunction: (∀x)(∀y)[ ¬(x and y are distinct fermions) ∨(x and y occupy different quantum states)]. This is just the original universally quantified material conditional rewritten as a universally quantified disjunction. So for a and b in particular, an exact verifier of ‘[ ¬(a and b are distinct fermions) ∨(a and b are in distinct quantum states)]’ is part of the verifier for the universal generalization. But the first disjunct is false in this case since a and b are distinct fermions. Recall that [a and b are distinct fermions] is also part of the verifier for the explanans. Thus, the exact verifier of the entire explanans must include a verifier for the right disjunct. And the exact verifier of the right disjunct is [a and b are in different quantum states]. This state is the same one that serves as the exact verifier for the explanandum. The explanandum is part of the content of the explanans: Every exact verifier for the explanans includes a verifier for the explanandum. And every verifier for the explanandum is included in a verifier for the explanans.16 Humean explanations violate CON under the second approach to verifying universal generalizations as well. Here the Pauli exclusion principle is exactly verified by the conjunction of its positive instances. The fusion of states verifying the instances of ‘x and y are distinct fermions and x and y are in different quantum states’ counts as part of the exact verifier of the explanans. So the fusion of [a and b are distinct fermions] and [a and b are in different quantum states] is part of any exact verifier for the explanans. And this state contains [a and b are in different quantum states] as a part. So the explanans contains the explanandum as a part on this characterization as well. This follows generally: Since the Humean takes laws to be universal generalizations and explains particular matters of fact by appealing to the laws along with antecedent conditions, the Humean’s explanations will be harmfully circular in the way CON specifies.17 While this version of the circularity objection depicts a semantic circularity, it differs from Hempel and Oppenheim’s objection to D-N explanations. Here, not every logical consequence of a sentence counts as part of its content. The content of ‘It is either raining or it is not raining’ is not part of the content of ‘George is in the philosophy department’ because exact verifiers of the latter do not contain exact verifiers of the former as parts. An exact verifier for the latter sentence is the state [George is in the philosophy department]. And exact verifiers of this form do not contain exact verifiers for ‘it is either raining or it is not raining’. ‘It is either raining or it is not raining’ has as exact verifiers states that verify each of its disjuncts, that is, the state [it is raining] or the state [it is not raining]. Thus, not every scientific explanation that proceeds via logical consequence will exhibit semantic circularity. The notion of content part in use here is stricter than the one Hempel and Oppenheim employ. Unlike Maudlin's and Armstrong’s charges of circularity, the Humean cannot avoid this semantic circularity by avoiding certain metaphysical relations or distinguishing between scientific and metaphysical explanation. There is only scientific explanation at work here. The claim that the explanans of Humean explanations contain their explananda as part of their content rests on only semantic notions. 5 Humean Responses to the Semantic Circularity Charge This section explores four ways the Humean can respond to the circularity charge above. The responses, however, do not dismiss the threat of circularity. 5.1 Smuggling in metaphysics through the back door? The Humean may respond that this characterization of circularity illicitly appeals to metaphysical assumptions in the background and thus fares no better than Armstrong or Maudlin’s characterizations. There are two ways to develop this concern. First, perhaps the exact verification relation too closely resembles Armstrong’s truthmaking relation. Armstrong takes states of affairs involving the mosaic to be truthmakers for Humean laws, and here the truthmaker semanticist takes the states concerning the mosaic to exactly verify the Humean laws. Given the similarity of these claims, one may worry that exact verification is a metaphysically heavyweight relation in disguise. While truthmaker semantics invokes states and a verification relation between states and sentences, we have remained neutral as to the nature of the states. This differs from Armstrong’s truthmaking relation in that his relation obtains between concrete states of affairs and sentences. Furthermore, truthmaker semantics is used to specify the content of sentences, not the constituents of the physical world that settle or necessitate the truth of sentences. An account of the content of sentences is valuable for metaphysicians and non-metaphysicians alike. And unlike the grounding relation, the exact verification relation is not posited as a metaphysically primitive relation, nor is it intended to be a metaphysically explanatory relation. Exact verification is a relation appearing in our semantic theory, not in our metaphysical theories. The second way the Humean may allege that I have smuggled illicit metaphysical assumptions into the circularity charge is in the notion of parthood that covers the truthmaker semanticist’s state-space. Recall that states can contain other states as parts and that there are fusions of states on this account. This parthood relation over the state-space is crucial for making sense of exact verification. It would be troubling if the account forced Humeans to automatically accept a metaphysics of parthood as metaphysical theories of parthood are controversial. However, this theory of truthmaker semantics does no such thing. It requires that the states in the state-space stand in parthood relations, but this does not commit one to a particular metaphysics of parthood holding among objects in general, or for there to be metaphysically fundamental parthood relations in the world. As remarked above, if the states and the state-space are characterized abstractly, this may not even be the same kind of parthood relation that one finds among concrete objects. Philosophers often take mereological theories to hold among only concrete objects (although Fine [2010] is an exception), in which case the states only stand in ‘part-like’ relations to one another, relations that bear structural similarities to mereological parthood relations. Accepting relations that have some structural similarities to the mereological parthood relation does not force the Humean to adopt any robust mereological theory. Thus, we have no reason to believe that the account of partial content used to formulate this concern relies on any metaphysical assumptions that would be controversial for the Humean. 5.2 Do anti-Humean laws fare any better? 5.2.1 Primitivist accounts of laws and those involving necessitation relations If the circularity charge poses a distinctive threat to the Humean, we must ensure that anti-Humean explanations do not suffer from the same circularity. Let’s consider a few anti-Humean views. Anti-Humeans who take statements of the laws to be verified by states involving necessitation relations among universals or primitive modal entities offer non-circular explanations as well. For the anti-Humean who takes laws to be necessitation relations among universals, the exact verifiers of the explanans ‘The Pauli exclusion principle is a law and a and b are distinct fermions’ will be the state [F-ness and G-ness stand in the nomic necessitation relation] (for the relevant universals F and G underlying this law) along with the state [a and b are distinct fermions]. The fusion of these states do not contain as a part [a and b are in different quantum states]. So there is no circularity. Suppose the anti-Humean primitivist about laws offers an explanation of why fermions a and b are in distinct quantum states. Such an anti-Humean can appeal to the fact that the Pauli exclusion principle is a law in her explanans. I rewrite this sentence as ‘LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]’. This sentence together with ‘a and b are distinct fermions’ constitute the explanans of an anti-Humean explanation for a and b’s being in distinct quantum states. ExplanansAH-P: LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]], and a and b are distinct fermions. ExplanandumAH-P: a and b are in different quantum states. Although universal quantifiers appear in it, the sentence ‘LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]’ is not a universal generalization, and the state [a and b are in different quantum states] does not appear in an exact verifier for the explanans. The primitivist takes the fact that the Pauli exclusion principle is a law of nature to be a brute fact in the world; thus, states involving the behaviour of individual fermions do not verify the statement. A primitive state such as [LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]] serves as the exact verifier for the first conjunct of the explanans. And the state [LAW[ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]]] along with the state [a and b are distinct fermions] comprise the exact verifier for the explanans. An anti-Humean who takes laws to be necessitation relations between universals will be able to avoid the circularity charge as well. Let’s consider the explanation of ‘Ga’ by ‘Fa’ and the law ‘all Fs are Gs’. Here, the anti-Humean can claim that it is the sentence ‘F-ness necessitates G-ness’ or ‘N(F, G)’, conjoined with ‘Fa’ will explain ‘Ga’. The verifier for ‘N(F, G)’ will involve universals standing in a higher-order relation to one another, and the state [Ga] need not be involved. We should note that while these versions of anti-Humeanism can avoid the circularity charge, their verifiers for the statement of the law are still somewhat obscure. It is still questionable whether we are justified in positing primitive law-states or primitive necessitation relations between universals and what they are like. The potential obscurity of these verifiers highlights a different explanatory burden for many anti-Humeans: the inference problem. As Lewis ([1983]) and van Fraassen ([1989]) noted, it is mysterious how anti-Humean laws, along with initial conditions, are supposed to entail particular matters of fact if they do not do so via a universal generalization.18 Nevertheless, this is a distinct issue from the one at hand. What matters in this context is that these versions of anti-Humeanism do not fall victim to the semantic circularity afflicting the Humean. 5.2.2 Counterfactual and dispositionalist accounts of laws It is more difficult to determine whether anti-Humean accounts that employ primitive counterfactuals and dispositional properties violate CON. There are two sources of this difficulty: (i) accounts of partial content have not been fully worked out for counterfactual conditionals, and (ii) counterfactual and dispositional anti-Humean accounts of laws can take different forms. Let’s consider a counterfactualist account of laws first. Marc Lange’s ([2009]) account analyses a law of the form ‘It is a law that all Fs are Gs’, and concludes that such laws are a set of counterfactuals of the form ‘p □→∀x[Fx⊃Gx]’ for relevant counterfactual antecedents, p.19 Relevant antecedents include ones that specify certain features of the possible worlds in question, such as initial conditions of the universe that are slightly different from the ones of our actual world. First, we should note that if the counterfactuals in question do not (along with the initial conditions) entail the fact that the explanandum holds, then the explanans of the explanation will not contain the explanandum as a part. However, if one of the law-constituting counterfactuals includes the conditional in which p picks out the initial conditions of the actual world, then it seems as though an explanation using that conditional may entail the explanandum and exhibit the same kind of content-containment as the Humean’s. This will be the case if the counterfactual contains the corresponding material conditional as part of its verifier. Even in cases where p does not pick out the initial conditions of the actual world, the anti-Humean will face a problem if the verifier for the counterfactual ‘p □→∀x[Fx⊃Gx]’ contains a verifier of the consequent as a part. In this case, the counterfactual will involve verifiers for ‘∀x[Fx⊃Gx]’. Thus, whether accounts of laws based on counterfactuals survive the circularity charge will depend on which counterfactuals count as law-like and what the verifiers for counterfactuals should be. For similar reasons, it is difficult to see where dispositional accounts of laws stand on violating CON. If we analyse dispositional properties counterfactually, then the dispositionalist will have to answer similar questions as the ones facing the proponent of counterfactual-based accounts of laws. Say x has disposition P if and only if: if x had stimulus conditions F, then x would have manifestation conditions G (Px if and only if Fx □→Gx). Here our explanation of ‘Ga’ is ‘(Fa □→Ga)&Fa’. If we understand the content of this counterfactual as entailing and as having as a part the material conditional Fa⊃Ga, then the dispositionalist’s account will face the same circularity as above. However, if instead of analysing dispositions in terms of such counterfactuals, we appeal to primitive potencies of objects, then an explanation appealing to laws whose verifiers are picked out by dispositional properties will not, presumably, contain the explanandum as part of its content. This is because there is no reason to think that the verifier for ‘Pa’, if P is a primitive power or potency, has as a part the verifier for ‘Ga’. 5.2.3 Can the Humean avoid the circularity charge by changing the content of her explanation? Unlike the Humean explanations in Section 4, the anti-Humean explanations that avoid the circularity charge (the accounts considered in Section 5.2.1) do not directly involve a universal generalization like ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ in the explanans. Some anti-Humeans will invoke a statement that a necessitation relation holds among universals. Others, like the primitivist, use ‘(∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a law’ in the explanans of their explanations. Should we consider modifying the Humean’s explanation to imitate the primitivist’s? Perhaps it is the fact that the universal generalization in question is a law, which is explanatory for the Humean. The Humean still faces circularity under this modification. For the proponent of the best systems account, ‘The Pauli exclusion principle is a law’ amounts to the conjunctive claim that a certain regularity obtains and that it belongs in the best system: ‘ (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] and (∀x)(∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a simple and powerful generalization in our best system’. And the exact verifier for this state will be the fusion of states that exactly verify the first and second conjuncts of this claim. Since the first conjunct just is the universal generalization, the verifiers for the universal generalization still factor into the exact verifier of the explanans, and the circularity charge proceeds as before. Alternatively, the Humean could maintain that only the second conjunct belongs in her explanation: L: ( ∀x)( ∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states] is a simple and powerful generalization in our best system. The problem here is that L will plausibly contain the original universal generalization ‘( ∀x)( ∀y)[x and y are distinct fermions ⊃x and y occupy different quantum states]’ as part of its content.20 First, any verifier for L will include a verifier for the universal generalization. This is because part of what it is to be a generalization of the best system is for the generalization to obtain. Presumably, a verifier for a sentence that predicates a feature onto X (here the feature of ‘being part of the best system’) includes a verifier for X existing or obtaining in the first place. Second, it is plausible that every verifier for the universal generalization is part of a possible verifier for L. Take the states verifying the universal generalization (states that verify the instances, possibly together with a totality state). Here it seems like there will be a possible state verifying L that will contain the verifiers for the universal generalization.21 In this case, the Humean will be led back to the original circularity charge. I have demonstrated that anti-Humean explanations need not violate CON, but this is not to say that every anti-Humean account can avoid the circularity charge. As we witnessed above, it is not clear whether anti-Humean laws invoking counterfactuals can avoid the charge. Moreover, if an anti-Humean insists that explanations involving laws must have the universal generalizations as part of the explanans, her explanation will be circular as well. Nevertheless, this provides greater impetus to insist that it is really the anti-Humean laws—instead of the universal generalizations they are supposed to entail—that carry the explanatory weight. 5.3 The over-generalization concern The Humean may take issue with the principle that the explanans of a scientific explanation should not contain the explanandum as a content-part. CON looks initially plausible, but one could claim that it fails as a criterion for scientific explanation because too many cases of purported explanations violate it; in other words, perhaps the argument against the Humean over-generalizes. In addition to the Humean’s explanations involving universal generalizations violating CON, there are many other cases of deductively valid arguments where the content of the conclusion is contained in the content of the premises. Are these arguments unable to reflect genuine scientific explanations as well? For instance, some explanations may proceed by way of disjunctive syllogism. Consider the following: Explanans: The ducks are either in the pond or on the shore, and the ducks are not in the pond. Explanandum: The ducks are on the shore. If taken to explain why the ducks are on the shore, this violates CON because the content of the explanandum is part of the content of the explanans, according to our definition of a content-part. Perhaps having to discard cases like this from counting as scientific explanations is too high a price to pay for CON. I do not think that CON is threatened by such examples. CON tells us that arguments like these do not themselves constitute scientific explanations. Nevertheless, the example above may point us in the direction of a scientific explanation; in this case, it is a sentence that explains the disjunction that does the real explanatory work. This accounts for why the example above looks like it constitutes an explanation of why the ducks are on the shore. The disjunction initially looks like it can explain the ducks’ location, but this is only because there is a fact that explains the disjunction; the fact that it’s a principle that ducks will reside wherever they have access to the most food is what is capable of partially explaining the location of the ducks. The treatment of this case is similar to the anti-Humean’s treatment of the law-like explanations from the previous section. While it looks like the fact that all Fs are Gs and the fact that a is F explain that a is G, what actually explains, a is G is what motivates the universal generalization in the first place. The universal generalization itself does not do the explanatory work. CON provides a constraint on a genuine scientific explanation: a genuine scientific explanation must not presuppose the content of what is to be explained in the conjunction of the facts doing the explanatory work. These cases demonstrate that in order for scientific explanations to avoid triviality, we should look for the fact that explains the relevant universal generalizations (or, in this case, the relevant disjunction) in the first place. There are still escape routes for the Humean we have not explored: In order to escape the circularity, perhaps the Humean can reject CON as an adequate criterion for circularity, or develop an alternative account of the content of universal generalizations—or develop an alternative account of partial content altogether. It is not clear what such an account of partial content will look like; but even if one attempt is successful, it will yield a surprising consequence for Humean theories of laws of nature. It will show that if Humean laws are explanatory, the Humean must make substantive claims about the nature of semantic content in order to ensure this result. 6 Conclusion This article offered an account of the circularity of scientific explanations invoking Humean laws. First, I discussed existing versions of the circularity charge against the Humean and acknowledged that the Humean can coherently reject the metaphysical assumptions needed to spell out these versions of the worry. I then argued that the Humean succumbs to a semantic circularity charge. The semantic circularity charge discussed relies on a notion of partial-content that does not invoke controversial metaphysical assumptions. Finally, I considered responses on behalf of the Humean to the semantic circularity charge and determined that they are unsuccessful. Appendix: Partial Content and Possible-World Semantics In this article, I used Fine’s theory of partial content to argue that the content of the explanandum of a Humean explanation is contained in the content of the explanans. Fine’s account of partial content uses truthmaker semantics. Here I provide Yablo’s ([2014]) alternative characterization of partial content, which uses possible-worlds semantics instead, and I aim to show that it yields the same results for Humean explanations. Yablo provides an account of partial content in the framework of possible-worlds semantics. Yablo’s account of partial content invokes the notion of a ‘subject matter’.22 He maintains that the content of sentence B is part of the content of another sentence, A, just in case the inference from A to B is both truth-preserving and ‘aboutness-preserving’. And an inference from A to B is aboutness-preserving if and only if A’s subject matter includes B’s, and A’s subject anti-matter includes B’s. As for notation, single quotes—‘S’, ‘T’, and so on—continue to pick out sentences. Angle brackets—⟨S⟩, ⟨T⟩, and so on—designate propositions, which are sets of possible worlds. Possible worlds are picked out as w, w′, and so on. Yablo ([2014], p. 45) maintains that B is part of the content of A if and only if: A entails B. A’s subject matter includes B’s subject matter. A’s subject anti-matter includes B’s subject anti-matter. To understand this notion of partial content, we need to know what a sentence’s subject matter (and subject anti-matter) is, and what it is for some subject matter (and anti-matter) to include another. According to Yablo, A’s subject matter is the relation one possible world, w, bears to another, w′, if and only if A has a truthmaker in w that it also has in w′. Yablo maintains that we can think of the subject matter as a set of potential truthmakers. Moreover, A’s subject antimatter is the set of potential falsemakers. A truthmaker for sentence A in w is a proposition obtaining in w that necessitates A’s truth.23 Not all propositions necessitating A will count as truthmakers for A. A’s truthmakers are the propositions that A holds ‘because of’, or ‘in virtue of’.24 Yablo does not mean to pick out any heavy-duty metaphysical notions by ‘because’ or ‘in virtue of’ here. He thinks we can determine which proposition the sentence holds ‘in virtue of’ by finding the proposition that exhibits the best balance of ‘naturalness’ and ‘proportionality’.25 A falsemaker for sentence ‘S’ in w is a truthmaker for S’s negation. Yablo ([2014], Appendix) discusses a few different characterizations of truthmaking. Below I present one characterization, which Yablo calls the ‘recursive view of truthmaking’: Atomic sentences: Distinct atomic sentences will have different kinds of truthmakers on Yablo’s apparatus. For instance, a good truthmaker for the sentence ‘Spot has less than six legs’ will be the proposition ⟨Spot has four legs⟩. A truthmaker for ‘The flower is red’ will be ⟨The flower is scarlet⟩. Conjunctions: The truthmaker for a conjunction like P&Q will be the union of propositions that are truthmakers for P and Q. Disjunctions: The truthmaker for disjunction P ∨Q will be the proposition that is the truthmaker for P or the one that is the truthmaker for Q (both can be truthmakers for P ∨Q). A truthmaker for negation ‘ ¬P’ will be a falsemaker for P. Such a proposition will entail that P is false. For instance, a truthmaker for ‘The sun isn’t shining in Casablanca’ is ⟨It is raining in Casablanca⟩. Universal generalization: The truthmaker for universal generalization ‘( ∀x)Fx’ will plausibly be the propositions ⟨Fa⟩, ⟨Fb⟩, and so on, together with a totality fact.26 Yablo’s truthmakers resemble Fine’s truthmakers in that many sentences have the same kinds of truthmakers: a truthmaker for a disjunction will be the truthmaker for one of its disjuncts on both Fine’s and Yablo’s accounts, for example. The main distinction between Yablo’s and Fine’s accounts is that for Fine, truthmakers are states, which we can think of as parts of possible worlds; while for Yablo, truthmakers are propositions, or sets of possible worlds. We can now understand subject-matter inclusion as follows: A’s subject matter includes B’s when the truthmaker for B cannot change without the truthmaker(s) for A changing. And the falsemaker for B cannot change without a corresponding change in the falsemaker for A. In the case of an explanation of the form ‘ (∀x)(Fx ⊃Gx)&Fa’ explains ‘Ga’ (where the universal generalization contains a material conditional), Yablo’s apparatus will have the same results as the truthmaker semantics account. The explanandum is part of the content of the explanans. First, the explanans entails the explanandum, so the inference is truth-preserving (it will satisfy Condition 1, above). The inference from ‘(∀x)(Fx ⊃Gx)&Fa’ to ‘Ga’ is also aboutness preserving. The subject matter of explanans ‘(∀x)(Fx ⊃Gx)&Fa’ will include the subject matter of ‘Ga’. Here the truthmakers for the first conjunct of explanans ‘(∀x)(Fx ⊃Gx)&Fa’ will be the truthmakers for the instances of the generalization ‘¬Fa ∨Ga’, ‘¬Fb ∨Gb’, and so on (in addition to a totality proposition), and the truthmaker for ‘¬Fa’. So the truthmaker must include a truthmaker for ‘¬Fa ∨Ga’. This truthmaker cannot be the truthmaker for the first disjunct ‘¬Fa’ because such a truthmaker will be a falsemaker for ‘Fa’, and we need a truthmaker for ‘Fa’ to be the truthmaker for the second conjunct of the explanans. So the truthmaker of the conjunction will be ⟨Ga⟩. Furthermore, A’s subject anti-matter includes B’s subject anti-matter when each falsemaker for B is a falsemaker for A as well. Here, a falsemaker for ‘Ga’ (a truthmaker for ‘¬Ga’) should also serve as a falsemaker for the explanans. This is because a falsemaker for ‘Ga’ will make it such that either ‘Fa’ must be false or it is not the case that all Fs are Gs.27 Footnotes 1 Although not every account of laws invoking dispositions falls wholly under anti-Humeanism. For instance, Demarest ([2017]) also discusses a view that is ‘anti-Humean in its ontology’ because it accepts fundamental powers or dispositions, but ‘Humean in its laws’ because it retains a best systems account of laws. 2 In the discussion of Humean accounts below, discussion will be limited to laws that are exceptionless regularities taking the form of universal generalizations. As a result, this discussion will not incorporate laws that do not hold universally, nor will it incorporate probabilistic laws. We need a separate discussion of whether Humean explanations invoking probabilistic laws of nature are circular. Thanks to an anonymous referee here. 3 See (Lewis [1983]; van Fraassen [1989]) for this charge against the anti-Humean. 4 In this article, states of affairs, as well as states in truthmaker semantics (introduced below), will be picked out using square brackets ‘[ ]’ in order to distinguish them from sentences, which are picked out using single quotation marks. 5 There are other ways to develop Maudlin’s concern that avoid grounding and the ‘in virtue of’ relation altogether. One way to understand the semantic formulations of the circularity charge discussed in Section 4 is as an attempt to develop Maudlin’s circularity charge without using metaphysically robust notions, like that of grounding. 6 The relata of the grounding relation are contested. For sentential-operator accounts of ground, see (Fine [2012b]). See (Rosen [2010]) for a fact-based account, and see (Schaffer [2009]) for an object-based account. I will speak in terms of fact-grounding, but none of the points raised will rest on that choice. 7 See (Miller [2015]) for a discussion of rejecting grounding in this context. For general scepticism about grounding, see (Hofweber [2009]; Daly [2012]; Wilson [2014]; Koslicki [2015]). 8 This option may not be open to someone who thinks the laws are ‘nothing over and above’ the mosaic. Sometimes one fact or state can supervene on another, while the former still exists over and above the latter. For instance, the fact that {Socrates} exists supervenes on the existence of Socrates, but the existence of the set still exists as something over and above Socrates. Thanks to an anonymous reviewer here. 9 For further discussion of the distinction between metaphysical and scientific explanation, see (Lange [2013]) for criticism of Loewer’s approach. See (Hicks and van Elswyk [2014]) for a response to Lange, and (Marshall [2015]) for a response to Hicks and van Elswyk. 10 Here I take causal explanations to consist of the citing of causal processes and interactions leading to the event in question. However, maybe some causal explanations cite a causal law in the background. If causal laws are understood as mere generalizations over the mosaic, then they will be subject to violations of CON. 11Ruben ([1990], p. 204) discusses irreflexivity in scientific explanation. Irreflexivity constraints for metaphysical explanation (in terms of grounding) can be found in (Rosen [2010]; Raven [2013]). While irreflexivity constraints in scientific explanations are not explicitly discussed very often, explanatory asymmetry is more commonly addressed (Salmon [1970], p. 71; Kitcher [1989]; Strevens [2008], Chapter 1). If we take explanations to be asymmetric (in other words, if A explains or helps to help explain D, then D cannot help explain A), then the irreflexivity of explanation will follow: if A helps explain A, then asymmetry is violated). 12 See (Yablo [2014], pp. 95–112) for a discussion of paradoxes of confirmation and content. 13 One way Yablo’s and Fine’s accounts differ is in their treatment of certain necessarily true sentences, such as ‘2 + 2 = 4’ and ‘3 + 3 = 6’. For Yablo, the content of a sentence will be the set of possible worlds in which it is true, along with the ‘subject matter’ of the sentence. Intuitively, a subject matter is a division across possible worlds that picks out different ways in which the sentence can be true or false. For instance, the sentence ‘Bob the table is blue’ can be true in different ways: it can be true because Bob the table is turquoise, navy, royal blue, and so on. On Yablo’s apparatus, it is difficult to distinguish the content of mathematical claims like ‘2 + 2 = 4’ and ‘3 + 3 = 6’. Both sentences are true in all possible worlds, and it does not seem as if these two sentences can be true in different ways. As explained below, for Fine, the content of a necessarily true sentence will be picked out by the states that verify it in each possible world. Different states will verify ‘2 + 2 = 4’ and ‘3 + 3 = 6’, namely, the states of [2 + 2 = 4] and [3 + 3 = 6]. If one has the intuition that these two mathematical sentences don't have the same content, an account based on truthmaker semantics will secure this result more straightforwardly than one based on possible-worlds semantics. While I am using Fine's account of partial content to develop the circularity objection here, I do not intend to take a stand on this issue. 14 However, it is notoriously difficult to fully account for the verification conditions of universal generalizations, and perhaps the fusion of states verifying the instances of a generalization is not sufficient for exact verification of the generalization: we may also need something similar to a totality state, a state that ensures that there are no other Fs that have been left out of the verifiers for the instances (for a discussion of totality states, see Yablo [2014], pp. 45–54; Fine [2012b]). We can include a totality state as part of the exact verifier of the universal generalization in addition to the states that verify the instances of the generalization. There are different ways to try to characterize this kind of state, but whether and how we incorporate totality states does not impact the points made here. 15 This picture accords well with a rejection of the standard universal quantifier of first-order logic in favour of restricted quantifiers. To illustrate how such an account would handle this example, we can use a restricted quantifier that ranges over ravens, and ‘All ravens are black’ would then be written as ‘∀x : x is a raven [x is black]’. Thanks to Cian Dorr for discussion of this point. 16 I think it’s plausible that the verifiers for universal generalizations are the same as the verifiers for their equivalent negative existentials, and I take both to include verifiers for the instances. But were the Humean to deny this, she may have a response to the circularity worry I raise. An alternative would be to take laws to have the same verifiers as negative existential sentences instead. For instance, if the verifier for the law ‘all Fs are Gs’ is the verifier for ‘ ¬(∃x)[Fx& ¬Gx]’ and this sentence is not verified by the instances of the universal generalization ‘ (∀x)[Fx ⊃Gx]’, then the Humean may avoid the circularity charge depending upon what she takes the verifiers for negative existentials to be. Three questions arise for someone inclined to take this route: (i) What reasons do we have to deny that ‘ ¬(∃x)[Fx& ¬Gx]’ includes verifiers for ‘ (∀x)[Fx ⊃Gx]’? (ii) Which states exactly verify negative existentials? (iii) Do we have independent motivation for taking laws to have the same verifiers as these negative existentials instead of universal generalizations? Thanks to two anonymous reviewers for helpful discussion here. 17 This discussion is restricted to semantics involving possible worlds and states, and excludes semantics involving impossible worlds and states. Fine ([unpublished]) wants to extend the space of possible states to impossible states as well. Admitting the (controversial) existence of impossible worlds and states allows the Humean to avoid violating CON in some contexts because there may be some verifiers for (∀x)[Fx ⊃Gx]&Fa that are not verifiers for ‘Ga’, namely, impossible states that are the fusions of verifiers for Fa and ¬Fa. However, invoking impossible worlds and states will not help to avoid every violation of CON; it will not help if we adopt the second understanding of universal generalizations where the universal quantifier is restricted. Thanks to Kit Fine and Dominic Alford-Duguid for discussion here. 18 Thanks to an anonymous reviewer and Siegfried Jaag for valuable input here. 19 See (Lange [2009], Chapter 1). 20 Not every Humean will be happy to appeal to L in their explanations. Marshall ([2015], p. 3158) states: [...] there is a good reason to think that, provided they endorse BSA [the best systems account], any Humean, no matter how they respond to the objection from explanation, should reject [the fact that All Fs are Gs is a law, together with a is F, explains a is G] Given BSA, the fact that ‘All Fs are Gs is a law’ is the fact that ‘All Fs are Gs’ 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. 21 It is not entirely clear that L will contain the universal generalization as a part. The second condition for parthood states that every verifier for C must be contained in a possible verifier for A. But for the Humean, whether every possible verifier for the universal generalization will be contained as part of a possible verifier of L is controversial. Consider a possible world with one fermion. In such a world, the falsifier for ‘x and y are distinct fermions’ and a totality state will presumably be the verifier for the universal generalization. But is there any possible state where this verifier will be included in the verifier for L? That is unclear and will depend on whether the Humean thinks there can be a law like L in worlds that are sparsely populated in certain respects—for instance, in worlds with a single fermion. Nevertheless, it would be odd if the Humean were required to rely on intuitions about such controversial cases to avoid the content containment. 22 Yablo’s account of subject matters is similar, but not identical, to Lewis’s ([1988]). For Lewis, subject matters of sentences are partitions across the space of possible worlds, picked out by equivalence relations. Yablo thinks it is more appropriate to appeal to similarity relations instead of equivalence relations to understand subject matters. 23 Yablo’s truthmakers (sets of possible worlds) are distinct from both Armstrong’s truthmakers (states of affairs) and Fine’s truthmakers (states or parts of possible worlds). 24 See (Yablo [unpublished]) for a discussion of this. 25 To explain naturalness and proportionality, Yablo ([unpublished]) writes: Suppose that ⟨T⟩ and ⟨T′⟩ both imply X, and are to that extent candidates for the role of truthmaker. Naturalness: ⟨T⟩ is more natural if it obtains in a more compact, principled set of worlds. ⟨I am a man⟩ is preferred to ⟨I am a man or a mouse⟩ as truthmaker for ‘I am a man or a mouse’. ⟨T⟩ is also more natural if goings-on in a more compact, well-defined region determine whether it obtains. ⟨That chair is empty⟩ is preferred to ⟨No one is ten feet tall⟩ as truthmaker for ‘No one in that chair is ten feet tall’. Proportionality: ⟨T⟩ is more proportional to ‘X’ if it involves fewer irrelevant extras in whose absence it would still imply ‘X’. Proportionality favours ⟨Sparky weighs 16 pounds⟩ over ⟨ is a black and white dog weighing 16 pounds⟩ as truthmaker for ‘Sparky weighs under 20 pounds’. 26 Yablo’s ([2014], Appendix) account of truthmaking for propositional logic differs from the recursive view presented here. Yablo will take the truthmaker for sentence ‘X’ to be a minimal model of ‘X’. A falsemaker for ‘X’ will be a minimal model of ‘ ¬X’, which is a minimal counter-model of ‘X’. This is different from the recursive view in certain respects. On the recursive view, P ∨P&Q will plausibly have both ⟨P⟩ and ⟨Q⟩ as truthmakers. But if we are taking truthmakers to be minimal models, the truthmaker for P ∨P&Q will only be ⟨ P⟩. I do not think that this difference will impact the discussion of explanations invoking universal generalizations below. 27 As in the discussion of truthmaker semantics above, we will need to revisit the circularity charge if we admit the existence of impossible worlds in addition to possible worlds. Yablo ([2014], Appendix) himself provides a characterization of impossible worlds. Acknowledgments I am indebted to Martín Abreu Zavaleta, Dominic Alford-Duguid, Harjit Bhogal, Cian Dorr, Katrina Elliott, Kit Fine, Vera Flocke, Ian Grubb, Yu Guo, Ronald Houts, Siegfried Jaag, Zee Perry, Betty Shumener, Ang Tong, Tobias Wilsch, Jessica Wilson, and very patient anonymous reviewers for many helpful comments. Thanks also to audiences at New York University, the University of Köln, the University of Toronto, and the University of Pittsburgh. References Angell R. B. [ 1989 ]: ‘Deducibility, Entailment, and Analytic Containment’, in Norman J. , Sylvan R. (eds), Reason and Argument , Berlin : Kluwer Academic , pp. 1191 – 43 . Armstrong D. M. [ 1983 ]: What Is a Law of Nature ? Cambridge : Cambridge University Press . Beebee H. [ 2000 ]: ‘ The Non-governing Conception of Laws of Nature ’, Philosophy and Phenomenological Research , 61 , pp. 571 – 94 . Google Scholar CrossRef Search ADS Bhogal H. , Perry Z. R. 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