On Vacuous Grounding:The Case Study of Ethical Autonomy∗

Revised Draft: Cite as such

Jack Woods

Abstract: I introduce and motivate the notion of vacuous ground-ing. The occurrence of a fact q in a set of grounds for some factp is vacuous when any fact could have done the work q doesin grounding p. This notion turn out to be useful in cashingout metaphysical intuitions about what grounds what. I startmy investigation with a lengthy case study: how to formulatethe autonomy of the ethical. This is the view that we cannot“get” an ethical fact from a natural fact. Following recent workby Barry Maguire, we can make sense of this by saying that noethical fact is fully grounded in natural facts and that any factpartially grounded by ethical facts is ethical. I show that this isfalse—natural facts can be partially grounded in ethical facts—but that natural facts cannot be non-vacuously grounded in ethi-cal facts. We can better characterize the autonomy of the ethicalfrom the natural in terms of vacuous grounding and resolve sev-eral other difficulties, such as characterizing Ronald Dworkin’sview of the law; distinguishing between basic and non-basic val-ues, and resolving a paradox of grounding due to Kit Fine andStephen Kramer.

∗Thanks are due to Derek Baker, Sandy Berkovski, Catharine Diehl, Jon Litland, Ben-jamin Kiesewetter, Nora Kreft, Tristram McPherson, Eliot Michaelson, Andreas Muller,Giulia Pravato, Mike Raven, Gideon Rosen, Karl Schafer, Nick Stang, Irem Kurtsal Steen,Teemu Toppinen, Pekka Vayrynen, some helpful referees, and the members of ThomasSchmidt’s metaethics colloquium and Teemu Toppinen’s metaethics reading circle for use-ful discussion and comments. Thanks are especially due to Barry Maguire for the discus-sion which initially provoked this piece.

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1 Introduction

Recently, there has been increasing interest among metaphysicians in a typeof explanation that has come to be known as grounding.1 Grounding claimsare meant to express a conception of metaphysical explanation—distinct fromcausal, teleological, or pragmatic explanations—which is also connected upin some way to notions of dependence. One way of expressing groundingclaims is by using expressions like in virtue of, as in the following examples:

Fred’s ball is red in virtue of being crimson.

This is a triangle in virtue of being a three-side closed plane figure.

Pain is bad in virtue of being unpleasant.

Grounding has become an important part of the metaphysician’s toolbox.Besides articulating particular explanatory relations, like those displayedabove, it finds another useful application in articulating views about what thefundamental explanations are—such as definitions of physicalism like thosein (Schaffer 2009; Dasgupta 2014b)—and claims about the explanatory iso-lation of some domains of facts from others such as the autonomy of ethicalfrom natural facts (Maguire 2015) or the autonomy of mathematical fromcausal facts (common, but see (Dasgupta 2014a) for a recent example.)

However, even given the recent proliferation of expository work on ground-ing, many features of this relation2 have remained under-explored, and conse-quently, these theoretical uses of grounding notions are open to counterexam-ples of the sort which cry out for technical improvement instead of abandon-

1By ‘grounding’, I mean to include various metaphysical explanatory relations whichmay or may not form a unified class (see (Bennett 2011) for discussion). Many if not allof these will have the relevant features for my discussion here and they will all admit thedefined notion of vacuity I introduce below. So we need not take a hard line on the natureof such relations. For simplicity, I will simply talk about grounding without worryingovermuch about the nature of the relation itself.

2Following (Rosen 2010) and (Audi 2012), I am using a characterization of groundingin terms of a relation between facts. It is also possible to characterize grounding in termsof an operator on sentences (as in (Fine 2010) and (Schnieder 2011) ). Nothing substantiveabout my proposal turns on this choice—it is easy to see how to modify it for the operatorcase—so I will bracket the issue and continue talking in terms of grounding relating facts.

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ment.3 To get a grip on this sort of problem, let’s consider an example—theautonomy of the ethical from the natural. This intuition, dating at least backto Hume’s Treatise, has it that the ethical facts—facts about what we oughtto (ethically) do—are in some principled way independent from the naturalfacts about the world.4

The idea that the ethical is ‘autonomous’ is an important part of manymetaethical views, especially contemporary non-naturalist views such as thosedeveloped in (Scanlon 2014). However, exactly what this is supposed to meanis nowise clear. Recent work by Barry Maguire shows that a metaphysicalcharacterization of autonomy, cast in terms of grounding relations, ably han-dles many cases and provides a non-trivial and interesting account of auton-omy. Unfortunately, Maguire’s proposal faces a number of counterexamples.But these counterexamples do not undermine the project entirely since theyall involve ethical facts playing an inessential side-conditional role in the ex-planation of a natural fact. Such counterexamples, while they scotch thecharacterization as currently developed, motivate a technical improvementwhich explains how the ethical facts do not figure centrally in the relevantexplanations.

My overall aim in this paper is to develop a technical innovation—the no-tion of vacuous occurrence—which allows us to avoid such counterexamplesin an elegant way. The idea of vacuous occurrence is modeled on Quine’snotion of vacuous constituents of an entailment relationship. In the caseof entailment, vacuous constituents does not contribute to the truth of theentailed claim. Since our subject here is with grounding, we are concernedwith explanation and, in particular, with cases where a constituent or fea-

3Again, the minor details of the grounding relation or family of relations are not impor-tant for my purposes here. It’s sufficient that we assume it is a transitive, asymmetric andminimal relation which holds between some facts and a fact when the former explain thelatter in the sense of metaphysical explanation. See (Rosen 2010) and (Raven 2012) forjustification of the coherence, intelligibility, and usefulness of the notion and (Fine 2010)for an argument that even if some intuitive grounding notions fails to be asymmetric,transitive, and minimal, we should define and work with a notion which is such.

4I make no claim that Hume actually thought that this or its converse was true; Irather doubt he did. It is, for instance, difficult to square this account of autonomy withHume’s account of ethical judgments as arising from sympathetic reactions. But that’s amatter outside the scope of this paper and the historical facts do not undermine interestin a position inspired by a misreading. Thanks to Karl Schafer for discussion.

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ture of a constituent of an explanation does not contribute—in a sense to bespelled out below—to the explanation of the grounded claim even though itspresence is necessary for the explanation to go through. Such a notion is ofsignificant interest. As I’ll show below, it allows us to capture how the ethicalstatus of the grounds might not be a central part of the explanation of thegrounded fact, which plausibly undermines the idea that the ethical statusof the grounds “transfers” to the grounded fact. It only does so when theethical status of the grounds is non-vacuous. But we can also put the notionto broader use, for instance in capturing how the content of certain groundsmight also fail to “transfer” to the explanation of the grounded fact—whichallows us to solve a puzzle about cycles of ground by pointing out that suchcycles involve content which is vacuous. And other uses besides.

I start by developing Maguire’s proposal and the just mentioned coun-terexamples to it (§2-3). I introduce, in significant detail, the notion ofvacuity and the derivative notion of vacuous grounds (§4). I show how togive a revised definition of autonomy in terms of vacuous and non-vacuousgrounding (§4.2). As all the counterexamples to Maguire’s proposal involveethical facts vacuously grounding natural facts, this shows that a sharpenedand extensionally adequate metaphysical characterization of autonomy canbe vindicated in terms of non-vacuous grounding. Besides making progresson a hoary and seemingly intractable problem, my characterization suggeststhat those who want to lean heavily on the intuitive autonomy of the ethicalmay be back in business. The autonomy of the ethical from the natural is,of course, only one example of the sort of autonomy thesis that one com-monly finds in the literature. But the development of the examples andthe resulting solution suggests a roadmap for better characterizations of anysuch autonomy thesis. I close by pointing to three additional cases wheredistinguishing between vacuous and non-vacuous grounds can help to resolvedifficulties in applications of grounding—the distinction between being basi-cally and non-basically valuable states of affairs, Dworkin’s contention thatthe legal is, at least partially, ethical, and Fine’s and Kramer’s Paradox ofGround (§5). The progress on these three sorts of problems shows the payoffof adopting the distinction between vacuous and non-vacuous grounds.

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2 Ethical Autonomy

The autonomy of the ethical is most simply articulated in terms of whetherwe can “get” claims about what we (ethically) ought to do from claims aboutwhat is the case (non-ethically) and conversely. Part of the problem in giv-ing an account of the autonomy of the ethical concerns how to interpret this‘get’.5 Interpreting it in logical terms (entailment, etc) does not provide acompelling articulation of the intuition. Establishing this is relatively simpledue to a pair of examples from Arthur Prior.6 Let p be a statement presumednatural and q a statement presumed ethical. If p or q is ethical, then p orq and not q entail p, showing that the ethical can entail the natural.7 If it’snot, then p or q and not p entail q, showing the natural can entail the ethical.If p or q is natural, then the ethical q entails the natural p or q. If p or q isethical, then the natural p entails the ethical p or q. Q.E.D.8

While there have been useful attempts to answer Prior’s objections, thelesson of the ensuing literature seems to be that there is no rigorous logicalway to characterize the intuitive notion of the autonomy of the ethical in amanner which avoids triviality.9 Likewise, one could fuss about the status ofmixed disjunctions like p or q. But the central point stands; in any ordinarysense of entailment, ‘oughts’ sometimes entail ‘is’s and ‘is’s sometimes entail‘ought’s.10

5I use ‘natural’ instead of ‘non-ethical’ so as to not exclude the possibility that some-thing could be both natural and ethical. See below for a more explicit discussion of whatI mean by ‘natural’.

6It also immediately follows from “ought implies can.” Prior’s examples are purer.7Showing that the ethical can entail the natural is a trivial addition to Prior’s point.

Prior focused on the ethical-to-natural entailment relations, but clearly his point holds forany pair of taxonomical categories.

8Similar and, to my mind, deeper examples of failures of the autonomy of the ethicalfrom the natural are presented in (Geach 1977). Since my interest here is in the conversedirection, I shall put these interesting examples to the side.

9See (Pigden 1989) for perhaps the best of these attempts. Pigden focuses on a def-inition of autonomy in favor of a certain form of conservativeness, but, as noted aboveand by Pigden, the resulting type of autonomy is hardly distinctive—it holds as much forhedgehoggery as it does for ethics.

10The best fancy versions end up tracking distinctions very similar to those tracked bythe grounding-style interpretation of the autonomy of the ethical from the natural we willconsider below. Such fancy version also tend to be focused on whether ‘is’ entails ‘ought’,rather than the converse. See the papers in (Pigden 2010). A rare exception is (Restall

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My concern in this case study is with a more recent attempt to explicatethe autonomy of the ethical; this attempt locates the autonomy in a pair oftheses concerning what the grounds of ethical and natural facts must be like.That is, that ethical facts never hold (solely) in virtue of natural facts and,conversely, that fully natural facts never hold (partially) in virtue of ethicalfacts.11 This appeal of this type of account, albeit in a slightly differentcontext, has been given voice in (Dasgupta 2014a, pg. 6):

[A]ll particle arrangements are “apt for causal explanation”: thequestion of what causally explains how they came to be arrangedlike that can legitimately be raised even if the answer is in somecases ‘nothing’.

Consider by contrast the fact that 1+2=3. Like the initial condi-tion, this lacks a causal explanation. But there is a difference. Ifsomeone asked you what causally explains why 1 and 2 came toequal 3, you would not just say ‘nothing’; you would start talkingabout the notion of causation and the nature of abstract objectsin an attempt to show that the question should not have beenraised in the first place.

Dasgupta is here pointing to a certain autonomy thesis—the autonomy ofthe mathematical from the causally efficacious (on which, see §5.3 below).12

At least some believers in the autonomy of the ethical from the natural thinkthat a similar response is appropriate for the question of what physicallygrounds the fact that murder is wrong or what ethically grounds the factthat Earth has a single moon—the question should not have been raised atall since there is no and could, by virtue of the difference in topic, be no suchexplanation. This sort of response is a far more satisfying explication thanthe mere failure of entailment. Note that such an account of autonomy isnot yet a fully satisfying explanation of why the ethical and the natural are

and Russell 2010) who reinterpret the intuitive notion of ‘is’ and ‘ought’. Even on theircareful defense of the claim that you can’t derive a genuine ‘ought’ from a genuine ‘is’,the converse fails rather dramatically, as they note.

11A fact is partially grounded by some collection of facts just in case that collection isamong the total grounds of the grounded fact.

12On which, see more below.

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autonomous any more than an account of mathematical and causal auton-omy is a full explanation of why the mathematical is autonomous from thecausal.13 Rather, grounding explications of autonomy are accounts of whatautonomy comes to. They are attempts to flesh out the skeletal cartoon-Humean slogan that the ethical is autonomous from the natural.

I will argue (§2.1) that making sense of the taxonomical project of dis-tinguishing the ethical from the natural requires we admit an intuitive senseof both these terms. Moreover, in order to give a plausible taxonomy, thenatural and the ethical must be at least partially distinct. Given these twopoints, the role of an autonomy thesis can be seen as clarifying and extendingour intuitive sense of ethicality to cover non-obvious cases. In §3, I argue thateven in the intuitive sense of ethical and natural just mentioned, natural factscan be partially grounded in ethical facts.14 This is consistent with allowingthat the grounds of a fact can matter in determining whether or not it isethical—we need only accept that sometimes the ethicality of the grounds ofa fact do not transfer to the grounded fact. In §4.1, I introduce the notionof vacuity and use it to explain why ethicality only sometimes transfers fromgrounds to what they ground. Roughly, I argue that vacuous grounds haveno effect on the ethical status of the fact they ground. I then explore the no-tion of vacuity in detail. In §4.2, I use vacuity to formulate a more plausibleaccount of autonomy in terms of grounding . And in section §4.3 I discuss

13The above “they’re simply about different things and thereby inapt to explain eachother” is my preferred way of formulating the (cartoon) Humean intuition. This, at leastin part, because it respects the rejection of brute necessities that Hume is also famous for.However, Maguire’s account of autonomy (and my precisification of it) are not inconsistentwith its simply being a brute fact that no ethical fact has a fully natural explanation. Notealso that Dasgupta uses a slightly different notion of autonomy—certain facts being inaptfor explanation at all—in an attempt to rescue the Principle of Sufficient Reason. Thecomparison with this and the sort of local autonomy theses I am interested is interesting,but outside the scope of this paper.

14I do not discuss the question of whether ethical facts can explain or partially explainnatural facts in the non-metaphysical sense of explanation as the issue is both far toocomplex and far more tenuous given the difficulties in the notion of ‘explanation’. See(McPherson 2008) for a careful discussion of justificatory autonomy, epistemic autonomy,and an argument that autonomy in this sense depends on the correct metaethics. Nor willI address in any detail whether ethical facts can be part of causal explanations of naturalfacts—though see (Shafer-Landau 2007) for a defense of the causal efficacy of the ethicaland (Harman 2012) for the classic objections. As will be made clear below; my cases aresignificantly different.

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the application of vacuity in formulating other autonomy theses, like thosementioned above.

2.1 The Grounding Account of Ethical Autonomy

For simplicity, we will restrict our discussion to a particular sense of ‘what weought to do’; I will use ethical as an adjective to pick out those propositionsand facts that have to do with what we ought to do which are underwrit-ten by facts about what is right and wrong.15 Likewise, I will use naturalto pick out those propositions and facts which have to do with the way theworld is.16 For reasons which will be discussed shortly, it would be too hastyto assume that ‘ethical’ and ‘natural’ pick out complementary properties ofpropositions—we want to leave open that some facts and propositions couldbe both (or neither) ethical and natural in the relevant sense.17 Whetherthere are such facts is an interesting and open question. The question ofthe relationships between ethical and natural facts is also bound up with thetaxonomical question of how, exactly, the natural and the ethical are distin-guished.

A taxonomical distinction between the ethical and the natural is impor-tant for many things, but it is essential to make sense of metaethical views.Metaethical positions frequently assume these terms have a clear sense inorder to characterize their positions—no ethical claims are true, or meaning-ful, or what have you; the ethical is not reducible to the natural, or it is, orit sometimes is; ethical claims are not descriptive, but expressive, prescrip-tive, etc. Each of these easily recognizable positions relies on a distinctionbetween the ethical and the natural; rather few have attempted to providea clear one. We make progress in understanding these views by precisify-ing the distinction between the ethical and the natural. Since my primaryaim in this section is to show that the ethical can ground the natural, I will

15These are examples; they are not meant to be exhaustive. Facts about, say, ‘kindness’and ‘benevolence’ may also be intuitively counted ethical facts. Anyways, the intuitivedistinction is clear enough. Thanks to a referee for discussion.

16I do not exclude the supernatural, the high-church metaphysical, or even the numericalfrom the natural—as I might in another context.

17One might worry here that ethical facts are about how the world is—they’re abouthow it is ethically. I could use substantively natural or something like that to indicatewhat I have in mind by the purely natural facts, but since I want to leave it open whethersome facts are both, I will not worry about this overmuch here.

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focus on a very strong form of ethical autonomy where ethical concepts arenot reducible to natural concepts, ethical properties not reducible to naturalproperties, and where there is no explanatory priority of the natural over theethical. Even on this strong form of autonomy, the ethical can ground thenatural; though, as I will show, such cases are all cases of grounding wherethe role played by the ethical is vacuous in a sense I will develop below.

The developed version of this strong form of ethical autonomy is due toBarry Maguire.18 He argues for two principles which jointly give a completecharacterization of the ethical domain as autonomous from the natural.

METAPHYSICAL AUTONOMY (MA): No ethical fact isfully grounded just by natural facts.

CONVERSE METAPHYSICAL AUTONOMY (CMA):Any fact partially grounded in an ethical fact is an ethical fact.19

Maguire’s characterization is not an implicit definition. We thus need toassume that ‘ethical’ and ‘natural’ have a pre-existing sense. We also needto assume that the domain of the ethical and the domain of the naturalare at least partially distinct. For suppose not: then MA implies that weneed some additional domain to ground the ethical, if any ethical facts arenon-fundamental. So we assume the independently plausible claim that theethical and the natural are at least partially distinct. But more needs to besaid. We need to suppose that there is some class of facts and propositionswhich are in some sense clearly ethical and a class of facts and propositionswhich are in some sense clearly natural in order to make sense of Maguire’scharacterization.20

18See (Maguire 2015).19Note that if the domain of the ethical and the natural are fully distinct, then we

could replace this with “No natural fact is partially grounded in ethical facts”. But sinceconjunctions are grounded in both of their conjuncts, this would require taking on theadditional constraint that no ethical fact is also natural. I do not wish to take a stand onthis issue though it seems more plausible to me to work with a more inclusive conceptionof taxonomy since conjunctions seem partially ethical and partially natural. We could alsodefine a notion of “totally ethical” and “totally natural”, but such complications woulddistract us here.

20There are claims that contain ethical terminology which might be considered naturaleven though they say something about what we ought to do. Consider Geach’s example:“Nobody ought to adopt the practice of doing something he ought not to at least twice

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It is not hard to find facts from each type. Let us write 〈φ, F 〉 to indicatethat an object or action like φ has the property of being F .21 Propositionssuch as 〈murdering Jones, wrong〉 then indicate that the action of murderingJones has the property of being wrong—and seem clearly in the camp of theethical. Besides the wrongness of murder, perhaps the permissibility of help-ing strangers, the goodness of giving your mother flowers on her birthday arealso good cases.22 There are also clearly natural facts. That certain electronsare spin-up, that I have acquired a teddy bear, and that John won a gametoday seem clearly in the natural camp. Note that this carving is by nomeans exhaustive; there are going to be many borderline cases (cleanlinessis next to godliness) and cases like conjunctive, conditional, and disjunctivefacts where it is not clear what we should say. Are they in one camp or theother? Both? Neither? Spoils to the victor? We need not worry much aboutsuch cases much; they are generally not clear cases one way or another so itis reasonable to let a principled taxonomical account decide them.

We will further assume that our notion of proposition and facts involvesstructured entities like those displayed above and that facts are simply truepropositions. This means that we can have distinct propositions which arenecessarily (intensionally) equivalent and even where one is, in some sense,ontologically reducible to the other (Rosen 2010). So, for example, 〈Freddy,triangle〉 is grounded in 〈Freddy, three-sided closed plane figure〉 and thetwo propositions are necessarily equivalent, yet they are not identical. Thelatter unpacks the former; their constituents and their manner of combina-tion are different. We also assume Rosen’s claim that when we can give areduction of a property F to another G in the sense of giving a real defini-tion or analysis—such as that provided for [triangle], then 〈a,F〉 is groundedby 〈a,G〉. Consequently, 〈Freddy, three-sided closed plane figure〉 grounds〈Freddy, triangle〉 but the latter does not ground the former.

every day.” (Geach 1977). Since this says nothing about what particular things we oughtto do or refrain from, Geach considers this claim morally vacuous, or, in his terminology,not morally significant. Given the plausible transitivity of grounding, this is no problemfor the examples I provide below.

21We could consider more complicated propositions, but the simple predicational formwill suffice for our purposes.

22We assume that some clearly ethical propositions are true for the purpose of thisdiscussion.

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Note that even this fine-grained conception of a fact is insufficient to guar-antee that necessarily equivalent, but intuitively distinct, facts, are distinct.We could, for example, have a case where 〈a, F〉 and 〈a, G〉 are intensionallyequivalent and where F and G are necessarily co-extensive simple properties.In such a case, it is not plausible that one “unpacks” the other in the waythat being a three-sided closed plane figure unpacks being a triangle.23 Andit is highly plausible that there could be such pairs of propositions—assume,for example, that [badness] is the sole fundamental normative notion andthat [badness] and [painfulness] are necessarily co-extensive. It is not plausi-ble that [painfulness] unpacks [badness], but it is plausible that 〈 a, bad〉 isdistinct from 〈 a, painful〉. This means that a proponent of autonomy needsto provide a hyperintensional—more fine-grained than intensional—criterionof property individuation if they are not to make substantive and implausiblemetaphysical and ethical claims. In order not to beg any questions againstthe proponent of autonomy, I thus assume that properties can and should beindividuated hyperintensionally in some way.24

Finally, for my purposes here, I do not assume that grounding facts in-clude all facts connecting properties involved in the grounds which mightbe needed to explain to someone why the grounded fact is explained by thegrounds. For example, I do not assume that Joe’s having done somethingunethical is totally grounded in both in the fact that he stole someone’s carand in the fact that stealing cars is unethical. Likewise, I do not assume thatA and A entails A ∨ B both occur in the grounds of A ∨ B. Settling whichfacts go in the grounds would require a discussion of how seriously and inwhat way we are to take take the concept of explanation which is built intothe grounding relation. And nothing in my initial argument turns on it. Iintroduce a technical device, grounding* relations, in the later portion of thepaper to finesse the issue.25 I do assume that grounding is a relation whichis asymmetric, transitive, and minimal—though, again, each of these prop-erties has been called into question. I’ll leave these challenges aside in thediscussion to follow since (1) as Rosen and Fine point out, if you don’t thinkgrounding has these features, we can define a useful and important relation

23See again (Rosen 2010) for a discussion of unpacking.24Thanks to an anonymous referee for pushing me on this issue and suggesting I indi-

viduate hyperintensionally.25See also the below discussion of grounding vs grounding* facts and, especially, the

discussion of complete vs incomplete grounding relations in §4.1.2.

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which does and (2) nearly all of my argument can be adapted to groundingrelations which fail to have these features.26

Assuming there are some clearly ethical and some clearly natural propo-sitions and, moreover, these details about the grounding relation, we caneasily make sense of the purpose of CMA and MA. They serve to constrainany extension of the properties of being a clearly ethical proposition and be-ing a clearly natural proposition to cover the non-clear cases. We need nothave an antecedently perfect characterization of the ethical and the naturalin order to accept MA and CMA—nor must we require that such a constraintguarantees a unique extension. MA and CMA articulate a constraint on whata successful account of the ethical—and the natural—would be, presumablyderiving from some intuitive grasp of what the ethical domain concerns.

It is worth noting initially that Maguire’s view implies that our intuitivegrasp of the topic of a proposition may fall victim to the circumstances of thegrounds of the proposition. Giving up our intuitions about propositions notclearly ethical or natural is not devastating. We can accept, perhaps, thatcertain borderline cases can be tidied up. However, this problem becomesmore serious when it bleeds over into reclassifying the core cases which moti-vated our criterion in the first place, such as the fact that John won a gametoday. Since CMA and MA are not meant to implicitly define the conceptsof the ethical and the natural, but rather to analyze them, strikes againstour intuitive grasp of these concepts are strikes against the classification.

As long as the ethical domain is interestingly distinct from the natural,Maguire’s autonomy principles do a significant bit of explanatory work. Forexample, Maguire convincingly argues that Prior’s examples of natural factsentailing ethical facts (Prior 1960) and ethical facts entailing natural factsdo not undermine the Humean autonomy intuition when it is explicated byCMA and MA. As Maguire points out, if (p or q) really is ethical, then pre-sumably it is because either p or q is ethical and true. So it will not be fullygrounded only by the natural component—it will also be fully grounded inthe ethical component. If it is natural, then even though it, in combination

26And, in fact, I suggest below (§5.3) a principled reason for weakening the irreflexiveproperty—which property follows from my assumptions here—to allow reflexive vacuousgrounding.

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with the negation of the natural disjunct, may entail the ethical disjunct, ithardly grounds it. So in neither counterexample to the logical version of theautonomy thesis do we have a counterexample to Maguire’s version of ethicalautonomy.

The nature of Maguire’s proposal can be best seen by a comparison.Consider the Moorean position articulated in (Rosen 2010) where moralfacts are not reducible to non-moral facts, but where they are grounded innon-moral facts like the distribution of happiness.27 This sort of position—paradigmatically a form of ethical intuitionism—violates MA since fully non-moral facts fully ground moral facts.28 The positions that acceptance of MApermit thus require a stronger conception of how the moral is autonomousfrom the non-moral; they imply that the moral somehow floats free fromthe non-moral, being neither reducible to it, nor even grounded in it. Suchstrongly intuitionistic positions are on the rise recently and with their rise,a demand for a clear statement of their characterizing claims becomes in-creasingly important.29 I do not myself hew to such a position. Rather, myaim is to attempt to articulate what a defender of such a position shouldsay regarding the distinction between ethical and natural facts and proposi-tions. Only with such an articulation can the merits (and demerits) of sucha position be clearly assessed.

27Note that this is distinct from the position on which Moore’s claim is merely aboutconcepts, but where the properties may be reducible (Gibbard 2003, 32). Such a positionis obviously also in violation of MA at least on plausible assumptions about the relationof properties and facts.

28Note that these sort of positions put the lie to the converse of Rosen’s Grounding-Reduction link. Even if we accept that real definition or analytic reductions imply ground-ing relations, we should not accept that grounding relations imply reductions. Moore’sposition as articulated by Rosen is a cogent picture of how the moral can could fail tobe reducible to the non-moral while being grounded in it. See also Jonathan Dancy’sdiscussion of resultance, his term for a relation much like grounding, in (Dancy 1993).

29For perhaps the most notable such position, consider that articulated by T.M. Scanlonin his Locke lectures. See (Scanlon 2014, Ch. 2). See also (Fine 2002) and (Cohen 2009)and for a defense of the general position, (Maguire 2015).

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3 Counterexamples to Converse Metaphysi-

cal Autonomy

MA and CMA do better justice to the intuition that the ethical is autonomousfrom the natural than the logical characterization torpedoed by Prior. Afterall, the grounds of some fact intuitively make it the case in the non-causalsense that that fact holds. This shape is a triangle in virtue of the fact thatit is a three-sided closed plane figure. Picking my nose on a date is rude invirtue of the fact that it is a violation of dating norms. At least one way ofcashing out Hume’s intuition seems to be that facts about what we oughtto do are not made the case by facts about what is the case—except, ofcourse, for certain facts about what is the case about what we ought to do.So Maguire’s characterization seems, at least prima facie, superior to otherproposals to explain the autonomy of the ethical on this score. But onlyprima facie, since CMA is false, as we will shortly see. For the rest of thissection, I will focus on CMA and intuitive cases of purely natural facts whichare partially grounded in ethical facts.

To see that CMA is false, consider the following game:

Truth or Dare A pair of players, in sequence, challenge theother to either answer a question or perform a dare. The firstplayer who fails to do the dare or lies loses.

Suppose Steve has won truth or dare today, playing against Beth and itwasn’t a tie. Moreover, suppose that he won by challenging Beth to go 30minutes without doing something unethical. When Steve goes to the bath-room, Beth takes twenty dollars from his wallet. As stealing is unethical,Beth thereby loses. In fact, his winning truth or dare today is explained bythe fact that Beth failed her dare and this, in turn, is explained by the factthat she did something unethical before 30 minutes passed.30 Since ground-ing is transitive, the particular ethical transgressions Beth committed areamong the grounds of the fact that Steve won. And such facts will be propo-sitions of the form 〈φ, Wrong〉 which, by the lights of the discussion above,

30Note that it might sometimes be difficult to determine who won truth or dare. Thisis not to the point; the loser of the game is the one who lies or fails their dare, whetherthis is discovered or not.

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are clearly ethical.

We might worry that facts like 〈φ, wrong〉 are co-extensive with factslike, say, 〈φ, not utility maximizing〉 which are themselves intuitively natu-ral. And, furthermore, that it is the latter, not the former, which groundthe fact that Steve won. But this seems strange. There are three optionshere that I see. The first would hold that these two facts are not only co-extensive, but identical. In this case, we still have a counterexample sincethe fact is intuitively ethical (as well as being intuitively natural). But wewould also have a problem with MA since presumably the grounds of Beth’saction failing to be utility maximizing would be fully natural and therebywe would have an ethical fact fully grounded in natural facts. The secondoption would hold that the 〈φ, wrong〉 is totally grounded in 〈φ, not utilitymaximizing〉, but that they are not identical. Note though that here we havea violation of CMA and a violation of MA. If 〈φ, wrong〉 totally groundsSteve’s win, then we have a violation of CMA since 〈φ, wrong〉 is ethical.And since 〈φ, not utility maximizing〉 grounds 〈φ, wrong〉, then we have aclear counterexample to MA since 〈φ, not utility maximizing〉 is natural.

Finally, we could have the view that Steve’s winning is grounded in 〈φ,not utility maximizing〉 which is co-extensive to 〈φ, Wrong〉, but where thislatter fact plays no role in explaining why Steve won. But given the natureof Steve’s dare, this seems paradigmatically a case where we would invokeethical properties in explaining something—it would be strange to have eth-ical facts around, but where they played no role in explaining things likethe fact that Steve won the game. In other words, it would be strange thatSteve’s speech act of daring managed to reference [wrongness], but the ex-planation of its frustration and Steve’s consequent win involved only [utilitynon-maximization]. At best, such views are seriously ad hoc. Ethical factswould be theoretical danglers on such a view, full of righteous content butexplaining nothing.31

Objections along the lines that the grounds of Steve winning the gameare really descriptive facts—like that Beth failed her challenge or did some-thing not utility maximizing—thus misfire as facts like that Beth failed herchallenge are themselves plausibly grounded in particular ethical transgres-

31Thanks to an anonymous referee for pushing me to clarify this point.

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sions and grounding is transitive.32 You might also think the fact that Stevewon—or, rather, the event of his winning—is caused by Beth doing ethicalthings, but it is also certainly an instance of grounding.33 Compare the factthat Bradley Wiggins won the final yellow jersey in the 2012 Tour de France.This fact is explained by the fact that Bradley Wiggins accumulated the leastamount of riding time overall. That’s just what it is to win the final Tourde France yellow jersey;34 likewise, failing to perform a dare explains losingtruth or dare. It is not merely brought about by it.

Note that Steve’s winning truth or dare is not intuitively an ethical fact.It seems a simple natural fact—that he won the instance of truth or darethat he and Beth were playing today. Moreover, this fact grounds anotherfact—that Steve won a game today. And since grounding is transitive, thefact that Beth did the particular wrongs she did are among the grounds forthe fact that Steve won a game today. But it is even more implausible thatSteve’s winning a game is an ethical fact. If we agree that there is a class ofobviously and clearly natural facts, 〈Steve, won a game today〉 is a member.I have not given a substantive argument for this intuitive taxonomical fact,of course, but it’s not clear that it needs one. These facts about Steve andhis winning a game are not about the ethical in any ordinary sense of ‘about’.

If truth or dare is a game we can play—and it is, I’ve played it—and iffacts like ‘Steve won truth or dare today’ and ‘Steve won a game today’ arenot ethical—and they’re not—then the natural is not fully independent fromthe ethical.35 Maguire’s taxonomy is incorrect. Maguire successfully usesCMA to avoid worries for the ethical status of mixed claims like conjunc-tions. Such cases are different from those we considered above; conjunctionscan be viewed as composed of the relevant facts. That is, we can see that Aand B as composed of 〈A〉 and 〈B〉 and it is not crazy to hold that a fact

32Thanks to Giulia Pravato and Catharine Diehl for valuable discussion on this point.33Whether you think these are cases of causation as well will depend on your view of

causation. But all I need for my point is that it is a case of grounding; if it is also a caseof causation, this is interesting but irrelevant.

34At least given that other conditions on legitimate performance, such as refraining fromperformance-enhancing drugs, are met.

35Although I used facts like ‘Beth did something unethical’ in my example, I couldsubstitute examples using particular ethical transgressions instead. Nothing turns on this,though unconvinced readers might want to try the examples both ways.

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composed of both ethical and natural components is simultaneously ethicaland natural. But there is no plausibility in holding that the fact that Stevewon a game today has among its constituents that Beth did something un-ethical.

Maguire’s claim that facts like 〈Steve, won truth or dare〉 are ethical factsis, perhaps, more plausible when facts are fully analyzed into their sets ofgrounds and where these grounds include an ethical fact. For example, sup-pose (a) 〈Steve, won truth or dare〉 could be fully analyzed in terms of theconstitutive rules of truth or dare and Steve and Beth’s actions, yielding areduction of it to the conjunction of its grounds; and (b) if when we havepropositions α and β where α analyzes β, then if α has a certain taxonomicstatus, so does β.36 Then since the grounds of 〈Steve,won truth or dare〉 in-clude ethical facts and since conjunctions presumably inherit the taxonomicstatuses of their conjuncts, it would follow that 〈Steve, won truth or dare〉is an ethical fact. If such a position were cogent, then the spread of theethicality of Steve’s dare to what it grounds would be more plausible.

But though even nothing I say below hinges on answering the question ofwhether propositions which are reducible to their ethical grounds are ethicalor not, this line of defense will not work. There are cases where a groundedfact cannot be reduced to its grounds, but where an ethical fact partiallygrounds a natural fact. Consider:

Unethical Bet John and Fred agree that ownership of the fluffyteddy bear, call it ‘Ted’, currently under dispute shall go towhichever of them does the most unethical things between 12am and 11:59 pm on June 2nd, 2014.

That Fred owns Ted has among its grounds at least the details of Fred andJohn’s unethical bet and the fact that Fred did more unethical things thanJohn did during the period specified. Note that the fact in question is a factabout ownership, not possession. That Fred possesses Ted can be explainedcausally on the basis of the beliefs of John, Fred, and so on about who wonthe bet; that Fred owns Ted does not have to do with who is believed to havewon the bet—but with who actually did win it. And, most importantly,there is absolutely no plausibility to claiming that the fact that Fred owns

36See §4.1.2 below for complications related to (a) involving the individuation of games.

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Ted can be completely analyzed in terms of its grounds—perhaps in contrastto my initial truth or dare example. Fred’s ownership of Ted could have beengrounded in many distinct sets of grounds while remaining one and the samefact (though see §4.1.2 for complications.)

Now, the fact that Fred owns Ted is not an ethical fact, even though itis related to some ethical facts—just as the fact that the moon is full is nota tidal fact, though it is related to some tidal facts.37 To hold otherwise isto hold that whether or not 〈Fred, owns Ted〉 is ethical depends on the factsthat make it the case that Fred owns Ted. But notice that parallel cases seemto be absurd. 〈Fred, owns Ted〉 doesn’t seem to be a backgammon fact evenif Fred won Ted in a backgammon game. If even if Fred won Ted by gettinga paper accepted at Biological Review before John did, Fred’s ownership ofTed doesn’t seem to be an academic fact. It seems far better to say that〈Fred, owns Ted〉 is a fact which is grounded by an ethical fact, but it is notitself a ethical fact. After all, it is not a fact about ethics, but a fact aboutthe outcome of a bet whose criteria for winning involves ethical properties.Neither is the fact that Fred will win the bet if he does more unethical thingsthan Ted between between 12am and 11:59 pm on June 2nd, 2014 an ethicalfact, even though it explicitly contains ethical terminology. No more an eth-ical fact, say, than the piece of economic advice ‘the more unethical the job,the more profitable’.

Or consider:

Explicitly Moralistic American Politics In America*, electedofficials must have lived an ethically impeccable life as a prereq-uisite for high office. Barack Obama is currently occupying highoffice in this place and was elected in compliance with all the rules.Barack Obama thus must have lived an ethically impeccable life.

That Barack Obama is currently occupying high office in America* is par-tially grounded by at least the facts that he was elected, is over 35, born inthe United States, and that no electoral rules were broken—as well as hisliving a ethically impeccable life. Note that there is no plausibility in the

37By my lights, the ownership fact is a conventional fact. Even if a fact confers rightsand responsibilities, even ethical ones, it is itself not an ethical fact simply by virtue ofthis. I could rerun the examples using some other outcome, but the point would be thesame. Thanks to an anonymous referee for discussion.

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thought that it is caused by the fact that he lived an ethically impeccablelife. But is the fact of his presidency an ethical fact? Intuitively not.38 Thestyle of examples I have been exploring can be multiplied; it is enough forme to point out that we have given three examples, each of which partiallygrounds a natural fact in ethical facts. The second example explicitly avoidsthe worry about reduction. The third example explicitly avoids the worryabout causation. And neither worry really arises for the first example, how-ever tempting it is to think so.

3.1 Upshot of the Counterexamples

The upshot is a general recipe for trouble for autonomy theses like CMA. Forany natural social fact which is grounded by decisions, contracts, practicesand the like, we can concoct a case in which these practices make reference tosome ethical fact without thereby making the resulting fact ethical. We don’tthereby change the intuitive taxonomic status of the outcome. The failureof CMA is important since a large percentage of the facts characterizing ourdaily lives are social. It is thus significant that natural propositions can begrounded in ethical propositions; that the autonomy of ethics holds in onlyone direction, if at all.

It would be a radical move to deny that the status of a fact as ethicalor natural is largely due to what the fact is about, especially because theintuition driving MA relied on there being a difference in subject matterbetween the clearly ethical and the clearly natural facts. Surely facts areindividuated, at least in part, by means of what they are about.39 And theimportance of MA—and MA is important, even if false—depends on the factthat there is an interesting sense in which the ethical and the natural aredistinct. Avoiding the sorts of counterexamples to CMA I’ve given here in

38A reader might worry here that ‘being president’ carries ethical duties and responsibil-ities over and above the ethical conditions necessary to get legitimately elected. This doesnot really affect the case—substitute any office you like which has conventional rules gov-erning who can hold it and make the ethical necessary condition only hold up to the pointof election or selection. Afterwards, perhaps, you can be as much like Lyndon Johnson asyou choose. Thanks to Derek Baker for suggesting this worry.

39The notion of aboutness I am employing could be made precise in a number of ways,but the intuitive notion suffices here.

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this way—that is, by broadening the scope of the ethical far past those factswhich are about ethics—threatens to undermine the importance of MA. Atthe very least, these examples put the burden on the Maguirean Taxonomistto give a characterization of the clearly natural which doesn’t fall victim toour sorts of examples.

Of course, we might think that the border of the ethical fact is so vaguethat we would be performing a service by replacing it with a more explicittaxonomy. Perhaps we should say something is an ethical fact just in caseit is clearly ethical in the sense sketched above or has clearly ethical factsamong its grounds. But this would secure the autonomy of the ethical bysignificantly diminishing its import. If, say, the outcome of the Humeanintuition that the ethical is independent from the natural is that the ethi-cal is guaranteed to be independent of the fundamentally physical, then itis unclear why we should be so interested in this fact. Better to leave theintuitive conception of ethicality as it is and acknowledge that ethical factscan partially ground natural facts—especially since I will shortly offer a re-vised account that does justice to our intuitive conception while avoiding mycounterexamples.40

4 Vacuous Grounds

My examples all have a certain form—we have some fact, such as the doingof unethical things in a certain time period, which fact partially groundssome social fact such as the possession of a teddy bear. Well and good; theseexamples do show that we can partially ground the natural in the ethical.But they also suggest a way to capture Maguire’s underlying insight andhence a more accurate taxonomical distinction between the ethical and thenatural. The occurrence of each ethical fact in the grounds for a naturalfact is vacuous in the sense that any fact, ethical or not, could have donethe work the ethical fact does in partially grounding the natural fact. Notethat this is different than overdetermination of ground, like we have when adisjunction is totally grounded in each true disjunct. The disjunction case isnot vacuous and, conversely, my examples above are not cases of grounding

40This response was suggested to me by Gideon Rosen in personal correspondence.

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overdetermination. Let me explain.41

4.1 Vacuous Occurrence

Suppose we have a natural fact grounded in an ethical fact. If, roughly, theethical ground is such that we could have substituted, mutatis mutandis, anatural fact and we would, very roughly, still have a grounding relationship,then we can say that the ethicality of the grounds is vacuous in this groundingfact. So now consider our case of truth or dare. The particular game is setup such that satisfaction of an ethical proposition is part of the conditionsof winning the game. But, in another context, the same game (or, at least, astructurally identical game. See §4.1.2 below) where it was not satisfactionof an ethical proposition, but of a natural proposition, which determinedthe winner of the game. Suppose Steve had dared Beth to not do anythinghilarious in the next half hour, for example. The fact that Steve won is thengrounded in purely natural facts. So the ethical nature of Steve’s dare is notsignificant for the explanation of the fact that Steve won a game today—anyfailed dare would have done as well. So the ethicality of the grounds is vacu-ous in the explanation. Intuitively, this should block the connection betweenthe ethical status of the grounds and the ethical status of the grounded factsince the role being played by the ethical status of the grounds plays no sig-nificant part in explaining the grounded fact. We now turn to explicatingthis claim.

The notion of vacuity being used here is modeled on—though not identicalto!—Quine’s notion of vacuous occurrence (Quine 1936). Given a true sen-tence φ, a constituent F in φ occurs vacuously in φ just in case systematicallyreplacing F with a grammatical equivalent—replacing names with names,predicates with predicates, sentences with sentences and so on—always pro-duces a true sentence. So, for example, p occurs vacuously in p ∨ ¬p sincesubstituting q for p for any q produces a true sentence.42 Likewise, q occurs

41This way of approaching the problem was inspired jointly by conversations withCatharine Diehl and a paper of Charles Pigden’s (Pigden 1989). Pigden is concerned withlogical autonomy and his solution is problematic—see (Maguire 2015)—but an analogueof his method works beautifully in the grounding case, as I will show.

42Quine’s original criterion was designed to capture logical truths; he thus moved tosimultaneous variation of non-logical constituents to avoid certain problem cases. Weignore this wrinkle since our use of vacuity is not aimed at capturing the notion of logical

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vacuously in p∨ q if p is true since any substitution for q will preserve truth.Vacuously occurring terms do not, in an intuitive sense, contribute to thetruth of a sentence since anything could do in their place. Whether or notthe sentence is true is thus, in the specified sense, indifferent to the truth ormeaning of the vacuous constituent. Quine’s notion of vacuity captures thesense in which the truth of a sentence can be irrelevant to the truth of whatit entails. We want to do something similar. We want to define a notion ofvacuity where some feature of the grounds is inessential to the explanation ofwhy the grounded fact holds. In our test case of ethical autonomy, we wantour notion of vacuity to capture how the ethical status of a ground can playno role in contributing to the ethical status of the grounded fact.

When, very roughly, we obtain a grounding fact by (uniformly) substi-tuting arbitrary facts for facts in the grounds of some grounding fact or arbi-trary structure-appropriate non-logical constituents of facts for constituentsof facts, those components we can freely swap out are inessential to thegrounding explanation. Anything structurally equivalent would have donein their place. This does not mean that we could simply delete them, butit does mean that their particular nature is not relevant. So, in particular,when part of the ethical status of the grounds is fixed by one of these con-stituents, but where any constituent, ethical or natural, would have doneinstead, then the ethicality of the ground is inessential—and, plausibly, theethical status of the grounded fact is then indifferent to the ethical status ofthis portion of the grounds. We now turn to a slightly more formal accountof vacuity.

4.1.1 Vacuity and Vacuous Grounds Formalized—a first pass

Let us represent a grounding fact in the following form: p← q0, q1, . . . wherep and the qs may be atomic or complex. We are presuming, remember, afine-grained structured conception of facts and propositions where they haveconstituents. Let pD/E be the result of systematically replacing a non-logicalcomponent D in p with the non-logical component E. Thus, if p = 〈a, F 〉,then pF/G is 〈a,G〉. If p = 〈a, b,=〉, then p=/G is 〈a, b,=〉 since = is logical. IfD does not occur in p, then pD/E = p. Now, we will say a component C occursvacuously in a grounding fact γ = p ← q0, q1, . . . when p ← q

C/D0 , q

C/D1 , . . .

truths.

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(denoted γC/D) holds for all structure-appropriate D in any context in whichthe qC/Ds are true.43 Otherwise, we will say that C occurs non-vacuously.Note that we have not modified the grounded fact, merely the grounds. Wewill say that q occurs C-vacuously in a grounding fact γ when there is acomponent C occurring in q such that C is vacuous in q. And we will saythat a T -proposition q, for T a taxonomic category like the ethical occursvacuously in a grounding fact γ when

1. There is a component C occurring in q such that for some structure-appropriate D, qC/D is not a T-fact

2. q occurs C-vacuously in γ.44

For example, that Obama did something which was unethical is an ethi-cal fact, but that Obama did something which is not hilarious a naturalfact. The former can partially ground—in our fictional, explicitly moralisticAmerica—that Obama is not a legitimate president, but it does so vacu-ously since q occurs C-vacuously (for C the property of being ethical) in thisgrounding fact. Systematic substitution of the property of being hilarious forbeing ethical would result—in any context in which the substituted groundsare true—in Obama being an illegitimate president, but because he fails the“presidents must live hilarious lives” side-condition instead of the “presidentsmust live ethical lives” side-condition.45

Now, consider truth or dare. The fact that Steve won it today it isgrounded in at least the fact that Beth did something unethical, that shewas dared to avoid doing something unethical, the rules of truth or dare, andthe history of gameplay. Call this grounding fact W . Consider W ethical/funny.The truth or dare grounding fact, now modified, would still go through in acontext in which the modified ground-propositions are facts—it would justmean that Beth failed to avoid doing something unfunny. And clearly funnyis arbitrary. So, the property of being ethical occurs vacuously in W .46

43This means we cannot swap out a property for an object, a property of properties fora first-level property, etc. Complications having to do with the structure of propositionsare inessential to my point and are thus omitted for brevity.

44The generalization from a single components to multiple components is straightfor-ward, if tedious. I leave it to the reader to supply the details.

45Thanks to Tristram McPherson for valuable discussion of this definition.46The reader might worry about cases like 〈Murder, is Wrong〉 grounding my correctly

believing that murder is wrong. This is, of course, non-vacuous if we hold that we cannot

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Now that we have formalized the notion of vacuity, we need to addressa few complications arising from our definition. This will result in a moreprecise definition that we will summarize after discussing the complicationsthat come up.

4.1.2 Complications with Vacuity: The Wronging Game

Consider games which essentially include an ethical component. Consider

The Wronging Game Players have one day to do as manythings as they can which are wrong. The player who does thelargest number of things which are wrong wins. Ties are settledby coin flip. In the case in which no one does anything wrong,there is no winner, only losers.

If, as sounds titularly plausible, ethical properties play a central, not merelyside-conditional, role in the wronging game, then the fact that Bob won thewronging game today is plausibly an ethical fact.47

Centrality can and should be reflected in the structure of grounds. Con-sider a standard grounding fact:

〈a, square〉 ← 〈a, rectangular〉, 〈a, equilateral〉

This grounding fact is plausibly complete in the sense that if we grasp whatit is for a to be rectangular and what it is for a to be equilateral, nothingfurther needs to be said in order to grasp what it is for a to be square. Itamounts to a reduction of the property of being square to the properties ofbeing equilateral and rectangular. The grounding fact

〈φ, immoral〉 ← 〈φ, not utility maximizing〉

is also complete, but, in contrast to the above, it is not intuitively connectedto a reduction. Adding in that actions which don’t maximize utility are im-moral would also, in a sense, be redundant since it is this connection which

perform the replacement in opaque contexts. Opacity is difficult, but in general the worri-some cases factor: correctly believing A is believing A and A being true. The conjunctivefact is ethical, so there is no counterexample here. Thanks to Andreas Muller for valuablediscussion on this point.

47Note that this claim, though useful for my purposes here, is not forced by the use ofvacuous grounding in order to settle the taxonomical spread of ethicality.

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the grounding fact in some way articulates. It is central to what it is for anaction to be immoral. Such redundant grounds should not occur in ground-ing facts.

The claim that x won the wronging game is grounded in that x wrongedthe most during the game articulates what it is to win the wronging game.So the fact that Bob’s winning the wronging game involves Bob’s wrong-ing the most during the game will not occur among the grounds of the factthat he won. The substitution will thus fail to show that the ethicality ofBob’s actions are vacuous since Bob’s winning is not grounded in that hedid, say, the most funny things. This is the right result—if it is central tothe wronging game that it is about wronging, then winning the wronginggame is an ethical fact and one which our account marks as such. This isno counterexample to the grounding account of autonomy, properly spelledout. Some facts about winning are essentially ethicality-involving and suchfacts are plausibly ethical (see also the discussion of Dworkin below in §5.2).48

There are other potentially similar ethicality involving facts. Consider

〈Steve, won this game of Truth-or-Dare〉

Suppose we perform our substitution, substituting hilarity for ethicality inits grounds and look to a world where these grounds are true. In that world,perhaps, we have a different game of Truth-or-Dare or, at least, one with adifferent history. So ethicality is non-vacuous in the above grounding claim—there are contexts in which the grounds hold but they do not ground theabove fact.49 What should we say about this case? Is the above particular-ized claim ethical or not? The answer to this problem will depend on how

48Note that this does not mean that, say, Bob’s wronging actions have the additionalsubstantive ethical property of figuring in his winning the wronging game. Such ethicalfacts are clearly derivative, not basic. See §5.1 below for a division of basic from derivativevalue; similar things can be said about the division between basic and derivative ethicalproperties. But discussion of this would take us too far afield. Thanks to Derek Baker forpressing this worry.

49An alert reader might note at this point that it’s possible that we could particularizethe grounds as well and they might worry that this means that we might get the extensionof vacuity wrong. But this seems to cause no problems for the above definition—sincewe are looking at all permissible substitutions, there will be grammatically appropriatesubstitutions where the grounds hold but the grounded fact does not. For example, eventhough the fact that this is actually water is grounded in the fact that this is H2O and

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finely we individuate objects cross-modally. I’m inclined to think that thefirst game of truth-or-dare I played when I was 11 could have gone much bet-ter than it did, for example; I’m also inclined to think that if the participants,location, and sequence of questions had been entirely different, it would havebeen a different game. Where to carve the line between identity-destroyingand identity-preserving changes is a difficult one, but not one unique to myproject. And given any reasonable way of drawing this line, there will befacts which are so finely individuated that their ethical grounds are centralto them—such as the fact that Steve what that particular ethical-dare in-volving game of Truth-or-Dare—and facts individuated coarsely enough thattheir ethical grounds are not central to them—such as the fact that Stevewon a game of Truth-or-Dare at all.

Now, if we carve the identity conditions of social entities like games(among other things) so finely that we could not have had the same game,suitably modified, then my definition results in ethicality non-vacuously oc-curring in the above example. Likewise, of course, if we “actualize” the factby writing the historical facts and conditions into it, like so

〈Steve, won this actual game of Truth-or-Dare which went so and so〉

But this is how it should be: treating all these properties as “built-in” to thetoken game as referred to in the grounded fact more or less eliminates themfrom the explanation of why the fact holds.50 And, so built-in, we shouldtreat the grounded fact as an ethical fact—or, anyways, there is little cost indoing so. But, on the other hand, if we individuate more coarsely, as we oftenshould, then it seems clear that ethicality can be vacuous in the above-citedfact—as this game could have differed with respect to at least some of the

even though substituting XY Z for H2O in the grounds results in a proposition which doesnot hold anywhere as this is never XYZ, H2O is not vacuous in the grounding claim:

〈this, water〉 ← 〈this, H2O〉

Simply substitute the property of having non-negligible mean kinetic energy instead andwe get

〈this, water〉 ← 〈this, non-negligible mean kinetic energy〉

both components of which can be true, but which is not legitimate case of grounding.50It’s worth reminding the reader here that we are assuming that facts are true propo-

sitions.

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dares which were dared.51

4.1.3 Complications with Vacuity: Grounding* facts

There are also cases with intuitively vacuous grounds which aren’t capturedby the above test because of inessential differences in the components of thegrounds.52 So, for example, consider a variant of Explicitly MoralisticAmerican Politics. One way a “president”, say Mitt Romney, would notbe a legitimate president is for them to have lived an ethically suspect life.But the natural fact of his illegitimate occupation of office (ρ) might be seenas grounded in the ethical side-conditions of legitimate presidency (Γ) andthe particular peccadillo which falsified the impeccability of Romney (say,stealing from small businesses. Call this fact ι). So, we have the groundingfact µ:

ρ← Γ, ι

but presumably µethical/funny yields the grounding non-fact

ρ← Γethical/funny, ι

because the property of being ethical doesn’t appear in ι, even though itappears in Γ. And, thus, the ethical status of the grounds is non-vacuous,contra our intuition (remember that we have assumed that the ethicality side-condition is a side-condition and thus (a) we should not assume that being alegitimate president is an ethical fact and (b) the ethical side-condition ap-pears in the grounds.) We can solve this problem by running our replacement-style criterion of vacuity on slightly expanded grounding facts (call themgrounding* facts) which include not only the grounds of some fact, but alsothe connections between properties which are not necessarily part of thegrounds. Let me explain.

51Note that the type-directed fact 〈Steve, won Truth-or-Dare〉 might be grounded ina token-directed fact 〈Steve, won the game of Truth-or-Dare that went thus and so〉, justas the existential fact 〈Steve, won a game of Truth-or-Dare〉 might be grounded in thefact that he won the particular game that he did. But, in both cases, even if we admitthat the token-directed or particular facts are ethical, the ethicality does not transfer tothe type-directed or existential facts since the token-directed or particularized elementsof these facts are vacuous in the intermediate grounding relationship. See §5.3 below formore detail on a related case.

52Note that we are dealing with propositions or facts, not the statements expressingthem. Synonymous expressions in co-referring statements thus cause us no difficulties.

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The fact that stealing from small businesses is unethical may not bepart of the explanation of Romney’s illegitimacy; it may be sufficient forthe grounds to contain merely that he stole from small businesses. But wecan define a new notion—a grounding* fact—which contains all of theseconnections between properties. The relevant grounding* fact for Romney isµ∗

ρ←↩ Γ, ι, ι is unethical

not µ.53 Our notion of a replacement extends seamlessly to this new, de-fined notion. Now, writing en for the additional components of the ground-ing* fact, we will say a component C occurs vacuously in a grounding* factγ∗ = p←↩ q0, q1, . . . , e0, e1 . . . when p← q

C/D0 , q

C/D1 , . . . (denoted γC/D) holds

for all structure-appropriate D in any context in which the qC/Ds and theeC/Ds are true. We will often speak of the modified grounding* fact holding,but by this we always mean that the grounding fact holds when the groundsand the additional “grounds” of the grounding* fact are true.

Note that µ∗ethical/funny is

ρ←↩ Γethical/funny, ι, ι is unfunny

and ρ ← Γethical/funny, ι as long as Γethical/funny, ι, and ι is unfunny are alltrue. So, we can easily accommodate our problem by requiring our ground-ing* facts to include all such connections between properties and by thendefining vacuity in terms of grounding* facts. Note that grounding* factsare not necessarily grounding facts—they violate a plausible minimality con-dition on grounding if grounding relations do not already include all relevantproperty connections.54 They are, rather, a defined formal notion which cor-responds to articulating all the connections between properties occurring inthe relevant grounding facts they arise from. Intuitively, grounding* facts arepedantically full explanations, leaving unarticulated no connection betweenproperties occurring in the grounds.

53I use ←↩ instead of ← to emphasize the difference between grounding facts and theextended defined notion of a grounding* fact.

54As mentioned above, I am officially agnostic about how full grounding explanationsshould be taken to be. If they are all pedantically full, the distinction between groundingand grounding* facts is slim to non-existent and we do not need our innovation. If not,then we do.

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We can give a slightly more precise account of how to get from a ground-ing fact to a grounding* fact in terms of our above discussion of completeand incomplete grounds. A grounding* fact contains all those connectionsbetween grounds which are not redundant in the sense of being articulated bythe grounding relationship itself. In our example above, that actions whichdo not maximize utility are immoral is redundant in the grounding of some-thing being immoral. But, in contrast, it is not central to the grounds ofbeing a legitimate president that stealing from small businesses is unethical.It is instead something that needs to be grasped in order to grasp the expla-nation of why Romney is not a legitimate president. And, given this, it willoccur in the grounding* fact derived from µ.

Consider, as further illustration, a problem suggested to me by SelimBerker.55 Suppose we have a case where p and q ground r for p and r ethicaland q natural. That is,

re ← pe, qn

and where the component C of pe which renders pe ethical is vacuous inthe displayed grounding fact. Now, suppose pe is not ungrounded, but isgrounded in a further ethical fact se.

re ← pe, qn and pe ← se so re ← se, qn

by the transitivity of grounding. But there is no guarantee that se contains Cas a component and, therefore, no guarantee that the consequence is vacuous.This is worrisome, but the required solution is clear. Consider a plausiblecase of this. Suppose 〈φ, wrong〉 is itself fully grounded in 〈φ, harmful〉.Then, plausibly, by the argument just given, the grounds of Fred winninga bet might also be that Sarah bet him to do something wrong (call thecollection of bet facts Σ) and he did something harmful.

〈Fred, won a bet〉 ← Σ, 〈Fred, did something harmful〉

But this is not vacuous by our simple substitution method since wrong doesnot occur in both the grounds of the fact we obtain by transitivity.

55I have modified Berker’s example slightly for ease of exposition. In particular, Berkeruses a case where the natural grounds are further grounded. Nothing turns on this; thesolution is the same.

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However, if we look at the grounding* fact corresponding to the obtainedgrounding fact, we need to add into the grounds that Fred did somethingwrong by doing something harmful or something akin—the just displayedgrounding act is incomplete in the sense sketched above and grasping theexplanation fully involves grasping that Fred did something wrong by doingsomething harmful.

〈Fred, won a bet〉 ←↩ Σ, 〈Fred, did something harmful〉, 〈harmful things are wrong〉

But then running the substitution on harmfulness shows that harmfulnessvacuously occurs:

〈Fred, won a bet〉 ←↩ Σ, 〈Fred, did something hilarious〉, 〈hilarious things are wrong〉

and, in the context in which hilarious things are wrong Σ and that Fred didsomething hilarious grounds that he won the bet. Similarly with the prop-erty of wrongness. This means that both the ethicality of the fact that Freddid something harmful and the ethicality of does not transfer to the fact thatFred won the bet, which is the result we want.

A similar problem crops up in other examples suggested to me by SelimBerker. Consider an instance of Truth or Dare where Steve dares Beth torefrain from doing anything which can be described with an English predicatestarting with the letter ‘W’. Beth does something wrong (call it ψ) andthereby loses. Steve’s win is plausibly then grounded in the fact that Bethdid something wrong and the terms of the game (Σ).

〈Steve, won Truth-or-Dare〉 ← Σ, 〈ψ, wrong〉

But then, prima facie, wrongness turns out to be non-vacuous in theexplanation since replacing it with hilarity does not return a grounding fact.

〈Steve, won Truth-or-Dare〉8 Σ, 〈ψ, hilarious〉

But only prima facie since, plausibly, the explanation is incomplete. Thegrounding* fact will include that Beth’s act can be described with an Englishpredicate starting with a ‘W’—as this must be grasped by someone in orderfor them to grasp the explanation of why Steve won.

〈Steve, won Truth-or-Dare〉 ←↩ Σ, 〈ψ, wrong〉, 〈wrong things, W-describable〉

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and, once this is in place, our replacement also returns that hilarity can bedescribed with an English predicate starting with a ‘W’.

〈Steve, won Truth-or-Dare〉 ←↩ Σ, 〈ψ, hilarious〉, 〈hilarious things, W-describable〉

And, given this, our substitution returns a grounding* fact which goes through(in suitable contexts in which all the grounds are true. For example, perhapsin one where English contains the word “wuffible” which is a synonym forhilarious).56

Note that the extra components of the grounding* fact do not include alinking claim like ‘harmful things are wrong’ when this connection is articu-lated by the itself. Consider, for example, the claim that my being friendlyis grounded in my smiling. If this articulates the connection between smil-ing and being friendly, then the grounding* analogue does not contain thatsmiling is a way to be friendly—that would be redundant.57 In contrast, inthe case above, we really do need to grasp that being harmed is a way to bewronged in order to grasp why Fred won the bet; this property connectionlies outside the connection articulated in the grounding relationship betweenFred harming someone, the game facts, and Fred winning the bet. Vacuous,note, in this grounding relation, but not in the connection between harmful-ness and wronging. This means that applying the transitivity principle canresult in vacuous grounds, but this is as it should be since we have elided bitsof structure that does crucial work in the explanation when we elide over themiddle-grounding links. Vacuous grounding is, after all, still grounding.58

Vacuity just blocks certain transfers of properties from grounds to what theyground.59

56Thanks again to an anonymous referee for raising this worry.57There is a potential issue with this example about whether or not this really is a case

of grounding since it might be contingent that grounding is a way to be friendly. Smiling,though, plausibly is necessarily a way of being friendly. See below.

58Confusion on this score seems to me to arise from the thought that the transitivegrounding claim is somehow more important than the intermediate grounding claims.Making too much of this thought, however, is a mistake; when we consider only the transi-tive grounding claim we lose important bits of structure that witness certain taxonomicaltransitions. See the next footnote for an example.

59Note that this means that cases of vacuous components will be fairly pervasive sincewe will obtain them in many cases when we elide over intermediate property-constitutivegrounding links like that connecting harmfulness and wronging. This is as it should be

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4.1.4 Vacuity correctly formalized

So, taking the above discussion and complications on board, we have thefollowing definition of vacuousness.

• Given a grounding fact γ = p ← q0, q1, . . ., the corresponding ground-ing* fact is γ∗ = p←↩ q0, q1, . . . e0, e1, . . .

• A component C occurs vacuously in a grounding fact γ = p← q0, q1, . . .when, given the es occurring in γ∗ = p←↩ q0, q1, . . . e0, e1, . . .,

p← qC/D0 , q

C/D1 , . . .

holds for all structure-appropriate D in any context in which the qC/Dsand the eC/Ds are true.

• q occurs C-vacuously in a grounding fact γ when there is a componentC occurring in q such that C is vacuous in γ.

• A T -proposition q, for T a taxonomic category like the ethical, occursvacuously in a grounding fact γ when

1. There is a component C occurring in q such that for some structure-appropriate D, qC/D is not a T-fact

2. q occurs C-vacuously in γ.

4.2 Vacuity applied to Autonomy

Having articulated the distinction between vacuous and non-vacuous grounds,we can now offer a formulation of ethical autonomy that captures what isplausible in Maguire’s core proposal, while avoiding the difficulties it faces.:

since, plausibly, in these cases components of the deeper ground don’t neatly transfer tothe grounded fact unless they already transfer in the initial case. Take, for example, acase where wronging is non-vacuously grounded in harming and wronging, in turn, non-vacuously grounds some further fact a is γ. Then that a is γ is ethical, by the fact thatwronging is ethical and it non-vacuously grounds a is γ even though harmfulness is vacuousin the transitive grounding fact

a is γ ← a is harmful

.

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(MAV ): No ethical fact is fully and non-vacuously grounded justby natural facts.

(CMAV ): Any fact partly and non-vacuously grounded by eth-ical facts is ethical.

This does a much better job at satisfying the intuitive autonomy of the ethi-cal from the natural and conversely. Additionally, though we allow groundingof natural facts by ethical facts, such grounding relations all involve cases ofvacuous grounding, where the ethical grounding fact plays no essential rolein the explanation of the natural fact. This is the result we want.

Our patched up principles also allows us to articulate a bold claim aboutthe nature of certain metaethical positions like the Moorean one sketchedabove (§2.1). Positions which ground, but do not reduce, moral facts to non-moral facts are in an important sense positions on which there are no purelymoral facts. Since the grounds of 〈Murdering, Wrong〉 are entirely naturalon this view and since the grounding is not vacuous, 〈Murder, Wrong〉 isactually a natural fact.60 This strikes me as correct; such positions reallydo have the consequence that morality has more to do with the intuitivelynatural than we would have thought. Reduction of the ethical to the natural,in the sense of there being a real definition of an ethical property in entirelynatural terms, is one way for the ethical facts to really be natural facts.But the grounds being entirely natural is another distinct and importantsense. Consider, analogously, positions that hold that mental facts are non-vacuously grounded in physical facts, but not reducible to them. It is by nomeans bizarre to say that such positions hold that mental facts are reallyphysical facts, that they claim a certain ontological priority of the physicalover the mental, even if our conceptions of the mental will never be reducedto the physical.61

4.3 Worries about Other Autonomy Claims

However, there are other autonomy claims which run into trouble, even givenvacuous grounding. Consider a case from Putnam:

60N.B. This does not mean it is not also an ethical fact. See above.61See, for example, positions such as that articulated in (Nagel 2000) which can easily

be formulated in terms of grounding.

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. . . let T be an actual (physically instantiated) Turing Machineso programmed that if it is started scanning the input “111,” itnever halts. Suppose we start T scanning the input “111 ,” let Trun for two weeks, and turn T off. In the course of the two weekrun, T did not halt. Is it not the case that the explanation ofthe fact that T did not halt is simply the mathematical fact thata Turing machine with that program never halts on the input,together with the empirical fact that T instantiates that program(and continued to do so throughout the two weeks)? (Putnam1979)

Putnam’s case appears to be one where a mathematical fact partially groundsan empirical fact. Given this, many approachs to autonomy in terms ofgrounding which accepts that the mathematical facts are autonomous fromthe natural facts will have to address such examples. Since the groundingfact also appears to be non-vacuous in this context, barring vacuous suchcases will not suffice to avoid the issue. Should we accept that the fact thatT did not halt is a mathematical fact?

There are a number of ways the defender of mathematical autonomymight address this case. First and foremost, we would need to think throughthe claim that any fact partially and non-vacuously grounded in a mathemat-ical fact is a mathematical fact. To my ear, this is significantly less plausiblethan the claim that any fact partially and non-vacuously grounded in an eth-ical fact is an ethical fact. After all, mathematical facts are regularly used toexplain, at least at an abstract level, natural facts. Abstract mathematicaldescriptions of natural phenomena are, after all, the coin of contemporaryscience. So it is by no means obvious that an account of mathematical au-tonomy needs to follow the template of ethical autonomy. Second, we mightthink that the mathematical facts are just abstract descriptions of physicalfacts that play a role in a causal explanation of the original physical fact andthat there is no real grounding fact here.62

62The question of how best to interpret such examples or similar cases like the expla-nation of Cicada life-cycles in terms of primality (Baker 2005) is far too large for thisproject. I hope to return to it elsewhere. Thanks to Selim Berker for discussion of causalinterpretations of such cases.

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Third, we might argue that in the appropriate and complete representa-tion of this grounding fact there are no mathematical facts. Such an accountwould face the problem of representing the Turing-machine facts in a non-mathematical fashion, perhaps drawing on the sort of quasi-logical analysis ofTuring-machine programs in (Buchi 1990).63 Finally, there are more plausiblealternatives to a stark-raving platonistic ideology where the mathematical isa distinct realm only mysteriously related to the non-mathematical. Struc-turalist or nominalistic accounts of mathematics would have little troubleswallowing that some empirical facts are also mathematical facts.

At any rate, that the mathematical is not autonomous from the naturalin the same sense as the ethical is autonomous from the natural seems tobe a reasonable view—and one which is best articulated using the notionof vacuous grounding I have articulated above. Even more than in the caseof the autonomy of the ethical, cases where Beth loses truth or dare byincorrectly summing 11 and 2 are not even prima facie counterexamples toa claim of mathematical autonomy. And similarly with respect to otherformulable autonomy theses.

5 Further Applications of Vacuous Ground-

ing

So we have seen that we can use the notion of vacuity and vacuous groundsto formulate a more interesting and more plausible version of Maguire’s au-tonomy thesis. This allows us to put forth a clear articulation of the stronglynon-naturalist position currently gaining currency among metaethicists—andconsequently to test its plausibility. And as mentioned in the introduction,the autonomy of the ethical from the natural is just one example of an auton-omy thesis: analogous views could be mooted for the autonomy of the mentalfrom the natural, the mathematical from the physical, the causal closure ofthe physical, and so on. And, for any of these cases, it is important to screenoff the sort of counterexamples I developed above in order to give such viewstheir most attractive dress.

63Such an attempt would have to face head-on the problem of eliminating the quantifi-cation over natural numbers which occurs in the most natural analyses of Turing-machineprograms.

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But, importantly, the notion of vacuity can also be used in other appli-cations of grounding. I will discuss three examples here, though I suspectapplications are everywhere. We will start by looking at how to use ground-ing and vacuity to distinguish Ronald Dworkin’s position—that moral factsplay a constitutive role in the explanation of legal facts—from inclusive legalpositivism, the view that moral facts play a role, but a non-essential role, inthe explanation of legal facts. We will then turn to the distinction betweenbasic and non-basic types of value. Finally, I will close by showing how touse vacuity to formulate a novel solution to Kit Fine’s and Stephen Kramer’sparadox of ground (Fine 2010; Kramer 2013).

5.1 Dworkin and Positivism

We can also use the notion of vacuity and our discussion of autonomy thesesto distinguish three views of legality—inclusive legal positivism, exclusivelegal positivism, and antipositivism of the sort endorsed by Ronald Dworkin(Plunkett 2012). Consider Dworkin’s view that legal facts essentially involvemoral facts.64 For Dworkin, the ethical character of a law is central to theexplanation of why it is the law, even though there is a large social componentto the explanation as well.65 It is not a mere side-condition built into thesocial fabric of a legal system. In contrast, inclusive legal positivism holdsthat ethical facts can play a role in the explanation of the law, but they donot play an essential role in the following sense:

. . . these moral facts (1) are not necessarily among the determi-nants of legal content in all legal systems, and (2) are not amongthe ultimate determinants of legal facts in the following sense:the moral facts owe all of their relevance to the legal facts to theobtaining of other, nonmoral social facts.(Plunkett 2012, 148)

And exclusive positivists hold that it is never the case that moral facts canplay a role in the explanation of the law.

64See, for example, (Dworkin 1978; Dworkin 1986). I do not endorse Dworkin’s accountof the law—rather, I’m merely pointing out that we can characterize his position cleanlyusing vacuity.

65See (Dworkin 1978) for the details of Dworkin’s two-factor account of legal facts.

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Put aside the notion of exclusive legal positivism. Distinguishing betweenthe two roles that ethical facts can play in grounding legal facts is of centralimportance to formulating the distinction between a view like Dworkin’s anda view like inclusive legal positivism. Consider, for example, the distinctionbetween the use of ethical facts in applying the “shocks the conscience” test(as articulated in Rochin v. California) and the use of ethical facts in de-termining that a contract employs unconscionable conditions. In the former,even though it is not part of the descriptive understanding of ‘due process’that actions which ‘shock the conscience’ violate it, nevertheless the ethicalfact that something shocks the conscience—note that this is not a psycho-logical claim—plays a role in determining whether or not something violateddue process. In the latter, the ethical conditions are more or less part of thelaw itself. The fact that the contract is void is partially grounded in the factthat it is unjustly lopsided, but the articulation of contract law could havebeen such that this was not the case—although luckily it is not.

For the antipositivist like Dworkin, ethical facts can play both roles. But,importantly, there are also always non-vacuous ethical grounds of any legalfact, in much the manner that the “shocks the conscience” property is in-tended to partially explain the failure of due process. As sketched above,such grounds will not be part of the complete grounds* of the legal factsince they are, in part, articulated by the grounding relationship itself. So,if Dworkin is right, then the role of the privileged ethical grounds is non-vacuous—swapping out the fact that something tickled the fancy for shock-ing the conscience would destroy the grounding relation since the legalitysomething which tickles the fancy is not eo ipso illegal.

Contract law, on the other hand, could have been such that contractsthat employed entertaining conditions, rather than unconscionable condi-tions, were void. Such conditions will be part of the grounds of the validityof a contract and thus be vulnerable to our substitution test. Of course,this kind of contract law would be silly, but descriptively there is nothing tokeep it from obtaining.66 Now, in contrast to Dworkin, the inclusive legal

66Note that we are presuming that the fact that such and so is contract law obtainsand, if Dworkin is right, that such and so law would be more or less ethical or just. Thiscan be spelled out as in the discussion above, but such complications do not make troublefor my general point about two ways the explanation of legal facts might involve ethicalfacts.

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positivist thinks that ethical facts only play the latter role exemplified inthe contract law example. And such facts will clearly occur vacuously in theexplanation of the holding of the law.67

The distinction between these views is almost tailor-made for the notionof vacuous grounds. Exclusive positivism can be explicitly taken to be theview that moral facts are never among the grounds of the explanation of alaw (though, of course, what we take to be moral may find its way there), in-clusive positivism as the view that ethical facts can occur among the grounds,but only vacuously, and antipositivism of a Dworkonian stripe demands theexistence of non-vacuous ethical facts in the grounds of every law.68 With-out our distinction between vacuous and non-vacuous grounds, we cannotcleanly distinguish cases of conventional practices where an ethical fact isinvoked as a side-condition and those where the ethical fact is central to theexplanation of the legal fact. “Relevance”, in the above quote, is a place-holder for the more precise sort of account allowed by the notion of vacuousgrounding. But given the notion of vacuity, we can easily distinguish inclu-sive legal positivism from antipositivism; the ethical grounds of legal factswill sometimes be non-vacuous for Dworkin (and there will always be somenon-vacuous ethical grounds) and such grounds will always be vacuous forinclusive legal positivism.

5.2 Basic and Non-basic Values

We can apply the notion of vacuity to make sense of another case in whichgrounds plausibly transfer certain properties to the grounded fact. Supposethat there are some states of affairs that are basically valuable. Perhaps thatI feel pleasure or pain, that something helps my fellows, etc. But considerthe fact that I will feel pleasure while wearing black Converse. Plausibly,this describes a valuable state of affairs, but one which is not basically valu-

67Of course the inclusive legal positivist can explain the use of the “shocks the con-science” property as well; they will just claim that this is an unarticulated side-conditionon the notion of due progress. Though I don’t think it conclusive, there is the grain of anargument against inclusive legal positivism in this claimed collapse of two seeming distinctphenomena. At best, there’s an epicycle here to be finessed.

68Putting matters this way shows that there is a fourth unoccupied position here—theidea that moral facts can sometimes, but do not always, play a non-vacuous role in theexplanation of the law.

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able. Now, following (Maguire 2015), we can make sense of this distinctionin terms of grounding. A non-basically valuable state of affairs is one whichis partially grounded in a basically valuable state of affairs. This will keepus from double-counting valuable states of affairs when we figure out ourreasons for action—that I will feel pleasure while wearing black converse andthat I will feel pleasure don’t intuitively count as two distinct considerationsin favor of acting a certain way.

So far so good. But problems creep in in ways familiar from above. Takean instance of Truth-or-Dare where I am dared to produce no value in thenext minute and, unfortunately, a feeling of pleasure at my near successwashes over me 55 seconds in. The fact that I lost this instance of Truth-or-Dare is partially grounded in the fact that I felt pleasure, but the fact that Ilost this instance of Truth-or-Dare does not even intuitively describe a non-basically valuable state of affairs. So, again, we have a failure of grounding-transfer—here, value properties of basically valuable states of affairs do notintuitively transfer to what they ground if the grounding is vacuous. This isin contrast with the case of my feeling pleasure while wearing black conversewhich is, intuitively, a valuable state of affairs. But note here that the value ofmy feeling pleasure is not vacuous—substitute mild ennui for pleasure and thegrounding relation fails in the relevant context. So, if we want to maintain adistinction between basic and non-basically valuable states of affairs in termsof grounding—and there is good reason to do so—then we are going to wantto use non-vacuous grounding to get the correct transfer properties of valuefrom basically valuable states of affairs to those they ground.

5.3 Paradoxes of Grounding

The above examples have been aimed at properties of grounds (ethicality,morality, value) which are external properties of the grounds or of the stateof affairs the grounds describe. And I have argued that we can get right howthese properties transfer from grounds to what they ground by invoking thenotion of vacuity. The final use of vacuity I will discuss is more direct—itinvolves the transfer of explanatory force itself.

Grounding itself, especially total grounding, has often been thought to beirreflexive. That is, p8 p for all p. However, as has been pointed out by KitFine and Stephen Kramer, we can get violations of the irreflexive property

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by invoking very minimal principles for grounding. Focusing on Kramer’sversion, in the context of the existential grounding principle (EG) that anexistentially quantified claim is to be (totally) grounded in each of its in-stances and propositional quantification, we can infer from the theorem ∃p pto ∃p p ← ∃p p by EG (∃p p being the existential generalization of ∃p p).But this is a straightforward violation of the irreflexive property.

But if we think that the reason to ban reflexive grounding relations isthat nothing explains itself, then we can use vacuity to give a more nuancedaccount of how grounding is irreflexive. This is because the most plausibleway to understand why something doesn’t explain itself is that the contentof a fact can’t explain the content of that fact. But note that any true propo-sition would do to explain that there are some and ∃p p is certainly a trueproposition. The key thing here is that there is nothing about the contentof ∃p p that is involved in the explanation; what is involved is simply that∃p p is itself a true proposition. Any other true proposition would do in itsplace. Note that this is not the claim that the fact that ∃p p is true grounds∃p p. Rather, it is merely the claim that any true proposition would havegrounded ∃p p—the content of the true claim doesn’t matter.

But we can capture this fact with our notion of vacuous grounding. Ifany proposition would have done as well as p to ground some claim q, thenp is vacuous in the grounding of q. We can then replace the naive irreflexiveproperty

[I] p8 . . . p . . .

with the sophisticated irreflexive property :

[SI] p8 . . . p . . . unless p occurs vacuously in p← . . . p . . .

and capture what the ban on irreflexive property really should be, therebyavoiding Fine and Kramer’s puzzle entirely.69 We can retain the extremelyplausible claim that existentially quantified claims are (totally) groundedin each of their instances—only now realizing that some such grounds arevacuous.

69I discuss this case more thoroughly in (Woods ms). In particular, I discuss therevarious objections to this style of approach and show how analogous cases of violations ofthe reflexive property can be treated similarly.

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6 Conclusion

I have introduced and argued for the usefulness of distinguishing vacuousfrom non-vacuous grounding. In the first half of the paper, I worked throughthe case study of an autonomy thesis—the autonomy of the ethical from thenatural. I argued there that in order for the claim that the ethical is au-tonomous from the natural to make sense, we need a distinction between theclearly ethical and the clearly natural. I then argued that on the intuitivecharacterization of this distinction—in terms of what the facts are about—the ethical can partially ground the natural, falsifying Maguire’s autonomyclaim CMA. I then showed how to revise CMA and MA, using the notion ofvacuous grounding, so as to avoid my examples. My revised version capturesthe spirit of Maguire’s proposal while avoiding the result that all social factsand maybe even some physical facts are potentially ethical facts, preservingour intuitive conception of the natural.

I then made precise the notion of vacuity. In particular, I defined a notionof vacuous occurrence over constituents of grounding facts, then extendedthis discussion to show how to characterize the notion of the taxonomicalstatus of a fact being vacuous in a grounding fact. Noting a number of com-plications in our definition, I defined a new notion—grounding* facts—andshowed how this new machinery avoided the complications with vacuity andthereby allowed us to better capture the intuitive idea of vacuity and vacuousgrounding. I closed this section by raising some worries about autonomy the-ses in areas other than ethics, such as mathematics. I concluded that whilethese worries were interesting, they did not undermine the use of vacuousgrounding in articulating a plausible and defensible version of the particu-lar claim that the ethical status of a fact varied with its non-vacuous grounds.

I went on to show that the distinction vacuous and non-vacuous groundingcan serve us well in other areas, distinguishing between cases like Dworkin’s—where ethical facts play a deep explanatory role—and cases like bets overethical facts, where they do not. Next, I showed how to use it to give an aptcharacterization of the distinction between basic and non-basically valuablestates of affairs and how to use it to develop a distinctive solution to KitFine’s and Stephen Kramer’s paradoxes of ground. The manifest usefulnessof this notion demonstrates, I believe, that rather than being an ad hoc posit,needed only to adequately articulate the idea of the autonomy of ethics, the

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distinction between vacuous and non-vacuous grounding is an important toolfor metaphysics.

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