28 June 2015 T. Scott Dixon Departments of Philosophy Ashoka University; University of California, Davis

Grounding and Supplementation (Final draft. Forthcoming in Erkenntnis.) ABSTRACT: Partial grounding is often thought to be formally analogous to proper parthood in certain ways. Both relations are typically understood to be asymmetric (and hence irreflexive) and transitive, and as such, are thought to be strict partial orders. But how far does this analogy extend? Proper parthood is often said to obey the weak supplementation principle. There is reason to wonder whether partial grounding, or, more precisely, proper partial grounding, obeys a ground-theoretic version of this principle. In what follows, I argue that it does not. The cases that cause problems for the supplementation principle for grounding also serve as counterexamples to another principle, minimality, defended by Paul Audi.

1. Introduction Suppose that we know what it is for a given fact x to be fully grounded by some facts.1 Roughly, it’s for these facts to be metaphysically prior to x and to provide a complete non-causal explanation of x. It is typically understood to require, among other things, that the facts metaphysically necessitate x (although metaphysical necessitation is not sufficient for grounding).2 I express full grounding claims as instances of the following schema. x is fully grounded by Γ, where ‘x’ is a singular variable and ‘Γ’ is a plural one.3 For example, the fact that grass is green and snow is white is fully grounded by the fact that grass is green together with the fact that snow is white. It is standard to define the notion of partial grounding in terms of full grounding as follows. 1

While I take the fundamental notion of grounding to be expressed by a predicate, and not an operator, what is said below will be of interest to those who work in an operator framework as well. See Correia 2010: 253–54 for a discussion of the difference between operationalist and predicationalist conceptions of grounding. It will also be of interest to those who work in a predicationalist framework but think that grounding may hold between things other than facts. See, for example, Schaffer 2009: 375–76. 2 See, for example, Witmer et al. 2005: 332, Trogdon 2009: 128 and 2013a, deRosset 2010: 91, Rosen 2010: 118, Bennett 2011: 36, Correia 2011: 3, Audi 2012b: 697, Fine 2012b: 1, Raven 2012: 690–91, and Bliss 2014: 147. It should be noted here that this claim has been challenged. See, for example, Schaffer 2010a and Leuenberger 2014. 3 I allow a single object to be assigned to any plural term, and assume that, for any Γ, there is an x such that x is among Γ. That is, I assume that there are no empty pluralities.

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Partial Grounding. x is partially grounded by Γ =df for some ∆, (i) x is fully grounded by ∆ and (ii) Γ are among ∆.4 If x is fully grounded by Γ, then x is partially grounded by Γ, since, for all Γ, Γ are among Γ. So the fact that grass is green and snow is white is partially (as well as fully) grounded by the fact that grass is green together with the fact that snow is white. But the converse does not hold. In other words, it’s not true, in general, that if x is partially grounded by Γ, then x is fully grounded by Γ. This is because sometimes a fact is partially grounded by some facts without being fully grounded by them. I call this type of grounding ‘proper partial grounding’, and define it as follows. Proper Partial Grounding. x is properly partially grounded by Γ =df (i) x is partially grounded by Γ and (ii) x is not fully grounded by Γ. In contrast to full grounding, if x is properly partially grounded by some facts Γ, then Γ provide a partial, but not full, non-causal explanation of x. For example, the fact that grass is green and snow is white is properly partially grounded by the fact that grass is green.5,6 Partial grounding is often thought to be formally analogous to proper parthood in certain ways. For example, both relations are typically understood to be asymmetric (and hence irreflexive) and transitive, and as such, are thought to be strict partial orders.7 The same is true of proper partial grounding. It is also asymmetric and transitive, and as such is a strict partial order. Proper partial grounding, in fact, is an even closer analog of proper parthood, in that a proper partial ground is guaranteed not to be a full ground, just as a proper part is guaranteed not to be an improper part. But how far does the analogy between proper parthood and proper partial grounding extend? Proper parthood is often said to obey the weak supplementation principle: 4

See Rosen 2010: 115, Audi 2012b: 698, Fine 2012a: 50, and Raven 2013: 194 for endorsements of this definition. See Audi 2012b: 698 and Fine 2012a: 50 and 2012b: 3–5 for more discussion on the difference between full and partial grounding. I have Fine’s notion of a strict ground in mind, rather than weak ground. See Fine 2012a: 51–53 and 2012b: 3–7. 6 In what follows, I allow singular as well as plural terms to fill the second argument places of each of the grounding predicates. Similarly, I allow singular as well as plural terms to flank the ‘is/are among’ predicate. I also allow the construction of complex plural terms from simpler singular or plural terms via lists. So, for example, ‘x, y’, ‘x, Γ’, and ‘Γ, ∆’ are all plural terms. 7 For explicit reference to the similarity of proper parthood and grounding in this respect, see Trogdon 2013b: 106. For characterizations of partial grounding as asymmetric, irreflexive, and transitive, or as a strict partial order, see Cameron 2008: 3, Rosen 2010: 115–16, Schaffer 2010b: 37, and Raven 2012: 689 and 2013. Correia (2010: 262 and 2011: 3–4), Fine (2010: 100), and Schnieder (2011: 451) provide inference schemas that characterize the behavior of grounding in an operator framework in an analogous way. 5

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(WSP) For any x and y, if x is a proper part of y, then there is some z such that z is a part of y and z and x do not overlap (share a part).8 According to (WSP), if a thing has a proper part, then it must have another part — one that is ‘separate’ from the first in a certain sense. It is natural to wonder whether proper partial grounding obeys an analogous supplementation principle. Informally, such a principle would say that if a certain fact x is properly partially grounded by a fact y, then x is also partially grounded by some fact z that is in an analogous sense ‘separate’ from fact y. In what follows, I argue that proper partial grounding does not obey such a principle. The cases that cause problems for the ground-theoretic analog of (WSP) also serve as counterexamples to another principle, minimality, defended by Audi (2012b). 2. A Supplementation Principle for Grounding In this section, I regiment the ground-theoretic supplementation principle that was informally introduced at the end of the last section. The basic strategy is to replace every occurrence of ‘is a proper part of’ in (WSP) with ‘properly partially grounds’, and every occurrence of ‘is a part of’ with ‘partially grounds’. But since the predicate ‘overlap’, which is defined in terms of parthood, also appears in (WSP), it will be helpful to pin down the ground-theoretic analog of this notion first. For x and y to overlap, in the mereological sense, is for x and y to share a part. Call the most natural ground-theoretic analog of overlap, whatever that turns out to be, groverlap. One might initially think that for x and y to groverlap is simply for x and y to share a partial ground. But this can’t be the end of it. Since parthood is reflexive, overlap is too, and so everything overlaps itself. Groverlap should also be reflexive. Since grounding is not reflexive, the definition of groverlap must be disjunctive; a clause should be added that ensures that x and y groverlap when they are identical. It would now seem that for x and y to groverlap is for them either to be identical or to share a partial ground. But this doesn’t secure the analogy either. The reflexivity of parthood also guarantees that, if x 6= y and x is a part of y, then x and y overlap. x is a part of itself, and also of y, and so they share a part. Since grounding is not reflexive, two other clauses should be added 8

For endorsements of this principle, see Simons 1987: 28 and 116, Olson 2006: 743, Uzquiano 2006: 431, Sider 2007: 69–70, Effingham and Robson 2007: 635, Varzi 2008: 110–11 and 2009: 599, Bohn 2009: fn. 3, McDaniel 2009: fn. 48, and Bynoe 2010: fn. 8. van Inwagen (1990: 39) and Lewis (1991: 74) actually endorse Uniqueness of Composition, but (WSP) follows from this principle along with the standard definition of composition.

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to the definition that ensure that x and y groverlap when one partially grounds the other. Thus groverlap is properly defined as follows. Groverlap. x and y groverlap =df (i) x = y, (ii) x is partially grounded by y, (iii) y is partially grounded by x, or (iv) there is a z such that x is partially grounded by z and y is partially grounded by z. It is now clear how to formulate the ground-theoretic analog of the phrase ‘z and x do not overlap’ that appears in (WSP). The ground-theoretic analog of (WSP), then, can be stated as follows. (WSG) For any x and y, if x is properly partially grounded by y, then there is a z such that x is partially grounded by z and y and z do not groverlap. In the next two sections, it will emerge that (WSG) is inadequate. In those two sections, I offer two counterexamples to (WSG). In each section, my strategy is to offer a case, and to argue that it is a counterexample to (WSG) by appealing to some plausible and commonly accepted principles governing grounding. It is worth considering both counterexamples, since they rely on different principles. As such, the reader may find one more convincing than the other. The merits and drawbacks of each of the counterexamples will be discussed in due course. 3. First Counterexample to (WSG) In this section, I develop the first counterexample to (WSG). I’ll start with the required principles, and then give you the case. The first principle is a schema commonly taken to govern the relationship between grounding and determinate and determinable properties. Let p[ϕ]q abbreviate pthe fact that ϕq, and let pΛx F (x)q be a singular term that abbreviates pthe property (of) being F q.9 (DG) If Λx F (x) is a determinate of Λx G(x), then, for any x, if F (x) then [G(x)] is fully grounded by [F (x)].10 9

I follow Fine 2012a: 67–68 in my use of the term-forming operator ‘Λ’. See Rosen 2010: 126, Audi 2012b: 686 and 689, and Schaffer 2012: 126–27 for endorsements of this principle. Schnieder (2006: 32–33) may endorse a version of this principle, formulated in terms of metaphysical explanation. It should be noted that Jessica Wilson (2012) discusses reasons to think this principle is false. I will not discuss the details of her argument here. Instead, I direct the reader who is sympathetic to Wilson’s criticism to the next section, wherein I develop a counterexample to (WSG) that does not rely on (DG). 10

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Less formally, according to (DG), any fact that ascribes a determinate of some determinable property to an entity fully grounds the fact that the entity has that determinable property. So, for example, [grass is colored] is fully grounded by [grass is green]. The next two principles of which I make use govern the interaction between grounding and conjunction. The second principle is commonly taken to provide a sufficient condition for being a ground of a conjunctive fact. (&I) If ϕ and ψ (and [ϕ] 6= [ϕ & ψ] and [ψ] 6= [ϕ & ψ]), then [ϕ & ψ] is fully grounded by [ϕ], [ψ].11 Less formally, (&I) says that a binary conjunctive fact is fully grounded by its two conjunct facts (as long as neither is identical to the conjunctive fact).12 So [grass is green & snow is white] is fully grounded by [grass is green], [snow is white]. The third principle provides a necessary condition for being a ground for a conjunctive fact. Fine gestures toward the principle I have in mind in the following passage. [C]onsider. . . the question of when a set of truths ∆ is a (strict full) ground for A & B. We naturally want to say that any grounds for A & B should be mediated through A and B; the conjuncts are the conduit, so to speak, through which truth to the conjunction should flow. (Fine 2012a: 63)

One way to regiment this idea in a predicate framework is as follows. (&N) For any x, if [ϕ & ψ] is partially grounded by x, then either x = [ϕ], or x = [ψ], or [ϕ] is partially grounded by x, or [ψ] is partially grounded by x. Less formally, (&N) says that any partial ground of a binary conjunctive fact is either identical to one of its conjunct facts or partially grounds one of its conjunct facts.13 So if something partially 11

Adapted from Correia 2010: 268 and 2011: 5, Rosen 2010: 117, Schnieder 2011: 449, and Fine 2012a: 58. The parenthetical restriction is necessary, since, if, for example, [p] = [p & q], then (&I) would yield the result that [p & q] is partially grounded by [p & q], or, equivalently, [p] is partially grounded by [p]. This would violate the irreflexivity of grounding. See Correia 2010: 268 for an endorsement of an operationalist version of this restriction. 13 Given a certain assumption, this principle may best be understood to apply only when ϕ and ψ are atomic sentences. Suppose that p, q, r, and s, and that [p], [q], [r], and [s] are pairwise distinct. Given this, it follows that [p & q], [r & s], and [(p & q) & (r & s)] are also pairwise distinct. By (&I), [(p & q) & (r & s)] is fully grounded by [p & q], [r & s], and so is partially grounded by each of them. And, while this is not guaranteed by (&I), if one thinks that [(p & q) & (r & s)] is also partially grounded by [q & r], a potential problem arises for (&N). This is because [q & r] cannot be identical to, nor partially ground, either [p & q] or [r & s]. Given that [p], [q], [r], and [s] are pairwise distinct, [q & r] is not identical to either [p & q] or [r & s]. And it is usually thought that, for x to be partially grounded by y, y must be explanatorily relevant to x. (See, for example, Correia 2010: 263, Schnieder 2011: 450, Audi 2012b: 693 and 699, Fine 2012a: 56 and 2012b: 2, Raven 2013: 198, and Dasgupta 2014: 4.) So, as long as [r] is not explanatorily relevant to [p & q], and [q] is not explanatorily relevant to [r & s], [q & r] does not partially ground either [p & q] or [r & s]. Thus, given the assumption that [p & q], [r & s] is partially grounded by [q & r], there is a violation of (&N). Thanks to Daniel Nolan for this example. 12

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grounds [grass is green & snow is white], it must either be identical to one of [grass is green] and [snow is white], or it must partially ground at least one of those facts. The fourth principle of which I make use is a transitivity principle commonly taken to govern partial grounding. (PT) For any x, y, and z, if x is partially grounded by y and y is partially grounded by z, then x is partially grounded by z.14 It is worth pointing out that Schaffer (2012) has argued against (PT). I will ignore such arguments for two reasons. First, Schaffer is, as far as I know, the only philosopher who rejects (PT),15 and there are a number of philosophers who dispute the effectiveness of Schaffer’s putative counterexamples.16 Second, though Schaffer rejects (PT) itself, he endorses a contrastive analog of it framed in terms of a four-place grounding relation. If grounding is governed by this contrastive, four-place version of transitivity, it is natural to wonder whether that relation is also governed by a contrastive, four-place analog of (WSG). I suspect that the cases to follow could act as counterexamples to a four-place analog of (WSG) as well, though I will not try to argue for that here.17 And now for the first counterexample. Suppose that [grass is green] is fundamental, i.e., suppose that it is not partially grounded by any facts. (I realize this is not a particularly good candidate for a fundamental fact. But fortunately, the case does not require the postulation of any fundamental facts. I make this assumption for the time being simply for ease of explication.) Given that being green is a determinate of being colored, it follows by (DG) that [grass is colored] is fully grounded by [grass is green]. Diagramatically, 14

This principle, or an operationalist version of it, is endorsed or follows from principles endorsed in Correia 2010: 262 and 2011: 3, Fine 2010: 100, 2012a: 56, and 2012b: 5, Rosen 2010: 116, Schnieder 2011: 451, and Raven 2013: 198. 15 Tahko (2013) provides reason to think that truth-grounding is not transitive, but provides a possible way out of saying that grounding is not transitive, by denying that truth-grounding is grounding. 16 See, for example, Litland 2013 and Raven 2013: 198–200. While Javier-Castellanos (2014) does not argue directly against the effectiveness of Schaffer’s counterexamples, he argues that Schaffer’s alternative — his contrastive analog of transitivity — is susceptible to counterexamples as well. 17 Gilmore (2009) argues that parthood is a four-place relation, and formulates a four-place analog of (WSP). Kleinschmidt (2011: 268–271) independently formulates such a principle. Either of these would be a good starting point for formulating a four-place analog of (WSG).

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[grass is colored]

[grass is green] Fig. 1

(In this diagram and those to follow, facts are represented by nodes and full grounding claims are represented by solid lines connecting nodes.) Now consider [grass is green & grass is colored]. By (&I), it follows that [grass is green & grass is colored] is fully grounded by [grass is green], [grass is colored], and so, by the definition of partial grounding, is partially grounded by each of them. Diagramatically, [grass is green & grass is colored]

[grass is colored]

[grass is green] Fig. 2

(In this diagram and those to follow, I indicate that x is fully grounded by Γ by connecting x to a solid box that encloses the facts among Γ. Given the definition of partial grounding, each fact in such a box partially grounds the fact to which the box is connected.) So [grass is green & grass is colored] is partially grounded by [grass is colored]. Moreover, I take it as obvious that [grass is green & grass is colored] is not fully grounded by [grass is colored].18 As a result, [grass is green & grass is colored] is properly partially grounded by [grass is colored]. Diagramatically, 18

It’s worth noting that one could get this result in another way. In particular, one could conclude that [grass is green & grass is colored] is not fully grounded by [grass is colored] on the basis of the fact that the proposition that grass is colored does not metaphysically necessitate the proposition that grass is green & grass is colored. As noted in the introduction, many are of the mind that full grounding implies metaphysical necessitation.

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[grass is green & grass is colored]

[grass is colored]

[grass is green] Fig. 3

(In this diagram and those to follow, proper partial grounding claims are represented by dotted lines.) Now suppose for reductio that (WSG) is true. Then, because [grass is green & grass is colored] is properly partially grounded by [grass is colored], (WSG) guarantees that there is a fact z that partially grounds [grass is green & grass is colored] and does not groverlap [grass is colored]. But since z must ground a conjunctive fact, (&N) guarantees that there are only four possibilities: (a) z = [grass is colored], (b) [grass is colored] is partially grounded by z, (c) z = [grass is green], or (d) [grass is green] is partially grounded by z.19 In each of these cases, z and [grass is colored] do groverlap. (a) Obviously, z cannot be [grass is colored], as clause (i) of the definition of groverlap would be satisfied. (b) Nor can [grass is colored] be partially grounded by z, as clause (ii) would be satisfied. (c) Nor can z be [grass is green]. Given that [grass is colored] is fully, and so partially, grounded by [grass is green], clause (ii) would be satisfied in this case as well. (d) Nor finally is [grass is green] partially grounded by z. Since [grass is colored] is fully, and so partially, grounded by [grass is green], [grass is colored] will be partially grounded by z by (PT). Thus clause (ii) would also be satisfied in this case. So in every case, z and [grass is colored] groverlap, contradicting the reductio assumption. Given (DG), (&I), (&N), (PT), and the definitions of partial grounding and proper partial grounding, then, (WSG) is false. 4. Second Counterexample to (WSG) In this section, I develop the second counterexample to (WSG).20 In order to do so, I rely on 19 Regarding possibility (d): while I have supposed that [grass is green] is fundamental, I have done so only for the simplicity it afforded in the presentation of the case. I recognize that being green is probably not fundamental. Furthermore, I want the case to be conceivably regarded as a counterexample to (WSG) even by those who deny that there are any fundamental facts. Thus I take this possibility just as seriously as the other three. 20 Thanks to Cody Gilmore and an anonymous referee for bringing this counterexample to my attention.

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three of the principles introduced in the last section, namely, (&I), (&N), and (PT). But I require two other principles as well. The first is a principle commonly taken to govern the relationship between facts and their doubly negated correlates. (∼∼I) If [∼∼ ϕ] exists (and [ϕ] 6= [∼∼ ϕ]), then [∼∼ ϕ] is fully grounded by [ϕ].21 The second principle I require, like (&N), provides a necessary condition for conjunctive facts. But while (&N) provides a necessary condition for being a partial ground of a conjunctive fact, the principle to follow provides a necessary condition for being a full ground of a conjunctive fact. (&A) For any x, if [ϕ & ψ] is fully grounded by x, then either (1) x = [ϕ] and x = [ψ], (2) x = [ϕ] and [ψ] is fully grounded by x, (3) x = [ψ] and [ϕ] is fully grounded by x, or (4) [ϕ] is fully grounded by x and [ψ] is fully grounded by x. Less formally, (&A) says that any single fact that fully grounds a conjunctive fact must ‘fully account for’ each of the conjunct facts, either by being identical to each of those facts, by fully grounding each of them, or by being identical to one and fully grounding the other. And now for the second counterexample. Suppose that p and that [p] is fundamental. (As before, the case does not require the postulation of any fundamental facts. I make this assumption simply for ease of explication.) By (∼∼I), it follows that [∼∼ p] is fully grounded by [p]. Diagramatically, [∼∼ p]

[p] Fig. 4

Now consider [p & ∼∼ p]. By (&I), it follows that [p & ∼∼ p] is fully grounded by [p], [∼∼ p], and so, by the definition of partial grounding, is partially grounded by each of them. Diagramatically, 21

See Correia 2010: 268, Schnieder 2011: 449, and Fine 2012a: 63 for endorsements of this principle or of an operationalist version of it.

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[p & ∼∼ p]

[∼∼ p]

[p] Fig. 5

So [p & ∼∼ p] is partially grounded by [∼∼ p]. Now it is less obvious that [p & ∼∼ p] is not fully grounded by [∼∼ p] than it is that [grass is green & grass is colored] is not fully grounded by [grass is colored].22 If one finds it obvious, all the better. But (&A) provides extra reason to think that [p & ∼∼ p] is not fully grounded by [∼∼ p]. According to this principle, any single full ground of a conjunctive fact must either be identical to each conjunct fact, fully ground each conjunct fact, or be identical to one and fully ground the other. Since [∼∼ p] is identical to one of the conjunct facts of [p & ∼∼ p], it must, according to (&A), either be identical to or fully ground the other conjunct fact, [p]. Since, however, [p] fully, and so partially, grounds [∼∼ p], these possibilities would violate the irreflexivity and asymmetry of partial grounding, respectively. This means that [p & ∼∼ p] is not fully grounded by [∼∼ p]. So, given the definition of proper partial grounding, [p & ∼∼ p] is properly partially grounded by [∼∼ p].23 Diagramatically, 22

Note that the claim that grounding implies metaphysical necessitation is of no help here either, since ∼∼ p metaphysically necessitates p. (See fn. 18.) 23 It is hard to hide the fact that this argument relies on the claim that partial grounding is irreflexive and asymmetric. And there are certainly some who would deny this. Jenkins (2011), for example, argues that grounding is not irreflexive. I will sidestep this issue nonetheless. After all, anyone who denies that (proper) partial grounding is irreflexive and asymmetric will presumably have no interest in the argument against (WSG) anyway, since she would not see an interesting analogy between proper parthood and (proper) partial grounding in the first place. After all, on their view, only the former is irreflexive and asymmetric, and so only the former would be a strict partial order.

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[p & ∼∼ p]

[∼∼ p]

[p] Fig. 6

Now suppose for reductio that (WSG) is true. Then, because [p & ∼∼ p] is properly partially grounded by [∼∼ p], (WSG) guarantees that there is a fact z that partially grounds [p & ∼∼ p] and does not groverlap [∼∼ p]. But since z must ground a conjunctive fact, (&N) guarantees that there are only four possibilities: (a) z = [∼∼ p], (b) [∼∼ p] is partially grounded by z, (c) z = [p], or (d) [p] is partially grounded by z.24 In each of these cases, z and [∼∼ p] do groverlap. (a) Obviously, z cannot be [∼∼ p], as clause (i) of the definition of groverlap would be satisfied. (b) Nor can [∼∼ p] be partially grounded by z, as clause (ii) would be satisfied. (c) Nor can z be [p]. Given that [∼∼ p] is fully, and so partially, grounded by [p], clause (ii) would be satisfied in this case as well. (d) Nor finally is [p] partially grounded by z. Since [∼∼ p] is fully, and so partially, grounded by [p], [∼∼ p] will be partially grounded by z by (PT). Thus clause (ii) would also be satisfied in this case. So in every case, z and [∼∼ p] groverlap, contradicting the reductio assumption. Given (∼∼I), (&I), (&A), (&N), (PT), asymmetry, and the definitions of partial and proper partial grounding, then, (WSG) is false. This last counterexample has an advantage over the determinate-determinable counterexample in that it relies only on principles governing the interaction of grounding and various logical operations. Further, it may be more appealing to someone, like Jessica Wilson (2012), who denies that, in general, ‘determinates ground determinables’. The determinate-determinables counterexample, however, may be of more interest to those with a coarse-grained conception of facts. Those of such a mind may be unwilling to accept the distinctness of [p] and [∼∼ p]. They would have a more difficult time, however, denying the distinctness of [grass is green] and [grass is colored]. This point will be important in what follows, in my argument against Audi’s principle minimality, since he 24

As before, taking possibility (d) seriously ensures that the assumption made earlier, that [p] is fundamental, is innocuous.

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has a rather coarse-grained ‘worldly’ conception of facts. 5. The Argument against Minimality It turns out that the cases depicted in Figs. 3 and 6 also act as counterexamples to a principle defended by Paul Audi, which he calls ‘minimality’. Minimality. For any x and Γ, if x is fully grounded Γ, then there are no ∆ such that (i) x is fully grounded by ∆ and (ii) Γ are properly among ∆, where Proper Inclusion. Γ are properly among ∆ =df (i) Γ are among ∆ and (ii) for some x, x is among ∆ and x is not among Γ. Audi defends minimality in the following passage. Should we hold the. . . principle that adding an arbitrary element to a full ground never yields another full ground? I believe we should. Intuitively, for something to be contained in a ground of some fact, it must actually do some work with respect to making that fact obtain. That is why, for example, the fact that the shirt is both maroon and cotton does not ground the fact that it is red; the fact that it is cotton does no work with respect to making it red. So it appears that a [full] ground of some fact must be minimal in the sense of containing only elements that jointly suffice to bring it about that the fact in question obtains[.] (Audi 2012b: 699, italics in the original)

In his argument for minimality, Audi only explicitly considers a case in which a fact y which is irrelevant to x, and so does not partially ground x, is added to the full grounds Γ of x. He does not consider one in which some y that does partially ground x is added to Γ. As I will show, the cases developed in the previous two sections include just such facts. So as not to belabor the point, and because Audi himself explictly endorses (DG), but not (∼∼I), I will frame the discussion in terms of the determinate-determinable counterexample only. Indeed, given Audi’s coarse-grained conception of facts alluded to at the end of the last section (see Audi 2012a: 103 and 110, and 2012b: 691 and 696–97), it seems unlikely he would accept the distinctness of [p] and [∼∼ p].25 To see that the case depicted in Fig. 3 is a counterexample to minimality, consider the following. It was established in section 3 that [grass is green & grass is colored] is fully grounded by [grass is green], [grass is colored]. And, while [grass is green & grass is colored] is only properly partially 25

For a good discussion of fine- vs. coarse-grained conceptions of facts, see Correia and Schnieder 2012.

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grounded by [grass is colored] alone, [grass is green & grass is colored] is fully grounded by [grass is green] alone. This follows from a plausible transitivity principle for full grounding. (FT) For any x, y, Γ, and ∆, if x is fully grounded by y, Γ and y is fully grounded by ∆, then x is fully grounded by Γ, ∆.26 To see why, recall that it was also established in section 3 that [grass is colored] is fully grounded by [grass is green]. By full transitivity, it follows that [grass is green & grass is colored] is fully grounded by [grass is green], [grass is green]. So [grass is green & grass is colored] is fully grounded by [grass is green].27 This result obviously requires a transitivity principle, and one that is stronger than (PT), introduced in section 3. As a result, one might have the Schafferian worries discussed there about (FT) as well. But it is worth noting that there is reason to think this full grounding claim holds independently of any transitivity principle, just given the nature of the facts involved. For starters, the existence of [grass is green] metaphysically necessitates the existence of the conjunctive fact, so it at least meets that necessary condition on full grounding. What is more, it seems right to say that (grass is green and grass is colored) because grass is green, and that this is a non-causal explanation, and a complete one at that. So, in addition to being fully grounded by [grass is green], [grass is colored] collectively, [grass is green & grass is colored] is fully grounded by [grass is green] alone. The resulting situation is depicted below. 26

I take this principle from Rosen 2010: 116. See Correia 2010: 262 and 2011: 3 for operationalist versions of this principle. The operationalist version is derivable in Fine’s system PLG as well. See Fine 2012b: 6. 27 As evidence that this last inference is sound, consider the following. It is standard in plural logic to define identity in terms of the ‘is/are among’ predicate. In particular, Identity. Γ = ∆ =df (i) Γ are among ∆ and (ii) ∆ are among Γ. (See McKay 2006: 122 and Oliver and Smiley 2013: 109.) In addition, it is plausible that (AX 1) For any Γ and ∆, Γ are among ∆ iff, for any x, if x is among Γ then x is among ∆. (See McKay 2006: 121.) I take it as obvious that [grass is green] is among [grass is green], [grass is green]. But it is also the case that [grass is green], [grass is green] are among [grass is green]. Because [grass is green] is the only thing among [grass is green], [grass is green], for any x, if x is among [grass is green], [grass is green], then x is among [grass is green]. So it is (AX 1) that guarantees that [grass is green], [grass is green] are among [grass is green]. By the above definition of identity, then, [grass is green] = [grass is green], [grass is green]. Thus, the footnoted inference is just an instance of good old-fashioned identity elimination.

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[grass is green & grass is colored]

[grass is colored]

[grass is green] Fig. 7 Since [grass is green] is properly among [grass is green], [grass is colored], there is a violation of minimality. Given (&I), then, and either (FT) or an independently plausible premise, minimality is false. Despite this counterexample, it is worth recognizing there is something to the concern Audi expresses in the passage above. Audi seems to be after a principle that if y ‘does no work’ in making x obtain, to use Audi’s phrase, then adding y to some Γ that fully ground x will not automatically result in a new plurality, y, Γ, that fully ground x. But he needs a principle that does not at the same time rule out cases like that depicted in Fig. 7. The following principle meets these conditions. Full Non-Monotonicity. If something is partially grounded by something (else), then it is not the case that, for any x, y, and Γ, if x is fully grounded by Γ, then x is fully grounded by y, Γ. (The initial antecedent is necessary to allow for the possibility of models with only fundamental facts.) Audi’s red cotton shirt example provides reason to reject the claim that full grounding is monotonic. After all, while [the shirt is red] is fully grounded by [the shirt is maroon], it is not fully grounded by [the shirt is maroon], [the shirt is cotton]. And full non-monotonicity does not rule out cases like that depicted in Fig. 7. It merely precludes adding an arbitrary fact y to some full grounds of x and have the resulting facts also fully ground x. And indeed, Audi (2012b: 699) explicitly endorses a principle relevantly similar to full non-monotonicity. But one needn’t add full non-monotonicity as an extra principle to one’s theory of ground. On models which include only fundamental facts, full non-monotonicity is vacuously true. And on any model which includes at least one non-fundamental fact, full non-monotonicity follows by the irreflexivity of grounding and the definition of partial grounding. 14

Proof. Suppose that a is fully grounded by Γ, and suppose for reductio that a is fully grounded by a, Γ. So there are ∆ (a, Γ) such that a is fully grounded by ∆ and a is among ∆. By the definition of partial grounding, a is partially grounded by a. But this contradicts the claim that grounding is irreflexive. So a is not fully grounded by a, Γ. And on models that include both a non-fundamental fact x and another fact that does not partially ground x, full non-monotonicity follows from the definition of partial grounding alone. Proof. Suppose that a is fully grounded by Γ, and that a is not partially grounded by b. Now suppose for reductio that a is fully grounded by b, Γ. Then there are ∆ (b, Γ) such that a is fully grounded by ∆ and b is among ∆. By the definition of partial grounding, a is partially grounded by b, contradicting the initial hypothesis. So a is not fully grounded by b, Γ. It seems, then, that the types of cases Audi is worried about in the passage above are already ruled out by other formal features of grounding.28 6. Conclusion I have offered two arguments against the ground-theoretic weak supplementation principle (WSG). The arguments have different virtues and do not stand and fall together. The main virtue of the first argument is that it will have force even for those who identify facts with their doubly negated correlates. The main virtue of the second argument is that it will have force even for those who deny that, in general, ‘determinates ground determinables’. I then showed that the cases on which these arguments are based also serve as counterexamples to Audi’s principle minimality, and that the types of cases to which Audi objects are already ruled out by other formal features of grounding.29

28

Similar remarks apply to a non-monotonicity principle governing partial grounding. Partial Non-Monotonicity. If something is partially grounded by something (else), then it is not the case that, for any x, y, and Γ, if x is partially grounded by Γ, then x is partially grounded by y, Γ.

Such a principle is just as plausible as full non-monotonicity. After all, while [the shirt is red and expensive] is partially grounded by [the shirt is maroon], it is not partially grounded by [the shirt is maroon], [the shirt is cotton]. Further, this principle, like full non-monotonicity, is vacuously true when there exist only fundamental facts. And it also results from other formal features of grounding, given the existence of at least one non-fundamental fact. 29 I’d like to thank Cody Gilmore for helpful suggestions and for reading numerous drafts of this paper. I’d also like to thank an audience at the 2014 Australasian Association of Philosophy Conference, an audience at a workshop in the Philosophy Department at the University of California, Davis, and an audience at the 2015 Central Division Meeting of the American Philosophical Association for helpful comments and suggestions. I would like to give special thanks to Aldo Antonelli, David Copp, Li Kang, Daniel Nolan, Gabriel Rabin, and Jonathan Schaffer.

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