1.a Property fixing

Consider the grounding problem between material coincidents.Footnote ⁹ We have a statue, ‘Statue,’ constituted by a lump of clay, ‘Lump.’ Statue is not Lump since the former, but not the latter, is a statue and cannot survive being smashed. But how can Statue and Lump differ given that they have the same parts (at least at the microlevel) arranged in precisely the same way? Put differently, Statue is a statue because of facts involving its grounds (its microstructure). But Lump, it seems, has the same grounds (same microstructure). So these facts also hold of Lump’s grounds. Thus, what explains that Statue is a statue is also true of Lump. Yet somehow, Lump is not a statue.

I appealed to materially coincident things. But I need not have. Suppose, as some have (Schaffer Reference Schaffer2010b), that the proper parts of the cosmos are grounded in the cosmos. Take me and the computer I am typing on. I am not this computer since I, unlike the computer, am a person and can think. But how can we differ given that we have the same metaphysical base? Put differently, given that the parts of the cosmos are grounded in the cosmos, I am a person because of facts involving the cosmos. But the computer has the same ground (it is part of the same cosmos). So these facts also hold of its ground. Thus, what explains that I am a person is also true of the computer. Yet somehow, the computer is not a person.

Here is another case. Many think that sums are grounded in their parts and sets in their members. So we have Quine, Carnap, their sum (‘Sum’), and their set (‘Set’). Now Sum is not Set since the former, but not the latter, is concrete and weighs n pounds. But how can Sum and Set differ given that they have the same metaphysical base? Put differently, given that sums are grounded in their parts and sets in their members, Sum is concrete because of facts involving its grounds (Quine, Carnap). But Set has the same grounds (Quine, Carnap are its members). So these facts also hold of Set’s grounds. Thus, what explains that Sum is concrete is also true of Set. Yet somehow, Set is not concrete.

The structure of these puzzles is the same. So the puzzle is not at all unique to the cases I have used in order to illustrate it. To see this more clearly, notice a principle undergirding these cases: that if grounded thing x is F because of some fact or facts involving its grounds, then for anything y with the same grounds, y is F. That is, if x is F because its grounds Γ are collectively some way G, then for any y, if what grounds y are Γ, y is F. More formally,

Using Fix to express the first of the above cases, Statue is a Statue because of facts involving its grounds. But these facts appear to hold of Lump’s grounds (same microstructure). And so from Fix, Lump is a statue. But Lump is not a statue. Hence, the grounding problem between materially coincident entities.Footnote ¹¹

Focusing on the grounding problem between materially coincident entities, note the lengths many have gone in trying to solve it. Appeal has been made to a four-dimensional view of persisting objects (Sider Reference Sider2001, sec. 5.8; Wasserman Reference Wasserman2002), material objects having non-material parts (McDaniel Reference McDaniel2001; Paul Reference Paul2002; Fine Reference Fine2008; Koslicki Reference Koslicki2008), to sortal and/or modal properties of coincident entities being brute (Bennett Reference Bennett2004; Moran Reference Moran2018), to a plenitudinous ontology (Bennett Reference Bennett2004, sec. 4), and to fictionalism about monadic modal properties (Sider Reference Sider2008a). That the grounding problem has led many to take seriously such solutions, all of which seem to preserve Fix (or some supervenience-style version of it), shows just how reluctant many are to reject this property fixing principle.Footnote ¹²

This reluctance is well-motivated. Without Fix, it may be that something is F in virtue of facts involving its grounds while something else, in spite of having the same grounds, is not. But this is hard to stomach. In talking about the grounding problem between persons and animals, Olson (Reference Olson2001, 345) says

Notice what the problem is. Why can Person, but not Animal, think given that whatever it is that is able to explain that Person can think—its microstructure, surroundings, and evolutionary history—is true of Animal? Since there is nothing unique here about Person and Animal, what Olson says is just as forceful when applied more generally by asking ‘why is x, but not y, F given that whatever it is that is able to explain that x is F is true of y?’Footnote ¹³

I have, here, focused on the grounding problem in order to motivate Fix. But, as my appeal to cases not involving materially coincident entities shows, we can motivate it independently (as I have already remarked and as these other cases show, the problem here is not at all unique to materially coincident things). Given our assumption that if something is grounded, then it exists and has the nature it does because its ground exists and has the nature it does, Fix becomes very plausible. Suppose that x is F because its grounds, Γ, are G. Suppose also that y is grounded in Γ. Now it would be a very strange thing if y were not also F. Why the difference, given that x and y are grounded in the same thing and that if something is grounded, then it exists and has the nature it does because its ground exists and has the nature it does? Here, for some weird and seemingly inexplicable reason, that the ground of x and y is some way results in x’s, but not y’s, being F. This is hard to believe.Footnote ¹⁴

1.b Similar principles

Before moving on, it is worth contrasting Fix with two other principles.Footnote ¹⁵ The first is given by deRosset (Reference deRosset2010, Reference deRosset, Hoeltje, Schnieder and Steinberg2013b). Following his notation, where ‘r’ names a raindrop, suppose that

1. 1. r is F because ϕ(r, t ₁, …, t _n)

where “all of the individuals involved in the explanans are denoted by exactly one term among r, t ₁, …, t_n, and ϕ says how the properties and relations in question are distributed over those individuals” (Reference deRosset2010, 79).Footnote ¹⁶ Now the constraint that deRosset puts on explanation says that good explanations do not have confounding cases. What is a confounding case? A confounding case of 1 is a case in which something r*, along with some other things, have the properties needed to “satisfy the explanans clause ϕ, but in which r* lacks F” (80). That is, r*, along with the other things, instance

$$ \exists {y}_1,\dots, {y}_n\exists x\left(\unicode{x03D5} \left(x,{y}_1,\dots, {y}_n\right)\ \&\sim \mathrm{F}x\right). $$

So if 1 is a good explanation, then according to deRosset’s constraint on explanation, the above existential statement is false. That is

$$ r\hskip0.35em \mathrm{is}\hskip0.35em \mathrm{F}\hskip0.35em \mathrm{because}\hskip0.35em \unicode{x03D5} \left(r,{t}_1,\dots, {t}_n\right)\to \forall {y}_1,\dots, {y}_n\forall x\left(\unicode{x03D5} \left(x,{y}_1,\dots, {y}_n\right)\to \mathrm{F}x\right). $$

Generalizing, we get

Now, Fix and Determination vary in important ways. First, Determination does not require, in the antecedent, that the things that satisfy the explanans clause ϕ ground the thing that is F. But Fix does, stating a necessary condition on cases where something is some way because this thing’s grounds are some way. Second, Determination has it that for anything whatsoever, if it, along with other things, satisfy the explanans clause ϕ, then it is F. This is not so for Fix. Instead, it has it that for anything whatsoever, if it is grounded in the things that ground x, then it is F.

These differences matter. deRosset uses Determination to argue against

(Explanation is very much in the spirit of one of the assumptions of this paper: that if something is grounded, then it exists and is the way it is because whatever grounds it exists and is the way it is). Here is the argument. Suppose that we have a raindrop r, tectonic plate t, where r, but not t, is transparent. Also suppose, consistent with Explanation, that

1. 2. r is transparent because R(u ₁, …, u _n),

where ‘R’ stands for some complex relation and none of the terms ‘u ₁,’ … ‘u _n’ denote r. That is, we have a claim of the form ‘r is transparent because ϕ,’ where “ϕ says how certain properties and relations are distributed over individuals other than r” (Reference deRosset2010, 82).Footnote ¹⁷ Now given Determination, it follows that 2 is true only if the following universal generalization is true

$$ \forall {y}_1,\dots, {y}_n\forall x\left(\mathrm{R}\left({y}_1,\dots, {y}_n\right)\to x\hskip0.35em \mathrm{is}\ \mathrm{transparent}\right). $$

But given that ‘R(y ₁, …, y _n)’ does not contain ‘x,’ this is equivalent to

1. 3. $ \exists {y}_1,\dots, {y}_n\left(\mathrm{R}\left({y}_1,\dots, {y}_n\right)\right)\to \forall x\left(x\hskip0.35em \mathrm{is}\ \mathrm{transparent}\right) $ .

Assuming then that 2 is a good explanation, R(u ₁, …, u _n) is true. So the antecedent of 3 is satisfied, from which it follows that everything, and so t, is transparent! But t is not transparent. So 2, which is the proposed explanation, fails. Given Determination, Explanation is false.Footnote ¹⁸

Fix, however, does not tell against Explanation (which I take to be a good thing). In order to infer from Fix that t is transparent, it must be that t is grounded in whatever grounds r. But it is not. Plausibly, what grounds r (its microstructure) does not ground t (r’s microstructure is not t’s). From Fix then, it does not follow that t is transparent. And so 2 does not fail and Fix does not threaten Explanation.

We have seen why this difference exists: in order to infer that t is transparent, Fix, unlike Determination, requires that grounding links exist between r, t, and u ₁, …, u _n (it requires that u ₁, …, u _n ground both r and t). Indeed, once we see the argument against Explanation from Determination, it is not hard to see what is needed in order to avoid having it that t is transparent: we need to add some condition or other that tells us how u ₁, …, u _n relate to r and to t (Saenz Reference Saenz and Raven2020, n30). And having it that u ₁, …, u _n ground r, but not t, is such a condition (recall footnote 13).Footnote ¹⁹ So Fix can be seen as supplying what is missing in the present case. But then Fix can be seen as supplying what Determination, here at least, omits.Footnote ²⁰

Here is another principle that Fix should be contrasted with. Where ‘F’ and ‘G’ are predicates that correspond to qualitative properties (the latter corresponding to the conjunction of all of y’s qualitive properties), Barker (Reference Barker2021) has put forward the following thesis:

That is, if x is F, y grounds x, and y is G, then necessarily, for any z, if y grounds z and y is G, then z is F.

There are several differences between Fix and Property Fixing Thesis (‘PFT’ for short).Footnote ²¹ First, there is a sense in which PFT is stronger than Fix. Notice that we are not told what explanatory role y’s being G plays with respect to how x is. PFT does not, for example, say or require that x is F because y is G. It is consistent with PFT that y’s being G does not explain x’s being F. So unlike Fix, PFT is not a constraint governing explanation. In this sense, PFT is stronger than Fix. The latter requires that x is F because y is G in its antecedent. PFT does not require this in its antecedent.

Second, there is a sense in which PFT is weaker than Fix. PFT restricts itself to qualitative properties. Because it does, it allows things that share a ground to be numerically, even if not qualitatively, distinct. But Fix does not restrict itself in this way. It holds for qualitative and non-qualitative properties alike. Because of this, as will soon be made clear, Fix does not allow things that share a ground to be numerically distinct. In this sense, PFT is weaker than Fix.Footnote ²²

Third, Barker and I view our respective principles differently. I have argued in favor of Fix. Barker does not argue in favor of PFT (indeed, he has communicated to me that he rejects it). He does argue that it follows from the following purity thesis: there are no fundamental facts about grounded things. But it is no part of Barker’s paper to accept this purity thesis (in fact, Barker seems more than open to rejecting it and, in his paper [Barker, Reference BarkerForthcoming] does reject it). Barker and I also put our principles to some different uses. To give just one example, Barker appeals to PFT in order to argue against the claim that grounded things are ontologically innocent with respect to their grounds. I, however, appeal to Fix in order to argue in favor of a stricture on grounding which, in turn, allows me to defend the claim that grounded things are ontologically innocent with respect to their grounds (see sec. 3.2).

1.c The argument

I will now show that Fix, along with some plausible assumptions, gives rise to a powerful stricture on grounding.

Making use of the third case discussed earlier, suppose for reductio that Quine, Carnap collectively ground their sum and their set. Now this sum weighs n pounds because its summands, Quine and Carnap, weigh n pounds. Given Fix, it follows from Quine, Carnap collectively grounding their set that this set weighs n pounds. But it does not. Sets, unlike sums, are not the kinds of things that have weight and so are not the kinds of things that weigh n pounds. And so, since Fix should not be rejected, we must reject that Quine, Carnap collectively ground their sum and their set.

To appeal to a case involving qualitatively identical things, and so to a case where things involved are as similar as can be (short of being one and the same thing), suppose for reductio that we have a Max Black world with two spheres, A and B, each of which are grounded in the same pure properties the Fs (some form of bundle theory holds here). Now A occupies region R because the Fs occupy R. Given Fix, it follows that B occupies R. But it does not, instead occupying R*. Since Fix should not be rejected, we must reject that the same pure properties can ground numerically distinct individuals.

Finally, to use our first case, suppose for reductio that what grounds Statue grounds Lump (their microstructure). Now Statue is a statue because of facts involving its ground. Given Fix, it follows that Lump is a statue. But it is not. Since Fix should not be rejected, we must reject that Statue and Lump have the same grounds.

Generalizing, where ‘α,’ ‘a,’ and ‘b’ are constants (with ‘α’ being a plural constant) and a is not b, here is the argument:

1. 1. α < a & α < b (for reductio)

2. 2. Fa & ~Fb (assume)

3. 3. Fa because Gα (assume)

4. 4. Fb (1, 3, Fix)

Since 4 and 2 are inconsistent, we must reject 1: α can ground at most one thing. Now since no special assumptions were made about α, a, and b, the reasoning generalizes: everything can ground at most one thing. And so we have a uniqueness principle for grounding, namely

(‘Uniqueness’ for short). Plainly put, if x and y share a ground, then x is y. Footnote ²³ This is the argument from Fix to a stricture on grounding. The argument involves three basic premises. Since the first is our assumption for reductio, we are left with 2 and 3.

Starting with 2, that a is some way that b is not is hard to deny. For one, there will always be a difference in identity: a, but not b, is identical to a. But pushing this to the side, notice that there are no plausible cases involving two or more things failing to differ. The most plausible concern qualitatively identical things. But even here, there are non-qualitative differences. Take again A and B. In a Max Black world, A, but not B, occupies R, can survive the destruction of B, and is closer to A than to B.

Moving to 3, the notion of ground at work in this paper has it that grounded things are the way they are because their grounds are the way they are. So this notion guarantees 3: that a is F because its grounds α are some way. Suppose though that we reject 3 and so deny that a is F because of how its grounds are (notice that this is consistent with Fix). Then the difference between a and b—that a is F and b is not—must be brute, and so not had on the basis of facts involving their grounds.Footnote ²⁴ And so, if we deny 3 but affirm Fix, then if a and b share a ground, then either they are identical or, if they are not identical, differences between them are brute. Though this condition is not quite as strong as what we get in Uniqueness, it is still weighty. And so the reasoning that otherwise leads to Uniqueness still packs quite a punch!

Uniqueness has a precursor. Nelson Goodman (Reference Goodman, Bochenski, Church and Goodman1956) famously pushed for the following nominalist dictum: no difference of entities without a distinction of content. Or in another work (Reference Goodman1958), no two entities are generated from exactly the same atoms. Or as it is most often put, no difference without a difference maker (Varzi Reference Varzi2008).Footnote ²⁵ For Goodman, a generating relation is the proper-part relation, the ancestral of set-membership, or their disjunction and an atom is something that is not generated. So where ‘x’ and ‘y’ range over generated objects, ‘z’ ranges over atoms, and ‘G’ expresses the generating relation, we have

Dictum notoriously eliminates sets.Footnote ²⁶ But it does not eliminate mereological wholes. It instead puts a restriction on them: no two wholes can have the same atoms as parts. This is, minus the restriction to atoms, the uniqueness of parthood (for an explicit rendering of this uniqueness principle and discussion of it, see sec. 2.b).

What should we think of Goodman’s definition of ‘generate’? If he is merely stipulating what he means by it, then as Lewis (Reference Lewis1991, 40) says, I can hardly object. But if Goodman was trying to capture a sufficiently general pre-theoretic notion of ‘generate,’ then I do object. Some things generate other things even though the former are neither proper parts nor ancestors of members of the latter. This seems especially true when it comes to properties. Perhaps physical properties generate non-physical properties, natural properties normative properties, chemical properties biological properties, and so on. Indeed, in describing his generating relation, Goodman uses words and phrases that would appear to be consistent with this list of things generating things. He talks of things being made up of other things and things that concoct and result in something else.

What should we think of Dictum? Granting Goodman his definition of ‘generate,’ it is objectionable since two or more wholes (Statue and Lump) can share all their proper parts and two more or more sets (my singleton and its singleton) can have the same atoms (me).Footnote ²⁷ Still, Dictum is on to something. The spirit of it, captured by the slogan “no difference without different makers,” seems true. How, though, can we make sense of this slogan once we reject Dictum? Here, I suggest appealing to Uniqueness. That is, it is grounding that this slogan holds true of and so Uniqueness that best captures it.

It is helpful to have a picture of the grounding patterns Uniqueness both does and does not allow. Where the arrow represents the grounding relation, if Uniqueness is true, then the left and middle diagrams are possibly instanced. The right is not.Footnote ²⁸

In ruling out the getting of many from one, Uniqueness makes grounding branches impossible. But it is consistent with grounding roots in that it permits the getting of one from (read distributively) many.

1.d Kinds of grounding

Some will protest. Assuming that there are logically complex properties, does not the property of being F ground the property of being F∨G, the property of being F&F, and the property of being ~~F? And if so, is not Uniqueness false?Footnote ²⁹

One way of responding to this worry is to go rogue by rejecting that the property of being F grounds these logically complex properties.Footnote ³⁰ This is, in fact, my preferred response. But here, I want to give a less drastic response by granting that there is grounding going on in these cases, but that the kind of grounding involved is different than the one involved in Uniqueness. This is not ad hoc since this way of responding is independently motivated. Recall one of the assumptions made in this paper: that grounded things are the way they are because their grounds are the way they are. But this assumption does not generally hold when logically complex properties are grounded in the above kind of manner. There is little temptation to say that, in general, the property of being F∨G is the way it is because the property of being F is the way it is (for example, there is nothing about the property of being F that explains that the property of being F∨G is logically complex, has two disjuncts, or is instantiated if the property of being G is). This is also why there is no analogous grounding problem when it comes to grounding these logically complex properties in this way. The property of being F∨G and the property of being ~~F are sortally and modally different (the former, but not the latter, is a disjunctive property and can be instantiated even if the property of being F is not). But in spite of both being grounded in the property of being F, there is no grounding problem. Why? Because we are not at all inclined to think that the property of being F∨G and the property of being ~~F must get their character from the property of being F (and so on more generally for other logically complex properties).Footnote ³¹

We started this paper with some reasonable, albeit non-neutral, assumptions about grounding. Given them, we are now able to motivate thinking that there are in fact two kinds of grounding: one that obeys these assumptions and so obeys Uniqueness and one that does not and so does not. We are familiar with the first kind: it is the kind found in cases where something metaphysically depends on some thing or things for its existence and nature.Footnote ³² We are also familiar with the latter kind: it is the kind found in cases where the logically atomic gives rise to the logically complex. This distinction is important. Going in for such a non-univocal conception of grounding might strike some as a cost. But it is a cost we must put up with. Unless we reject that some things metaphysically depend on other things for their existence and nature, it looks like we must accept that there is some joint or other in grounding. There is, on the face of it, a natural difference between grounding as ‘dependence’ and grounding as, for lack of a better term, ‘logical realization.’Footnote ³³ Part of that difference is manifested in how things ground and how things are grounded (as made clear in the previous paragraph). And so a failure to account for this difference is a failure to capture what seems to be a natural joint in grounding. Because of this, we need to be careful not to conflate these two kinds of grounding. More to the point, we need to be careful in using instances of one kind of grounding as counterexamples to a principle of the other.

1.e A milder alternative

Perhaps Uniqueness is too severe. In having it that the sharing of a ground suffices for the identity of the grounded, some may think so. So why not have it that what suffices for identity is the sharing of all the full grounds and not merely some full ground (Saenz Reference Saenz2015, 2210)? Where the objects ranged over by x and y are grounded, here is a milder principle

(‘Mild Uniqueness’ for short). Plainly put, if x and y share all their grounds, then x is y. Now Mild Uniqueness makes possible certain cases of grounding that Uniqueness does not. Here are two.

Mild Uniqueness is consistent with grounding-skips since, as seen in the left diagram, c is, but b is not, grounded in b. And it is consistent with grounding-branches since, as seen in the right diagram, b ₂ is, but b ₁ is not, grounded in a ₂. So Mild Uniqueness is consistent with something grounding more than one thing. Many might see in this a reason to prefer it to Uniqueness. Still, it alone will not do. We need Uniqueness.

Given the premises of our argument for Uniqueness, two things cannot share a ground. But Mild Uniqueness permits two things to share a ground. So it is inconsistent with these premises. And it follows from this that, given our argument for Uniqueness, Mild Uniqueness alone will not do.

Now Fix played a big role in getting Uniqueness. But then maybe the right move is to opt for a milder version of it. Where ‘Δ’ is a plural variable, consider

$$ \forall x\forall \Gamma \left(\left(\Gamma < x\hskip0.35em \&\hskip0.35em \mathrm{F}x\hskip0.35em \mathrm{because}\hskip0.35em \mathrm{G}\Gamma \right)\to \forall y\left(\forall \Delta \left(\Delta <x\leftrightarrow \Delta <y\right)\to \mathrm{F}y\right)\right). $$

That is, if x and y have all the same grounds, then if x is F because its grounds are collectively G, y is also F. Dropping Fix and going in for this milder version gets us Mild Uniqueness but not Uniqueness. But we should not drop Fix.

Recall the grounding problem between Statue and Lump: how can Statue, but not Lump, be a statue given that what explains that Statue is a statue (some fact about its microstructure) is also true of Lump? Notice that this problem remains whether or not Statue and Lump share every one of their full grounds. Once we see that they share a full ground (their microstructure), the problem rears its ugly head. Or considering the above branching diagram, suppose someone claimed that b ₁ is F because a ₁ is G but went on to claim that b ₂, in spite of being grounded in a ₁, is not F. Do we not have a grounding problem here? How can it be that what explains that b ₁ is F is also true of b ₂ and yet, somehow b ₂ is not F? And this problem remains even if b ₁ and b ₂ do not share all their grounds (as they do not). Their sharing a ground is enough to generate a grounding problem. And so this milder version of Fix is, even if true, too mild to do justice to our difference making intuitions.

Consider an even milder version of Fix:

This says that if it is necessary that x and y have all the same grounds, then if x is F because its grounds are collectively G, y is also F.Footnote ³⁴ That this version of Fix is insufficient is even easier to see. Possibly, what grounds Statue does not ground Lump (there are worlds where Statue but not Lump exists). And possibly, what grounds Lump does not ground Statue (there are worlds where Lump but not Statue exists). So it is not necessary that they have exactly the same grounds. Still, the grounding problem between them remains: their sharing a ground (their microstructure) suffices for it to be problematic that they differ. But then we should keep Fix. And if Fix, then Uniqueness.

2.a Transitivity

Suppose that grounding is irreflexive. Then not only is Uniqueness inconsistent with

it entails that all cases where x grounds y and y grounds z are cases where x does not ground z. Uniqueness thus has it that grounding is intransitive. Though Uniqueness permits chains of grounding, it does not permit grounding to skip a joint in the chain. This result is strong (and for some highly objectionable). In having it that grounding is intransitive, Uniqueness goes beyond what those in the literature have committed to when they reject Transitivity (Schaffer Reference Schaffer, Correia and Schnieder2012; Rodriguez-Pereyra Reference Rodriguez-Pereyra2015).Footnote ³⁵ Given that they rely on counterexamples to Transitivity, they can at most conclude that grounding is not transitive.Footnote ³⁶

Suppose though that one wanted to hold on to Transitivity. What premise or assumption of our argument for Uniqueness should they reject? Fix. But for reasons already given, this is a hard sell. Assuming that y is F and that z is not F, if x grounds y and y grounds z, then on pain of thinking that z is F, we should not think that z is the way it is because x is the way it is. But then we should not think that z is grounded in x. Being grounded in x is one thing. Being grounded in something that is grounded in x is another. And this is exactly what Uniqueness, by way of Fix, brings to light.Footnote ³⁷

2.b Ontology

Uniqueness has several implications when it comes to ontology. Here I discuss those having to do with priority monism, sums, sets, and qualitativism.

Priority Monism. According to priority monism, the cosmos grounds all its proper parts (Schaffer Reference Schaffer2010b). So if Uniqueness is true, then priority monism is not.

How should the priority monist respond? She can say that it is how the cosmos is, and not simply the cosmos, that grounds its proper parts. So that the cosmos is me-ish, bears me-ness to here, or is here-me-ish, grounds me and that the cosmos is you-ish, bears you-ness to there, or is there-you-ish, grounds you. This may or may not work. It certainly needs fleshing out. But even if it does work, notice what Uniqueness has done. A virtue of priority monism is supposed to be its offering a sparse base on which to ground all the cosmos’s proper parts. In so far as ontological simplicity is measured in terms of the number of things at the fundamental level (more on this below), in comparing priority monism with priority pluralism, the former is economically preferable. However, if what grounds the proper parts of the cosmos are facts involving the cosmos such that for each proper part (or maybe for each atomic part), there is some way the cosmos is that uniquely grounds this proper part, then the number of things at the fundamental level will have greatly increased: one for each part of the cosmos. So understood, priority monism’s economical advantage is either lost or not as impressive as we thought.Footnote ³⁸

The priority monist can say something else. What the cosmos grounds are not its parts taken individually, but its parts taken collectively. So neither me, you, nor any other proper part of the cosmos is grounded in the cosmos. But we are taken collectively! This construal of priority monism yields a nice symmetry between it and standard accounts of priority pluralism. The latter says that the proper parts of the cosmos (or facts involving these parts), taken collectively, ground the cosmos. The mirror view of this is simply that the cosmos grounds its parts collectively. Be this as it may, that pluralities can be grounded is a radical thesis (or is at least something many do not accept).Footnote ³⁹ Uniqueness thus yields something of import in potentially forcing the priority monist to accept something that many think is false when it comes to the nature of grounding.Footnote ⁴⁰

There are other views that, on the face of it, share with priority monism the thought that, for some collection of things, all are grounded in one. One particularly interesting view involves God and claims that all of creation, and so each created thing, is grounded in God (Sullivan Reference Sullivan2015, 279; Craig Reference Craig2017, chap. 1; Pearce Reference Pearce2017; Hamri Reference Hamri2018). So understood (but perhaps we should not understand it this way; perhaps this description of this view is elliptical for some other grounding claim), this view must go. Given Uniqueness, nothing can ground more than one thing. And so, even if God is the cause of all else, God is not the ground of all else.Footnote ⁴¹

Sums and Sets. The following seems attractive: all it takes to ground a sum and a set of some things are those things. Thus, if we have some things, then we have them grounding their sum and their set. Since this contradicts Uniqueness, something must give.

Perhaps sets should give? Assume then that there are sums but no sets. So for any plurality greater than one, Uniqueness has it that such a plurality grounds and only grounds a sum. But then it follows that for anything collectively grounded in two or more things, it is a sum! This seems false: structured material objects (like chairs) and structural properties (like methane) are grounded in pluralities and neither kind of thing is a sum. So maybe sums should give? Assume then that there are sets but no sums. So for any plurality at all, Uniqueness has it that such a plurality grounds and only grounds a set. Thus, all grounded things are sets! This too seems false: there is more to derivative reality than sets. There are also material objects and properties, neither of which we are compelled to identify with sets.

Perhaps what should give is that sums and sets are grounded only in their parts and members? If so, then the popular thought that sums and sets are “ontological free lunches” should no longer be believed. Either sums and sets must go or they cannot be had merely on the basis of their parts or members.Footnote ⁴²

Uniqueness and that wholes are grounded in their proper parts entails a well-known uniqueness principle (where ‘PP’ expresses proper parthood and the objects ranged over by x and y are composite objects):

$$ \forall x\forall y\forall z\left(\left(\mathrm{PP} zx\leftrightarrow \mathrm{PP} zy\right)\to x=y\right). $$

Uniqueness and that sets are grounded in their members entails another well-known uniqueness principle (where ‘∈’ expresses set-membership and the objects ranged over by x and y are sets):

$$ \forall x\forall y\forall z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right). $$

That we, in part, get these principles from Uniqueness is not surprising. As discussed earlier, Uniqueness is one way of capturing the slogan that there can be no difference without different makers. And given this slogan and the claim that what “makes” wholes and sets are, respectively, their proper parts and members, these principles follow. But, as seen a few paragraphs ago, we have been given reason to think that what “makes” wholes are not their proper parts and that what “makes” sets are not their members. And what follows from this is not that these principles are false, but that they should not be seen as genuine instances of the above slogan. (Indeed, once we grant that wholes are not grounded in their proper parts, it is no longer obvious what the problem is with thinking that two wholes can have the same proper parts. If what grounds wholes are various facts involving their proper parts, then it can be that some facts involving these proper parts ground one whole and that some other facts involving these same parts ground another whole [Saenz Reference Saenz2015]. This, because it posits different grounds for different wholes, is consistent with Uniqueness.)

Qualitativism. As I am understanding it, this is the view that at the fundamental level, there are only qualitative facts and that these facts ground individuals. But there is an argument from Uniqueness against it:

1. 1. A Max Black world (a world with two indiscernible spheres) is possible.

2. 2. If a Max Black world is possible, then if Uniqueness is true, qualitativism is false.

3. 3. If Uniqueness is true, qualitativism is false. (1, 2)

4. 4. Uniqueness is true.

Hence,

1. 5. Qualitativism is false. (3, 4)

Though some have denied it, 1 is reasonable (Adams Reference Adams1979). And we can assume 4 in the present context. So it all hangs on 2. But 2 is true. In a Max Black world, spheres A and B are qualitative duplicates. But then if Uniqueness is true, and so true in a Max Black world (Uniqueness is not a contingent truth), A and B cannot share a ground. But then, on pain of sharing a ground, A and B cannot be grounded in their qualitative natures. And from this it follows that qualitativism is false. Thus, 2.

In talking of the grounds of the fact that A exists and of the grounds of the fact that B exists, Dasgupta gets close to following this line of reasoning. Though he mentions nothing like Uniqueness, it seems to be lurking behind the following quote:

How does Dasgupta avoid this worry with qualitativism? Not by doubling down and affirming with grounding orthodoxy that something can ground two or more things. Rather, he thinks that the qualitativist should go in for the grounding of pluralities. So what the qualitative facts ground are not A and B individually (or that A exists and that B exists), but A and B collectively (or facts involving A and B taken collectively). And going this route is tantamount to rejecting 2. Given Uniqueness then, we have reason to accept that if qualitativism is true, then pluralities are grounded. Since I reject that pluralities can be grounded, I see in this conditional a reason to reject qualitativism. But whether you tollens or ponens, Uniqueness has bite in potentially forcing the qualitativist, like the priority monist, to something most reject when it comes to grounding.

Grounding Problems. Unsurprisingly, we can construct a grounding problem for the above views. I am a person. My computer is not. We thus differ sortally. But given the standard way of understanding priority monism, and so given that my computer and I have the same ground in the cosmos and are the ways we are because of how the cosmos is, how is it that we can differ in this way?Footnote ⁴³ Or take qualitativism. A, the first iron sphere in a Max Black scenario, occupies region R. B does not. But given qualitativism, how can it be that they differ in this way? Something about those qualitative facts that ground A explains that A occupies R and yet, in spite of B being grounded in the very same facts, B does not occupy R. This is puzzling.

That we have these grounding problems is not surprising. As we saw at the beginning of this paper, once you have it that two things can share a ground, then given Fix and some plausible assumptions, it becomes puzzling how such things can differ. That is, once you reject Uniqueness, then given Fix and some plausible assumptions, you have a grounding problem.

2.c Ontological simplicity

Many now think that when it comes to ontological simplicity, derivative things cost no more than their grounds.Footnote ⁴⁴ And if true, then a good methodology would be a bang-for-your-buck one where one gets the most one can at derivative levels from the least one can at more fundamental ones (Schaffer Reference Schaffer, Chalmers, Manley and Wasserman2009, 361).

I find this methodology attractive. But we need to be careful. As I hope the reader now sees, things can get out of hand if one does not wield it wisely. There are two ways to get a bang for a buck. The first is captured in the diagram on the left, where the bang proceeds linearly. The second in the diagram on the right, where the bang proceeds branchly.

But on pain of contradicting Uniqueness, the bang-for-buck methodology should not be wielded in ways that result in branchy bangs. If you want the biggest bang for your buck, you must proceed linearly, yielding tall and slim, and not wide and fat, trees.

Given this, if a theory has n things in any given non-fundamental level, the level below it must have n grounds (or more if overdetermination occurs). The theory cannot, as concerns the number of grounds, slim as we go down. It can, however, slim in the number of things. Consider a theory that has only two fundamental things, a ₁ and a ₂. From them we are, in principle, able to ground three things: a ₁ may ground one thing, a ₂ another, and a ₁, a ₂ a third. So in spite of their being three grounded things and so three grounds (a ₁ and a ₂ and a ₁, a ₂), the theory slims from three things to two as we go down.Footnote ⁴⁵ This then is a way a theory can slim as we go down or branch as we go up. So where n is the number of things we start with, the maximal number of things we can ground from n is 2 ⁿ-1. Any theory that grounds more than this from n things contradicts Uniqueness.

Uniqueness explains a well-known trade-off. In general, and ontologically speaking, the more fruitful a theory is the less economical and the more economical a theory is the less fruitful. But what is the coin by which, in general, increasing one decreases the other? The answer, I think, is found in Uniqueness. Consider the ontologies discussed in section 2.b, each of which violate this trade-off. In them we were able to get so much from so little. Now if we want to rid ontology of this magic, we need Uniqueness. If we want a theory whose fruits are many, Uniqueness says that we need a theory less economical than one whose fruits are few (the exception being cases where a theory has not maximized the work its more fundamental things can do). So Uniqueness is the coin that explains the tradeoff. If you start off with one fundamental thing but want, at the next level, two derivative ones, you must widen the base. More generally, if at level L you start with n things but want, at L+1, more than 2 ⁿ-1 things, you must widen L. No surprise then that the more fruitful a theory is the less economical and the more economical a theory is the less fruitful.

In addition to constraining a methodology and explaining a trade-off, Uniqueness shows us why the following objection to this methodology does not work (Schaffer Reference Schaffer2015, 656–58; Turner Reference Turner2016, 382–85). Take an atomistic model of classical mereology which generates 2 ⁿ-(n+1) composites from n atoms. Contrast this with ‘doubled mereology,’ according to which for every non-atomic thing that exists according to classical mereology, there is its double composed from exactly the same parts.

Now doubled mereology is supposed to tell against the bang-for-buck methodology. The thought is that in spite of getting a bigger bang from the same buck, doubled mereology is worse off as concerns simplicity than is classical mereology on account of its involving redundant things; composites that do no more work than their classical twins. But whether you have this seeming or not, appealing to doubled mereology in order to tell against this methodology is misguided and we can see why. Doubled mereology is inconsistent with Uniqueness. And that it is explains why some might be inclined to think that it yields redundant things. The doubled entities that doubled mereology yields appear redundant on account of having the same grounds as their classical twins. Absent this sharing of grounds, there would be no obvious redundancy — grounding numerically distinct composites in numerically distinct atoms does not, on its own, give us reason to think that these composites are redundant with respect to each other. Assuming then that there is no sharing of grounds by composites, there is no reason to think that such composites are redundant. We only get doubled mereology and its redundant composites because we are assuming that these composites share a ground. But we should not. By means of Uniqueness, grounding prohibits such multiplying. So because this kind of multiplying is prohibited by grounding, doubled mereology should not be used to tell against the present bang-for-buck methodology. And if it should not be used to tell against this methodology, then the reason it gives to think that derivative things cost more than their grounds is not a reason that should move us. So given what is built into grounding, it is right to look to maximize one’s bang from one’s buck.