The fundamental: Ungrounded or allgrounding?
Abstract
Fundamentality plays a pivotal role in discussions of ontology, supervenience, and possibility, and other key topics in metaphysics. However, there are two different ways of characterising the fundamental: as that which is not grounded, and as that which is the ground of everything else. I show that whether these two characterisations pick out the same property turns on a principle—which I call “Dichotomy”—that is of independent interest in the theory of ground: that everything is either fully grounded or not even partially grounded. I then argue that Dichotomy fails: some facts have partial grounds that cannot be complemented to a full ground. Rejecting Dichotomy opens the door to recognising a bifurcation in our notion of fundamentality. I sketch some of the farreaching metaphysical consequences this might have, with reference to bigpicture views such as Humeanism. Since Dichotomy is entailed by the standard account of partial ground, according to which partial grounds are subpluralities of full grounds, a nonstandard account is needed. In a technical “Appendix”, I show that truthmaker semantics furnishes such an account, and identify a semantic condition that corresponds to Dichotomy.
Keywords
Fundamentality Grounding Truthmaker semantics HumeanismThe question of what is fundamental in the world is of intrinsic interest. Moreover, an answer to it can guide and constrain what we say about other central metaphysical topics.
In discussions of ontology, it is a common view that the fundamental needs to be considered as carrying a heavy weight, while the nonfundamental can taken to be lightweight. Such a differential treatment may be articulated in a variety of ways: as the claim that fundamental properties are universals and nonfundamental ones mere classes (Armstrong 1978; Lewis 1983); or that fundamental facts are states of affairs and nonfundamental facts mere propositions (Armstrong 1997); or as the general methodological maxim that Ockham’s razor should be wielded only against putative fundamental entities (Schaffer 2015).
I hold, as an a priori principle, that every contingent truth must be made true, somehow, by the pattern of coinstantiation of fundamental properties and relations. The whole truth about the world \(\ldots\) supervenes on this pattern. (Lewis 1999, 292)
The roles of fundamentality visàvis ontology and visàvis supervenience are related. One obvious bridge between them is provided by Armstrong’s doctrine of the ontological free lunch: “what supervenes is no addition to being” (Armstrong 1997, 12).
[T]he fundamental actual concrete objects should be freely recombinable, serving as independent units of being \(\ldots\). Thus each should be, in Hume’s words, ‘entirely loose and separate’. (Schaffer 2010, 40)
[W]hatever the (contingent) fundamental elements of the world are, they are open to free modal recombination. (Bennett 2017, 190)
The roles of fundamentality visàvis supervenience and visàvis possibility complement each other in fully determining the space of possibilities: the possibilities are all and only the recombinations of the fundamentalia.
The question of what is fundamental raises a prior question, which I shall be focussing on here: what is it for a fact, object, or property to be fundamental? It is natural to take fundamentality to be a matter of something’s place in the structure of the world. Such structure is, in turn, naturally articulated in terms of a certain relation. Indeed, once we have a term for such a relation—“ground”, by now entrenched in philosophical discourse—then using it to characterize fundamentality may not merely be optional, but required by the principle of ideological parsimony.^{1}
What is fundamental may also be called the “foundations.” In a house, the foundations can either be described in terms of what is below them—namely, no other part of the house—or in terms of what is above them—namely, every other part of the house. Since we are accustomed to think of grounds as lower than what they ground, there are, correspondingly, two obvious strategies for defining the fundamental in terms of ground: as that which is ungrounded, or as that which grounds everything else. I shall say that these are, respectively, characterizations of the fundamental as the “ungrounded” and as the “allgrounding.”
Perhaps the structure of reality is not given just by one relation, but rather by a family of relations which are not usefully amalgamated into one: the “building relations”, in the terminology of Karen Bennett (2017). If so, the same strategies for defining fundamentality remain available, as long as each of these relations is suitably vertically directed, with the relata on one side being thought of as lower than those on the other side. The fundamental can then be thought of as the “unbuilt” or as the “allbuilding.” This is so regardless of whether ground itself is or is not among the building relations, as in the views of Bennett and of Jessica Wilson (2014), respectively.^{2}
As I noted above, fundamentality is supposed to play a certain theoretical role in discussions of ontology, supervenience, and possibility. Between them, the two characterisations seem to account for its fitness to play that role. That the fundamental is ungrounded explains why it should be taken to be ontologically weighty, and why the fundamentalia do not modally constrain each other; that the fundamental is allgrounding explains why the nonfundamental need not carry an ontological cost, and why the fundamentalia modally constrain the nonfundamentalia.
Both styles of characterization are ubiquitous in the literature. The implicit assumption seems to be that they pick out the same notion. But the tenability of this assumption is underexplored. Bennett and Michael Raven (2016) deserve credit for contrasting the two characterisations explicitly, and exploring how they are related. In Sect. 1, I shall extend one of Bennett’s results. This discussion sets the stage for the main thesis of this paper: that the two characterisations may diverge.
My key move involves rejecting the orthodox account of the relationship between the notions of full ground and partial ground: that something is a partial ground iff there is something else together with which it is jointly a full ground. It follows from this that everything is either fully grounded or not even partially grounded—a principle I shall call Dichotomy. I give reasons to think that Dichotomy is false, and that the orthodox account of partial ground should be rejected—a thesis that is of independent interest for the theory of ground. If I am right, the notion of fundamentality bifurcates. Some potential metaphysical consequences of this bifurcation are sketched in Sect. 5. In the technical “Appendix”, I identify a mereological condition that corresponds to Dichotomy in Kit Fine’s truthmaker semantics for ground.
1 Two conceptions of the fundamental

f is Bfundamental =\(_{df}\) there is no full ground for f.
An entity is thus Bfundamental just in case it is at the bottom of the hierarchy generated by the relation of ground.
Bfundamentality may then be proposed as an explication of the notion of fundamentality. On the face of it, this is a plausible proposal, and does not call for extensive motivating commentary. I shall discuss its merits later, by comparing and contrasting it to a rival explication.

f is Bfundamental* =\(_{df}\) there is no \(\Gamma\) and building relation R such that \(R(\Gamma ,f)\).
If ground is a building relation, then everything that is Bfundamental* is Bfundamental. If ground is the only building relation, then the converse is true as well.
I shall now turn to the second conception of the fundamental, as the allgrounding. The idea is, roughly, that the fundamental entities are such that everything in the world can be accounted for in terms of them—that they are “complete”, in Bennett’s terminology. In my further discussion, I shall take the fundamentalia to be facts. However, much of the discussion will carry over, mutatis mutandis, to entities in other ontological categories.^{4}
It is tempting to say that some facts are allgrounding iff every fact is grounded in some of them. But there cannot be any such facts, provided some fact is Bfundamental. A remedy that suggests itself is to make an exception for the fundamental facts themselves—they need not be grounded, either by themselves or by any other facts. So we could say that the fundamental facts are those that provide a ground for every nonfundamental fact. However, this clearly cannot serve as a definition of fundamentality: as such, it would be circular.
To obtain a more careful formulation of the characterization of the allgrounding, I shall introduce a groundtheoretic analogue of the familiar notion of a supervenience base. Say that \(\Gamma\) is a grounding base iff for every f that does not belong to \(\Gamma\), there is \(\Gamma ' \subseteq \Gamma\) such that \(\Gamma '\) is a ground for f.^{5}
Being a grounding base is a good explication for what it is for some facts to be complete, and thus to be candidates for being the fundamental facts. However, it is a condition that the fundamental facts need to satisfy collectively, and does not give us an explication of f’s being a fundamental fact without further work.^{6}
It will not do say that f is fundamental iff it belongs to some grounding base. Since the plurality of all facts vacuously satisfies the definition of being a grounding base, that proposal would count everything as fundamental.

f is Afundamental =\(_{df}\)f belongs to every grounding base.
The idea is that whatever full account of the world we offer, f will be an indispensable part of that. Using terminology from Raven (2016, 609), we might also say that f is “fundamental qua ineliminable”. I shall adopt Afundamentality as my characterization of the fundamental in terms of allgrounding. (I shall say a bit more about this choice in the final two paragraphs of this section.)
Bennett (2017, 112) shows that Bfundamentality entails Afundamentality—that “[e]very independent entity is in every complete set”. As it turns out, the converse entailment holds as well, such that our two notions are equivalent:
Proposition 1
Suppose that ground is irreflexive. Thenfis Bfundamental\(\Gamma\)\(\Leftrightarrow\)fis Afundamental.
Proof
\(\Rightarrow\): Suppose that f is not Afundamental. Then there is some grounding base \(\Gamma\) to which f does not belong. It follows that there is \(\Gamma ' \subseteq \Gamma\) such that \(\Gamma '\) is a ground for f. Hence f is not Bfundamental.
\(\Leftarrow\). Suppose that f is not Bfundamental. Hence there is a ground \(\Gamma\) for f. By Irreflexivity, \(f \not \in \Gamma\). Now let \(\Delta\) be all the facts. Clearly, \(\Delta\) is a grounding base. Consider \(\Delta \setminus \{f\}\) (i.e. the settheoretic difference of \(\Delta\) and \(\{f\}\)), and let g be an arbitrary fact. If \(f = g\), then \(\Gamma \subseteq \Delta \setminus \{f\}\) is a ground for g. If \(g \ne f\), then \(g \in \Delta \setminus \{f\}\). So \(\Delta \setminus \{f\}\) is a grounding base. That is, f does not belong to every grounding base, and is not Afundamental. \(\square\)
The result that the two characterizations coincide is robust: it continues to hold if we let fundamentality depend on several building relations, rather than the one relation of ground, as I shall now show.
To generalize Afundamentality, we first introduce the notion of a building base. Say that \(\Gamma\) is a building base iff for every f that does not belong to \(\Gamma\), there is \(\Gamma ' \subseteq \Gamma\) and a building relation R such that \(\Gamma '\) stands in R to f.^{8}

f is Afundamental* =\(_{df}\)f belongs to every building base.
Notice that the definitions of the starred concepts are like those of the unstarred ones, except that the role of ground is now played by the disjunction of all building relations—the relation \(\bigvee \mathcal {B}\) that holds between \(\Gamma\) and f just in case there is some R among the building relations \(\mathcal {B}\) such that \(\Gamma\) stands in R to f. That relation \(\bigvee \mathcal {B}\) may well fail to satisfy the criteria for being a building relation. For example, it may not be asymmetric even if all building relations are asymmetric, and it may not be transitive even if all building relations are transitive. But provided that every building relation is irreflexive, \(\bigvee \mathcal {B}\) is irreflexive as well. The proof of Proposition 1 thus straightforwardly carries over to yield a generalized result:
Proposition 2
Suppose that every building relation is irreflexive. Thenfis Bfundamental* \(\Gamma\)\(\Leftrightarrow\)fis Afundamental*.
Propositions 1 and 2 only rely on the assumption of irreflexivity. Other standard assumptions, such as asymmetry or transitivity, are not needed; nor is the assumption that chains of ground end in ungrounded facts. Even if there is an “infinite descent” of ground, then Bfundamentality and Afundamentality coincide.^{9} However, it needs to be acknowledged that Afundamentality may not be a good explication of fundamentality in terms of the allgrounding in such a situation: the Afundamental things will not, collectively, be a grounding base. Each of them will be ungrounded, but they will not be jointly allgrounding.
It is not clear what should count as fundamental in the sense of allgrounding in a world of infinite descent.^{10} Perhaps the most plausible thing to say is that the notion is not defined relative to such a world. At any rate, I shall set such scenarios aside for the rest of this paper. Many philosophers take them to be incoherent anyway, and will not object to this restriction.^{11} Fans of infinite descent may already be open to the idea that characterizations of the fundamental in terms of being ungrounded and being allgrounding may diverge. But even for them, there is something new in the rest of the paper: the claim that the two characterizations can come apart even in the absence of infinite descent; and that they can come apart even though both are defined, rather than one being defined and the other not.
2 Disambiguating definitions of fundamentality
Afundamentality and Bfundamentality are stated using a generic notion of ground. Among the various distinctions in the family, at least one deserves attention in the present context: that between full ground and partial ground. I will discuss this distinction in more detail in a subsequent section; for now I only assume that ‘partial ground’ is used inclusively, to apply to full grounds too.
When we characterise the fundamental as the ungrounded, we presumably have the notion of partial ground in mind: only facts that are not even partially grounded are wholly ungrounded. It is natural to relate fundamentality as ungroundedness to a certain hierarchical conception of the world: an entity is fundamental, in the relevant sense, if and only if nothing is below it in the hierarchy. But this hierarchical structure of the world is typically articulated by the relation of partial ground—a binary relation that we can represent in a graph. (The relation of full ground, being plural on the lefthand side, does not lend itself to convenient pictorial representation.)
When we characterise the fundamental as the allgrounding, in contrast, we presumably have the notion of full ground in mind. In metaphysics, we are interested in identifying collections of facts which are complete, in the sense of accounting for everything. As the term ‘complete’ suggests, we are looking for full grounding bases—where ‘ground’ in the definition of a grounding base is read as ‘full ground’. The fundamentalia should provide more than merely partial grounds for everything else. This seems to be required by the theoretical role of fundamentality visàvis supervenience: why should a mere partial grounding base modally fix everything else?
To relate this discussion with our previous results, it will be useful to extend our terminology. I shall say that f is strongly Bfundamental iff no \(\Gamma\) partially grounds it; and weakly Bfundamental iff no \(\Gamma\) fully grounds it.^{12} Since any full ground of a fact is also partial ground, strong Bfundamentality entails weak Bfundamentality, as we might expect.
Further, I shall say that f is weakly Afundamental iff it belongs to every full grounding base; and strongly Afundamental iff it belongs to every partial grounding base. Since full grounds are partial grounds, every full grounding base is a partial grounding base, and hence strong Afundamentality entails weak Afundamentality. While the natural notion of completeness, which provides a constraint on the fundamentalia collectively, is the stronger one—being a full grounding base—membership in every strongly complete base is a weaker condition than membership in every weakly complete one.
The natural notions of fundamentality for individual facts are strong Bfundamentality and weak Afundamentality—marked in bold in the figure. Inspection of Fig. 1 confirms that strong Bfundamentality entails weak Afundamentality. However, the converse entailment has not been shown to hold. In the rest of the paper, I shall discuss under what conditions it does hold.
3 Full ground and partial ground
One of our candidate explications of fundamentality—strong Bfundamentality—is defined in terms of full ground, and the other—weak Afundamentality—in terms of partial ground. We need to take a closer look at these relations.
The distinction between full grounds and partial grounds does not map neatly to the various words in philosophers’ English that the idiom of ground is supposed to regiment—‘in virtue of’, ‘because’, ‘explains’, ‘accounts for’ and their cognates may all be used to express claims of partial or of full ground.^{13} The contrast is analogous to the one between a partial cause—which is often simply called “cause”—and a full cause—something that is typically too complex to cite.^{14} The distinction can be elucidated by examples. Let us suppose that Anna’s being my niece is grounded in Martin’s being my brother and Anna’s being Martin’s daughter. This is a case of full ground: the grounds fully account for what is grounded. It also makes sense to say that my being an uncle is grounded in Anna’s being Martin’s daughter. But here, we would have a case of partial ground: the ground contributes to what is grounded. Furthermore, it is a case of merely partial ground: the partial ground is not also a full ground. Likewise, my having a mass is a partial ground for my attracting the earth. But it is a merely partial ground, since a full ground would also include further facts, for example the earth’s having mass and the holding of the law of gravitational attraction.
There is a reason why in ordinary usage, locutions in the relevant family are often indicative of a connection of partial ground only. Once we go beyond toy examples like the above, we would often find it hard to specify full grounds. What is the fact that France is a nation grounded in, for example? Presumably, we could all cite, or at least gesture at, several partial grounds—facts about its internal organization, about international law, or, at a deeper level, about the intentions of its citizens. But it would be very difficult indeed to cite any full grounds—especially so if necessitarianism about full ground holds, that is, if the conjunction of the grounds needs to strictly imply what is grounded. For then the grounds will typically also include an indefinite number of facts relating to background conditions, or about the absence of potentially interfering factors. But even on a nonnecessitarian conception, full grounds are often too large to specify.
As has often been remarked, the notion of ground is closely related to that of metaphysical explanation. We are certainly familiar with the concept of a partial explanation. When we ask for an explanation of some fact, we may get an answer that partly satisfies what we are asking for. (Indeed, we typically do not request more than a partial explanation in the first place.) A merely partial ground does not fully account for the fact, but goes some way towards it. Or, to use another paraphrase, we can say that to be a partial ground for a certain fact is to contribute to its holding.
On what is arguably the orthodox account, put forward by Correia (2005, p. 60) and Rosen (2010, p. 115) among others, partial ground is defined in terms of full ground. In a slogan: a partial ground is a part of a full ground. Usually, ‘part’ is not understood in the strict mereological sense here. Rather, something is a partial ground if and only if it is one of some things that are, together, a full ground. Or to put it differently: to a be a partial ground is to be completable to a full ground.
Presumably, such a definition is not intended to be merely stipulative, concerning the use of a technical term. As I pointed out, many pertinent ordinary locutions are supposed to express partial ground. I shall thus treat the orthodox account as asserting a substantive claim about the relationship between full ground and partial ground.
The account is usefully broken down in two claims: that partial grounds are extractable—being part of a full ground is sufficient for being a partial ground—and that partial grounds are completable—being part of a full ground is necessary for being a partial ground. I shall not discuss the extractability condition.^{15} My concern will be with the question whether partial grounds are completable.
Or in the terminology introduced earlier: if f is weakly Bfundamental, then it is strongly Bfundamental.
If ground is dichotomous, then all of our four notions of fundamentality coincide, as inspection of the figure reveals. In particular, it will turn out that the explications of the fundamental as the ungrounded and as the allgrounding boil down to the same thing.
4 Merely partially grounded facts?
In general, a partial F need not be a part of a full F. If my employer gives me a partial reimbursement for my expenses, it does not follow that someone else will reimburse the remainder, alas. Likewise, the world is full of examples of partial success in the absence of full success. The question then is whether in the specific case where F is ‘ground’—more precisely, ‘ground for g’, for some given g—there always needs to be a full F.
[There is] a natural partial notion of ground for which a partial ground need not always be part of a full ground. One might wish to say, for example, that the truth that P is a partial ground for knowledge that P, even though there is nothing one might add to P to obtain a strict full ground for knowledge that P (as in the view of Williamson 2000). (Fine 2012a, p. 53)
I shall not discuss whether the truth of P should indeed count as a noncompletable partial ground of knowledge that P, on Williamson’s theory. Whether it is or not, it seems to me that the “knowledge first” theory does not provide us with a counterexample to Dichotomy.^{16} Certain facts about Fred’s brain, about his environment, and about how he causally relates to the environment are modally sufficient for Fred’s knowing that P.^{17} While Williamson does not use the idiom of ground himself, it is natural to say that those facts, or perhaps some relevant subclass thereof, jointly ground Fred’s knowing that P. What Williamson denies is that any (possibly disjunctive) brain conditions, or any (possibly disjunctive) environmental conditions, are necessary as well as sufficient for knowing that P. His claim is that knowing that P cannot be analysed with P as a conjunct, not that facts involving that property are ungrounded.^{18}
Similar remarks apply to cases that involve causation or action rather than knowledge. That c and e both happen may be a noncompletable partial ground of c’s causing e; and that my arm goes up may be a noncompletable partial ground of my raising my arm. In neither case, however, do we have good reason to think that there are no full grounds of the relevant facts.
I am, however, sympathetic to the idea that Dichotomy fails, and shall now sketch two potential counterexamples. The first one involves an imaginary scenario, and does not rely on specific metaphysical commitments; the second one turns on certain theoretical ideas about totality.
Consider the following scenario. In worlds \(w^+\) and \(w^\), a determinable quality, to be called “schmarge”, is instantiated. Schmarge is akin to electric charge in displaying a positive–negative polarity. The space of its determinates is simpler, though—it has only two rather than infinitely many determinates. We may call them “positive schmarge” and “negative schmarge”, but like in the case of electric charge, these labels are entirely arbitrary. Positive schmarge is no more similar to positive charge, or to positive growth, than it is to negative charge, or negative growth, respectively.
In worlds \(w^+\) and \(w^\), there are no infinitely descending chains of ground. Moreover, those worlds are mereologically atomic. Atoms tend to cluster in stable aggregates, which we may call “molecules”. The laws of nature of \(w^+\) and \(w^{}\) are such that a molecule has schmarge if and only if it is composed of an even number of atoms, all of which have the property F. That property is fundamental in both worlds, and every atom that has it does so contingently. Among the molecules with schmarge, some are positive, and some negative. However, for any given molecule with schmarge, it is a brute fact what its polarity is. There is no law linking its polarity with its composition, or with any of its other intrinsic features. (To be sure, there is a law linking the distribution of positive and negative schmarges with a field of intermolecular forces, but that field is generated by the positive and negative schmarges, rather than explaining them.)
Positive and negative schmarge exhibit a distinctive failure of supervenience. In \(w^+\), molecule a has positive schmarge. But in world \(w^{}\), a has negative schmarge. Yet every fundamental fact of \(w^+\) obtains, and is fundamental, in \(w^\)—except possibly the fact that a has positive schmarge, whose status as fundamental or nonfundamental shall not be prejudged. Likewise, every fundamental fact of \(w^{}\) obtains, and is fundamental in \(w^+\)—except possibly the fact that a has negative schmarge.
Prima facie, the schmarge scenario is metaphysically possible. Positive and negative schmarge do not supervene on the fundamental properties of their atomic parts, but their instantiation by a whole still has a certain configuration of the parts as a necessary condition.
It seems to me that the scenario describes a plausible possible counterexample to the dichotomy of ground. When describing it, I took care not to prejudge the question whether \([G^+a]\), the fact that a instantiates positive schmarge, is grounded in something, and if so, by what. We shall now ask that question.
It seems to me that \([G^+a]\) does not have any full grounds. There do not appear to be any suitable candidates around to play that role. Yet I would argue that \([G^+a]\) is partially grounded. Let [Ha] be the fact that a is composed of an even number of particles each of which has F. It is then plausible that in world w, \([G^{+}a]\) is partially grounded in [Ha], or perhaps by [Ha] together with a pertinent fact l concerning the laws of nature of w. After all, [Ha] and l together go some way towards explaining \([G^{+}a]\); and it is natural to say that \([G^{+}a]\) obtains because [Ha] does, or in virtue of [Ha] obtaining.^{19}
The world is determined by the facts, and by these being all the facts.
The addition of “these being all the facts” is prompted by the need to account for some negative and universally quantified truths. On one way of spelling out this idea, there is a plurality \(\Gamma\) of firstorder facts, and there is also a further fact t (for “totality”): that \(\Gamma\) are all the firstorder facts. The totality fact t thus entails each one of the facts in \(\Gamma\).
According to the most developed account of this kind, presented in D. M. Armstrong (2004, 72), the fact t involves a relation of alling or totalling. Specifically, it is the fact that the mereological fusion of all firstorder facts stands in that relation to the property of being a firstorder fact.^{20}
Supposing that this is correct, we can now ask what, if any, relationships of ground obtain between \(\Gamma\) and t.
The view under consideration belongs to the logical atomist tradition. The facts in \(\Gamma\) are taken to be logical atoms. In the key of ground, this can be expressed by saying that they are ungrounded. Specifically, none of them is grounded in t, either fully or partially.
Even if \(\Gamma\) consists of all the firstorder facts, it is possible for all the facts in \(\Gamma\) to obtain without t: there could be further firstorder facts, after all. This point was emphasized by Russell, and it presumably prompted Wittgenstein to add “and by these being all the facts” to “the world is determined by the facts”. On the widely held assumption that grounds necessitate what they ground, it cannot be the case that \(\Gamma\) is a full ground for t.
What is left open so far is whether \(\Gamma\) is a partial ground for t. It seems to me that it is: t is a complex fact which entails the obtaining of all members of \(\Gamma\); indeed, the members of \(\Gamma\) are naturally taken to be constituents of the complex fact.^{21} Hence t holds in part because the facts in \(\Gamma\) do, or in virtue of the facts in \(\Gamma\) obtaining.
But in the ontology in question, there is no plausible candidate for a full ground of t. We have seen that \(\Gamma\) is not—this is the reason why the totality fact has been introduced in the first place. Moreover, apart from t and the members of \(\Gamma\), there is not really anything else in the logical atomist’s ontology.^{22}
This second potential counterexample to Dichotomy could be resisted by adopting a different, purely exclusive account of the totality fact: as the fact that at most the actual firstorder facts obtain—a fact that does not entail any of them, and is arguably not partially grounded in any of them. We need to note, though, that this is indeed a different conception of totality facts than the inclusive one that is adopted in the relevant literature. Moreover, the change in conception may incur costs. When discussing the two conceptions, Kit Fine notes that he is “inclined to prefer [the inclusive conception] on theoretical grounds” (2012a, 62).^{23} Fine’s and Armstrong’s theoretical considerations—discussion of which would lead us too far afield—may or may not turn out to be weighty. The important point, though, is that if their account of totality were rejected to save Dichotomy, that principle would turn out to have revisionary consequences. This confirms my suspicion that the standard account of partial ground, which entails Dichotomy, is not a harmless regimentation of our talk of ground. Rather, it embodies substantive metaphysical commitments—a theme I shall take up again in the next section.
I started this section by reporting a putative counterexample to Dichotomy due to Kit Fine. While I do not find the example itself convincing, I wish to draw on Fine’s work to bolster my claim that Dichotomy may fail. Fine suggests his example in the course of distinguishing various different notions of partial ground that are definable in his semantics. At a minimum, his framework can be used to show that a notion of partial ground that violates Dichotomy is logically wellbehaved.
The semantics involves some entities—“verifiers” or “truthmakers”—standing in mereological relations. As I show in the “Appendix”, Dichotomy is closely related to the holding of the socalled “Weak Supplementation” principle among the verifiers. Such a principle rules out models, inter alia, where something has just one proper part. The totality fact t, in the above example, is naturally taken to have a verifier which has the fusion of \(\Gamma\), the firstorder facts, as a proper part, but no parts which do not overlap that fusion.
When we compare “red thing” and “colored thing” we find that the latter is contained in the former, but we cannot specify a second thing that could be added to the first as an entirely new element, i.e., one that would not contain the concept, colored thing. (1981, 112)
Another precedent is provided by Whitehead, on whose theory open interiors are proper parts of topologically closed regions, even though there are no boundaries which could serve as supplements.^{24}
In the more recent literature, further potential counterexamples to Weak Supplementation have been explored. It would be beyond the scope of this article to survey that work.^{25} Some of these examples might be adapted to challenge Dichotomy directly; others indirectly, via Fine’s mereological models for ground discussed in the “Appendix”.
5 Weak and strong fundamentality
If I am right that Dichotomy may fail, we need to distinguish between weak and strong fundamentality. While it is beyond the scope of the present paper to explore the ramifications of such a move, I shall close by sketching one way in which it might be significant.
Weak and strong fundamentality will split the theoretical role described at the beginning of this article. Weak fundamentality constrains ontology: accepting only the strongly fundamental entities is not enough to meet one’s commitments. Likewise, it is weak fundamentality that limits modal recombination: everything supervenes on the weakly fundamental. In contrast, it is strong fundamentality, at best, that guarantees modal recombination.
Recognising that these roles are played by different properties may prompt us to reassess a number of metaphysical theses and arguments. Consider Humeanism in metaphysics. To a first approximation, contemporary Humeanism holds that there is a modally recombinable basis for everything. The paradigmatic representative is David Lewis, whose “pointillism” takes the recombinable basis to consist of instantiations of perfectly natural qualities at spacetime points. But the view comes in a number of different versions. The members of the basis may be objects, or properties, or facts, or entities of yet another category. The key relation between the basis and the rest may be supervenience, truthmaking, dependence, ground, building, or yet something else. Some view in the vicinity of Humeanism is shaping the view of a number of contemporary metaphysicians, regardless of whether they label their view as “Humean”.
The principle that the members of the basis are recombinable is a powerful one: it has been used to argue that there is no such thing as a structural universal (Lewis 1986), or a totality state of affairs (Cameron 2010), and that facts involving the building relations are themselves built (Bennett 2017, 190). Moreover, the principle has played a key role in recent arguments for prioriy monism, the view that one and only entity is fundamental (Cameron 2010; Schaffer 2010). Priority monism is compatible with the principle: if modal recombinability is just the lack of necessary connections among distinct members of the basis, then a basis consisting of one element is trivially recombinable. Its proponents argue that given certain other plausible background views, it is the only view compatible with it.^{26}
 (1)
The fundamental facts are modally recombinable.
 (2)
The fundamental facts form a grounding base.
But if the argument of this paper is right, premise (1) in the master argument for Humeanism plausibly holds if we are talking about Bfundamentality, and premise (2) if we are talking about Afundamentality. For the argument to go through, Bfundamentality and Afundamentality need to coincide. But they may not, as I have argued in this paper. If that argument has succeeded, Humeanism is no longer mandatory, and the set of options for our fundamental world view has correspondingly expanded.
Footnotes
 1.
For a recent defence of the strategy of using grounding to characterise fundamentality, see Mehta (2017).
 2.
Wilson uses uppercase “Grounding” for ground, and “grounding relations” (lower case) for the building relations. I take it that that difference is merely notational.
 3.
The explication matches “the strongest sense of independence” that Bennett contrasts with “notions of independence \(\ldots\) that are indexed to a particular building relation”: “A thing (fact, property, what have you) is independent in this strongest sense just in case nothing builds it in any way whatsover; it is not the output of any building relation.” (Bennett 2017, 112)
 4.
The discussion does not carry over, at least not immediately, to a discussion of the relationship between the ground structure among facts and the fundamentality of entities that are constituents of these facts is in question—the main theme of Raven (2016). For that reason, the frameworks of this article and that of Raven’s cannot be straightforwardly compared.
 5.
Compare Bennett (2017, 109): “[T]he complete entities are those that build, not everything, but everything else” (emphasis hers).
The notion of a ground base is one conjunct in Kit Fine’s notion of a distributive ground for everything (Fine 2012a, p. 54). Moreover, given plausible assumptions, we can show that \(\Gamma\) is a grounding base just in case for every f, there is \(\Gamma ' \subseteq \Gamma\) such that \(\Gamma '\) is a weak ground, in Fine’s sense, for f.
 6.
Bennett (2017, 123) claims that “independence [Bfundamentality*] seems to me better suited as the central concept of fundamentality.” Her reason is that in a “flat” world, where no facts stand in the relation of ground, we count everything as fundamental because it is ungrounded, despite its failing to ground anything else. This argument seems to me to show only that Bfundamentality is the central distributive, rather than collective, concept of fundamentality. Suppose that we are asked, of all the facts in a flat world, why they are, collectively, the fundamental facts. It then seems to be a good answer to point out that nothing else fails to have a ground among them.
 7.
 8.I think that the notion of a building base matches the general notion of completeness that Bennett has in mind:
The more intuitive notion of completeness is \(\ldots\) one that quantifies over building relations: the complete set at a world is the set whose members build, in one way or another, everything else. (Bennett 2017, 110)
 9.
 10.
A rival characterization to my Afundamentality—one arguably favoured by Bennett, although she is not explicit on the matter—is belonging to some minimal grounding base. A detailed comparison between these two proposals would be technical and somewhat orthogonal to the main point of the present paper. I shall merely point out that provided ground satisfies an infinitary version of transitivity (“Cut” in Fine 2012b), then it can be proved that every fact that belongs to some minimal grounding base is Afundamental. Moreover, facts that are Afundamental but do not belong to any minimal grounding base are, intuitively, good candidates for being fundamental. In short: the notions will typically coincide, and when they do not, I think Afundamentality is preferable.
 11.
 12.
Following Litland (2018), I here use a notion of partial ground that is plural on the left.
 13.
For relevant discussion concerning ‘because’, see Schnieder (2011, 450).
 14.
 15.
For some examples that might be taken to motivate the rejection of that condition (although they are put forward for a different purpose by the author), see Schaffer (2012, 126–9).
 16.
Here I am indebted to comments by David Liggins.
 17.
In the words of Williamson (2000, 80): “[L]et \(\alpha\) be any case in which a prime condition C obtains, and D and E respectively the strongest narrow condition and the strongest environmental condition which obtain in \(\alpha\). Then it is plausible that the conjunction D \(\wedge\) E entails C”.
 18.
A necessary and sufficient condition would be a disjunction of conjunctions of internal and external conditions, which is not in general equivalent to a conjunction of a disjunctive internal condition and a disjunctive external condition.
 19.
The distinctive kind of explanation in question is metaphysical explanation. Metaphysical explanation does not depend on the conversational context or our epistemic situation. We can thus rule out a contrastive reading of the demand for an explanation of \([G^{+}a]\), along the lines of “why does a have positive schmarge rather than negative schmarge?”
 20.
Armstrong uses the term ‘state of affairs’ where I use ‘fact’.
 21.On Armstrong’s theory (2004, 72–73), they may not be: parts of a mereological sum are not in general constituents of that sum. But the very fact that they are not—that the first relatum of the alling relation is a mereological sum, not some more structured entity—creates trouble for Armstrong’s theory. He proposes to use alling in the analysis of universally quantified statements, as follows:

“All As are Bs” is true iff there is something that alls A and also alls \(A \wedge B\).
But this analysis gives absurd results. Consider “All parts of the universe have a mass of at least 1000 kg,” which is falsified by people, pens, and particles, among others. But it comes out as true under the analysis: there is something, namely the universe, that is the fusion of all parts of the universe, and hence stands in the alling relation to “being a part of the universe”; and that very thing is also the fusion of all parts of the universe that have a mass of at least 1000 kg, and hence stands in the alling relation “being a part of the universe that has a mass of at least 1000 kg”.
Things get worse. “All parts of the universe have a mass of less than 1000 kg” also comes out true, under the assumption that the universe has a decomposition into things that each have a mass of less than 1000 kg. (A decomposition into protons, neutrons, electrons, and kindred particles would do the trick.) Together, these two general claims propositions entail the absurd claim that nothing is a part of the universe.
The problem would not arise with classes in the place of fusions.

 22.
Armstrong accepts higherorder states of affairs, but they are supposed to be a “free lunch” given t; in our terminology, they are grounded in t, rather than being grounds of t.
 23.
Fine eschews appeal to totality facts in favour of appeal to totality claims, but that difference is immaterial here.
 24.
At least, this is so on the formalization of Whitehead’s theory in Clarke (1981). See Simons (1987, 97–98) for some discussion.
Drawing on an unpublished talk of mine, Jessica Wilson (2018b, 2019) mentions such putative failures of weak supplementation in connection with partial grounds that are not parts of full grounds.
 25.
See Varzi (2016) for references.
 26.
Cameron uses Humeanism to argue against totality states of affairs, as just noted, and he argues further that a plurality without a totality state of affairs would not be base in the relevant sense. Schaffer’s argument from Humeanism to monism has too many moving parts to be summarised here; for a useful discussion, see Raven (2016).
 27.
For a detailed discussion of alternative ways of motivating something like Humeanism, see Wang (2016).
 28.
For a careful discussion of these three notions and their relationship, see Hovda (2009).
 29.
See Kleinschmidt (2017) for discussion.
 30.
I often leave relativization to a mereological frame implicit.
 31.
When giving the truthconditions for claims of ground, I shall use settheoretic vocabulary, due to its familiarity and convenience. (Though I shall sometimes write f instead of \(\{f\}\) to avoid clutter.) Officially, the relation of ground is plural on the left, rather than relating a set of facts to a fact.
 32.
Fine says “weak ground”, suggesting that a species of ground is being defined, where I say “protoground”, suggesting an auxiliary device is being introduced. There is no need to address the question what the status of weak ground is.
 33.
To adapt the proof of Proposition 3, note that any g with \(0 \in g\) will be zerogrounded, and thus fully grounded; and that whenever \(\Gamma\) is a full protoground for f and \(0 \in g \in \Gamma\), then \(\Gamma \setminus \{g\}\) will also be a full protoground for f.
 34.
Varzi (2014, 48) claims that “everything is identical with the fusion of its proper parts” is entailed by “no composite things have exactly the same proper parts” together with “any nonempty collection of things \(\ldots\) has a fusion.” However, in a structure consisting of b and its sole proper part a, b is not the least upper bound of the set \(\{a\}\) of its proper parts. Varzi defines a fusion as “something that has all those things as parts and has no part that is disjoint from each of them”. In that sense, both a and b are fusions of the proper parts of b, such that we cannot speak of “the fusion” here.
 35.
Dixon (2016a) discusses a supplementation principle for ground, but in a context where partial grounds are defined as parts of full grounds, and where Dichotomy is thus guaranteed to hold; the question he addresses is whether a partial ground f can be turned into a full ground by adding further facts that do not overlap f, in a sense of overlap that he defines.
Notes
Acknowledgements
Thanks to audiences in Manchester, Varano Borghi, Barcelona, and Edinburgh. For comments and discussion, I am also grateful to Umut Baysan, Darragh Byrne, Scott Dixon, Stephan Krämer, David Liggins, Roberto Loss, Fraser MacBride, Neil McDonnell, Fiona Macpherson, Louis de Rosset, Joshua Schechter, Bruno Whittle, Jonas Werner, Alistair Wilson, as well as anonymous referees. This work was supported by the Carnegie Trust for the Universities of Scotland; the John Templeton Foundation [Grant Number 40485]; and the Arts and Humanities Research Council [Grant Numbers AH/J004189/1, AH/M009610/1].
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