# What Grounds What Grounds What

What Grounds What Grounds What Abstract If there are facts about what grounds what, are there any grounding relations between them? This paper suggests so, arguing that transitivity and amalgamation principles in the logic of grounding yield facts of grounding that are grounded by others. I develop and defend this view and note that combining it with extant accounts of iterated grounding commits us to seemingly problematic instances of ground-theoretic overdetermination. Taking the superinternality thesis as a case study, I discuss how defenders of this thesis should respond. It emerges that our discussion puts pressure on superinternalists to make an interesting qualification to their view: to only regard as a fundamental metaphysical law a version of the superinternality thesis that is restricted to minimal and immediate grounding. I. INTRODUCTION If there are facts about what grounds what, are there any grounding relations between them? Plausibly so. Suppose we want to explain why some portion of physical reality grounds the existence of a city. One good strategy is to trace the steps in the grounding chain between that portion of physical reality and the city: first, certain physical facts involving particles ground facts about larger physical objects (atoms, molecules…), which ground certain chemical facts, which ground facts about basic biological entities, which ground facts about organisms … (passing through the psychological and sociological realms) … and that is why this portion of physical reality gives rise to a city. We can conduct a more systematic discussion by connecting the question to issues in the logic of grounding. Two plausible claims then heave into view. Suppose that a fact A grounds another fact B and that B grounds a third fact C. Assuming that grounding is transitive, it follows that A grounds C. First claim: A grounds C in virtue of these mediating links in the grounding chain from A to C. It is because A grounds B and B grounds C that A grounds C.1 Suppose now that A grounds C and B grounds C. Assuming that the grounds of a fact always combine to yield another of its grounds—a principle that Fine calls amalgamation (Fine 2012: 57)—it follows that A and B together ground C.2 Second claim: A and B together ground C in virtue of the facts that A grounds C and B grounds C. It is because A and B individually ground C that they combine to yield a ‘bigger’ ground for C. These intuitively plausible claims suggest that the transitivity and amalgamation principles, assuming them to hold, yield facts of grounding that are grounded by others. Metaphorically speaking, the amalgamation principle allows us to ‘widen’ facts of grounding and the transitivity principle allows us to ‘lengthen’ them. The above claims suggest a picture on which the ‘narrower’ and ‘shorter’ facts of grounding are prior to the ‘wider’ and ‘longer’ facts of grounding that the transitivity and amalgamation principles yield when applied to them. I label this intuitive picture minimalism—a more precise formulation of minimalism is provided in Section III, and refined in Section IV. It is standard in the grounding debate to distinguish, on the one hand, immediate from mediate grounding connections (Fine 2012: 50–1) and, on the other, minimal from non-minimal grounding connections (Scott-Dixon 2016). These distinctions are firmly rooted in our intuitive thought about grounding—and explications of them will be offered in Section IV. Intuitively, an immediate grounding connection between two facts is direct; it does not need to be mediated by other grounding connections. That the apple is green directly grounds that the apple is coloured. But fundamental physical facts only indirectly, or mediately, ground the existence of cities. Where some facts minimally ground another, they do so without superfluity: the former facts suffice to ground the latter, and would not have grounded it if anything were removed from them (or, at least, not in the same way—see Section IV).3 The concepts of minimal and immediate grounding are absolute. Metaphorical talk of grounding connections being ‘longer’ and ‘wider’ expresses comparative cousins of these concepts: the longer grounding connections are less immediate than shorter ones, and wider grounding connections are less minimal. This paper aims to support minimalism and explore its bearing on other theories of iterated grounding—that is, theories that tell us when and how facts of grounding are grounded.4 A programmatic idea that underlies the discussion is that our theory of iterated grounding should be partly driven by our logic of grounding. This view would be strengthened if other logical principles, besides the transitivity and amalgamation principles, systematically yield derivative facts of grounding, but I leave discussion of this sort of extension to the project for another occasion.5 I focus on the transitivity and amalgamation principles because, firstly, we do well to start our inquiry focussing on the clearest cases we can find, and these strike me as the clearest cases in which logically derivative facts of grounding are grounded by more basic facts of grounding. Secondly, minimalism is, metaphorically speaking, the view that the big facts of grounding derive from the small. That we readily identify two dimensions along which facts of grounding can be big in this metaphorical sense suggests a degree of conceptual unity between them. The discussion proceeds as follows. Section II introduces some concepts and notational conventions. Section III tries to formulate minimalism with more precision and to motivate it. Section IV discusses a couple of problem cases for minimalism, and responds by incorporating the concept of a grounding path into minimalism's formulation. There are, in the literature, some general proposals about how facts of grounding—that is, all facts of grounding—are grounded by other kinds of fact.6 None of these entail either of the two claims we opened with so, if minimalism is correct, there are instances of iterated grounding that are not entailed by extant theories of it. It is not obvious that these general theories coexist peacefully with these instances. The worry is that combining any of the extant theories with minimalism commits us to problematic instances of ground-theoretic overdetermination. I explore this issue in Section V, using the superinternality thesis, defended by Bennett (2011) and deRosset (2013), as a case study. There are, indeed, several theories of iterated grounding on the market. I focus on the superinternality thesis because it is a prominent theory of iterated grounding and because discussing it does not require us to introduce controversial metaphysical concepts (such essentialist concepts), besides ground-theoretic concepts. This makes discussing the superinternality thesis comparatively straightforward, and I indicate the extent to which Section V’s discussion generalizes to competing theories of iterated grounding. Section VI provides concluding remarks. II. PRELIMINARIES Grounding is a concept of non-causal and explanatory determination, often expressed in metaphysical discussions with the phrase ‘in virtue of’. To say that a fact fully grounds another is to say that the former is wholly responsible for the latter, in the relevant non-causal sense. Where a fact partially grounds another, the former helps to bring it about that the latter obtains, but may not suffice to do so (for more by way of introduction to the topic of grounding, see Bliss and Trogdon 2014; Correia and Schnieder 2012; Rosen 2010: §1 and 2). I assume a broadly Finean background theory of grounding, although the discussion should generalize to other comparable frameworks. Adopting Fine's notation, I use the symbol ‘<’ as a sentential connective to express the concept of full grounding.7 ‘<’ takes variably long lists of sentences on its left and single sentences on its right. I use capitalized letters ‘A’, ‘B’… as sentential variables and capital Greek letters ‘Γ’, ‘Δ’ … as plural variables ranging over lists of sentences. Although listing sentences may not be the same as referring to a set of sentences, I will occasionally use familiar set-theoretic concepts in reasoning about lists. Like Fine, I assume that (plural) sentential quantification is intelligible. The general form of claims of full grounding is, then, as follows:   \begin{equation*}{\rm{\Gamma }} < {\rm{A}} \end{equation*} A claim of this form may be read: A wholly in virtue of the facts in Γ. The facts of grounding are just those facts, or truths, that have the above form (I use ‘fact’ interchangeably with ‘truth’). A fact of iterated grounding is a fact of the form ‘Γ < (Δ < A)’. A claim of iterated grounding is a claim to the effect that such a fact obtains. And a theory of iterated grounding is a theory that entails a claim of iterated grounding. We can contrast two kinds of theory of iterated grounding. What I will call the general theory investigates how and when facts of grounding are grounded by facts that are not facts of grounding. The central question for the general theory is whether grounding is a fundamental phenomenon and, if it is not, how facts of grounding are grounded by more fundamental matters. The general theory contrasts with the special theory, which investigates grounding relations only within the class of facts of grounding. The central question of the special theory is whether facts of grounding ever ground others. More precisely, say that a theory of iterated grounding is general just in case every claim it entails of the form ‘Γ < (Δ < A)’ is such that no member of Γ is a claim of grounding; and special just in case every claim it entails of the form ‘$${\rm{\Gamma}} < {\rm{}}({\rm{\Delta}} < {\rm{A}})$$’ is such that every member of Γ is a claim of grounding. The two claims that we began with constitute a rudimentary special theory. A main thesis of this paper is that, whatever our general theory, the logic of grounding should guide our special theory.8 Distinguishing the special and general theories allows for a division of labour in the theory of iterated grounding. Suppose G is the set of all and only the facts of grounding and we have a special theory according to which there is a certain set K meeting the following conditions: a) K⊂G b) $$\forall {\rm{A}}( {\rm{A}} \in {\rm{G\ }}\& {\rm{\ }}\sim ( {{\rm{A}} \in {\rm{K}}} ) ) \to \exists {\rm{\Gamma }}({\rm{\Gamma }} \subseteq {\rm{K\ }}\& {\rm{\ \Gamma }} < {\rm{A}})$$ c) $$\forall {\rm{A}}( {{\rm{A}} \in {\rm{K}}} ) \to \sim \exists {\rm{\Gamma }}({\rm{\Gamma }} \subseteq {\rm{G\ }}\& {\rm{\ \Gamma }} < {\rm{A}})$$ Condition (a) says that K contains only facts of grounding; (b) that every fact of grounding that is not in K is grounded by members of K; and (c) that no member of K is fully grounded by facts of grounding. Intuitively, the special theory we are considering claims that K forms a minimal grounding base for the facts of grounding. We then turn to the general theory. Suppose we aim to show that every fact of grounding is grounded by some other kind of fact (an essentialist fact, say). To establish this, we need a general theory that provides plausible grounds for the members of K, but we do not even need to discuss the other facts of grounding, because our special theory provides an account of how these are grounded by the members of K. Grounding is transitive, so whatever grounds the members of K will also ground the facts of grounding that are not in K. But our theory of what grounds the members of K need not apply directly to the facts of grounding not in K. For all we know at the outset, it might only provide grounds for these in conjunction with our special theory. Perhaps the best general theory directly applies to all facts of grounding, but that it does is a substantive thesis about iterated grounding, which the thesis that there are no fundamental facts of grounding does not obviously entail. It is, then, sensible to develop the special theory before the general theory; for only with a special theory in hand will we have a clear idea of which facts we need our general theory to directly apply to. III. MINIMALISM: FORMULATION AND MOTIVATION The two claims we opened with seem immediately plausible. Insofar as they suggest the general view that the facts of grounding to which the transitivity and amalgamation principles are applied are prior to those the principles yield, minimalism is a plausible default view. Still, it is instructive to trace minimalism's plausibility to that of some more general metaphysical ideas. The idea that grounding is transitive can be made precise in various ways. A strong transitivity thesis is as follows (outer universal quantifiers are omitted for readability, where it is obvious how to add them): Trans: If Γ < A and A,  Δ < B then Γ,  Δ < B I assume that grounding is indeed transitive in this strong sense. It might be suggested that Trans is trivial, since ‘<’ should be understood as the transitive closure of immediate grounding, which is trivially transitive (see Fine 2012: 51). In my view, this definition of ‘<’ is neither mandatory nor attractive. It is not mandatory because our intuitive concept of grounding is robust—we do not have to introduce ‘<’ as the transitive closure of immediate grounding. The definition is not attractive because it rules out the possibility of dense grounding, understood as follows: Γ densely grounds A iff $${\rm{\Gamma }} {< }{\rm{A }}\& {\rm{}}\forall {\rm{\Delta }}( {{\rm{\Delta }} {<} {\rm{A}} \to \exists {\rm{{\rm K}}}\exists {\rm{B}}( {( {{\rm{\Delta }} < {\rm{B}}} )\& ( {{\rm{{\rm K}}},{\rm{\ B}} {<} {\rm{A}}} )} )} )$$ If Γ densely grounds A, there are no links of immediate grounding between Γ and A, and so Γ does not bear the transitive closure of immediate grounding to A. So, if ‘<’ expresses the transitive closure of immediate grounding, the pair of conditions specified by the definition of dense grounding can never both obtain, and so dense grounding is impossible. But dense grounding seems possible, so we should resist the analysis of ‘<’ in terms of immediate grounding. In any case, it seems to me that the discussion below does not depend on any thesis about the epistemic status of Trans (or of the amalgamation principle, discussed below). I take the question of whether Trans is trivial to be orthogonal to the issues discussed here.9 The first component of minimalism says that Trans tracks grounding connections: it licenses the inference of a derivative fact from facts that ground it, or would do if they obtained. This idea is naturally expressed by the following thesis: Trans-Ground: If Γ < A and A,  Δ  < B then (Γ < A),  (A,  Δ < B) < (Γ,  Δ < B) It follows from Trans-Ground that if A < B and B < C then (A < B), (B < C) < (A < C), which is the first intuitive idea we started with; but Trans-Ground also applies in cases that are not one-one. Trans-Ground is analogous to principles connecting grounding to the logical constants—that, for instance, if A and B then A, B < (A and B) and that if Fa then Fa < Something is F. These principles tell us that certain classical inference rules (conjunction introduction, existential generalization) track grounding connections. Trans-Ground is similar, except the inference rule being claimed to track grounding connections is not classical but one in the logic of grounding. We can motivate Trans-Ground by appealing to the plausible idea that general facts are grounded by their more specific realisers. Consider the following plausible grounding claims: 1) Ball is scarlet < Ball is red 2) A < (A or B) 3) Fa < There are Fs These claims are aptly described using the locution ‘one way (among others)’. One way for the ball to be red is for it to be scarlet. One way for it to be true that A or B is for it to be true that A. And one way for there to be Fs is for a to be F. We tend to think that where some fact realises another, in this broad sense, the former grounds the latter; alternatively put, where some fact is a specific way for another fact to obtain, it is plausible that the latter's obtaining is grounded by the former's. Note that the locution ‘one way (among others)’ does not require, for its applicability, that the contrasting other ways be metaphysically possible; it can apply when there is only one metaphysically possible way for a given fact to obtain. On the standard account of the grounding of disjunctions, if A then A < (A or Not-A), whatever the modal status of the fact that A. If it is necessary that A then the fact that A or Not-A is necessarily not grounded by the fact that Not-A. Yet, the disjunction is still less specific than the disjunct, in the relevant sense. One way for A to ground C is for A to ‘go via’ B. Another (perhaps not metaphysically possible) way would be for A to ground C immediately. More generally, one way for it to be the case that Γ, Δ < B is for it to be the case that Γ < A and A, Δ < B and so—by Trans—Γ, Δ < B. The fact that Γ, Δ < B seems in the relevant sense less specific than the facts that Γ < A and A, Δ < B: we know more about the grounding hierarchy if we know that Γ < A and A, Δ < B than if we just know that Γ,  Δ  < B −  just as we know more about the apple if we know that it is scarlet than if we know merely that it is red. This asymmetry of informational content, I suspect, helps explain the plausibility of the first intuitive claim that we opened with. Trans-Ground, if it holds, is a general principle governing grounding connections; it is a generalization of the form ‘∀Γ∀A(φ → (Γ <  A))’, where ‘φ’ is a schematic sentence letter. I will use the label metaphysical law for all and only truths of this form, and their natural language equivalents.10 Examples 1–3 also seem law-like: all disjunctions are grounded by their true disjuncts; all facts of the form ‘… is red’ are grounded by facts about determinates of red; and all existentially general facts are grounded by their instances. But there is a notable difference between 1–3 and the grounding claims implied by Trans-Ground. The facts on the right of 1–3 each have a form, such that it is a metaphysical law that facts of that form are derivative. We can look just at the facts on the right of 1–3 and, as long as we know the relevant metaphysical laws, we will be in a position to know that these facts are derivative and how, broadly speaking, they are grounded.11 By contrast, it does not have to do with the form of the fact that A < C that it is grounded, as per Trans-Ground, by the facts that A < B and B < C. Not all facts of the form ‘Γ <  A’ are grounded in this manner, since some grounding connections are immediate. Does this disanalogy threaten my strategy of trying to motivate Trans-Ground by appealing to analogies between the facts of grounding it implies and examples 1–3? Not obviously. It is not obvious that whether and how a fact is grounded—and which metaphysical laws apply to it—can always be ‘read off’ from its form. What is more important, it seems to me, is whether the ‘one way (among others)’ locution applies and it seems evident that one way for A to ground C is for A to ground B and for B to ground C. That we cannot tell whether A grounds B and B grounds C if we only know that A grounds C is not obviously relevant. Turn now to minimalism's second component. The amalgamation rule is as follows (Fine 2012: 54–7): Amalg: If Γ < A and Δ < A … then Γ, Δ… < A Amalg tells us that collections of facts that fully ground A always combine to yield another full ground for A. The most obvious way to articulate the intuitive claim that Amalg yields derivative facts of grounding is as follows: Amalg-Ground: If Γ < A and Δ < A … then (Γ < A), (Δ < A)… < (Γ, Δ…  <  A) This captures the idea that Amalg, assuming it to be valid, licenses the inference of a derivative fact from facts that ground it, or would ground it if they obtained. Like Trans-Ground, Amalg-Ground is motivated by plausible general ideas. First, the same general-specific considerations apply here: we paint a richer picture of the grounding hierarchy if we say that A < C and B < C and hence—by Amalg—A, B < C than if we say merely that A, B < C. Amalg-Ground is also an instance of the plausible idea that macroscopic facts are grounded by microscopic ones. We tend to expect that facts about composite objects and pluralities are grounded by facts about their parts and sub-pluralities; as Kim puts it, this is ‘the metaphysical principle underwriting the research strategy of microreduction and the method of microreductive explanation’ (Kim 1994: 67). Assuming that Amalg itself holds, we naturally think that the ground-theoretic relations that a plurality of facts enters into can be determined by the ground-theoretic relations entered into by its subpluralities—in particular, that the facts of grounding yielded by Amalg are grounded by the those that Amalg is applied to. A further idea that motivates both Trans-Ground and Amalg-Ground (assuming Trans and Amalg themselves to hold) is that the fundamental facts in a given domain are those that suffice to minimally characterize that domain (Schaffer 2009: 377). Given Amalg and Trans, a host of facts of grounding (mediate and non-minimal ones) supervene on the class of immediate and minimal facts of grounding. More generally, Trans and Amalg fix the less immediate and minimal facts of grounding on the basis of the more immediate and minimal ones. It is then plausible to conjecture that the more minimal and immediate facts of grounding ground the less minimal and immediate ones. Supervenience is distinct from grounding, but it is often plausible that the presence of supervenience relations indicates an underlying grounding relation (Kim 1993: 167). And the supervenience in the case at hand is not mutual: we cannot recover the smaller facts of grounding from the bigger ones. We cannot, for instance, infer from the fact that Γ, Δ < A that either Γ < A or Δ < A. This has to do with the fact that there is nothing distinctive about the form of facts of grounding that determines whether they are derivative, by the lights of Trans-Ground or Amalg-Ground. IV. GROUNDING PATHS Our discussion so far has been friendly to minimalism. But minimalism, understood as the conjunction of Trans-Ground and Amalg-Ground, has plausible counterexamples.12 Begin with a problem for Trans-Ground. Given that x is scarlet < x is red and that x is red < x is coloured, Trans-Ground implies that SCA: (x is scarlet < x is red), (x is red < x is coloured) < (x is scarlet < x is coloured) SCA tells us that the fact that x is scarlet < x is coloured is grounded by the facts that x is scarlet < x is red and x is red < x is coloured. But intuitively, an object's being scarlet grounds its being coloured directly. Being scarlet is a determinate of both being coloured and being red. The relevant metaphysical law here is that if F is a determinate of G, then whenever something is F, its being F makes it G. No mention of intermediate grounding steps is made in stating this law and, intuitively speaking, it does not seem that x is F grounds x is G by virtue of grounding other determinates of G.13 Trans-Ground appears to yield dependencies when intuition says there are none.14 The second problem case targets Amalg-Ground. Consider the fact that A, B < (A or B or (A and B)). Given that A < (A or B or (A and B)) and B < (A or B or (A and B)), Amalg-Ground tells us that AMA: (A < (A or B or (A and B))), (B < (A or B or (A and B))) < (A, B < (A or B or (A and B))) AMA tells us that it is because A grounds the fact that A or B or (A and B), and B grounds the fact that A or B or (A and B), that A and B together ground A or B or (A and B). But this seems wrong. Rather, A, B jointly ground the disjunctive fact by virtue of grounding its third disjunct. To reinforce the point, consider a different disjunction: C or D or (A and B). We might reasonably think that the explanation for why A, B ground A or B or (A and B) should be the same as the explanation for why A, B ground C or D or (A and B). Both problem cases trade on the fact that one fact can ground another in different ways or, as I will say, via different paths. This notion is analogous to that of a causal path and can be illustrated with an example: A < (A or (A and B)) In a sense, A grounds the fact that A or (A and B) twice over: A fully grounds A or (A and B) ‘via’ the latter fact's left disjunct and partially grounds it (in combination with B) ‘via’ the right disjunct.15 Distinguishing paths between A and A or (A and B) enables us to make sense of this sort of idea. Grounding paths can be represented formally by structural equation models for grounding (Schaffer 2016), just as structural equation models for causation can represent causal paths. Representing such models graph-theoretically, causal paths appear as edges connecting nodes representing causally relevant variables in the model (see Weslake forthcoming: §2.). I will not try to provide a rigorous formal representation of paths here; nor will I provide anything like a complete metaphysics of paths, or answer every good question that arises about them. My aim is just to say something illuminating about how paths bear on the theory of iterated grounding and about how path-sensitive notions of grounding interact with more familiar ones.16 To that end, introduce a triadic locution ‘<p’. ‘<p’ is like ‘<’, except that it has a subscripted argument place for path-denoting terms. I will understand the subscripted argument as singular, using ‘p’, ‘p0’, ‘p1’… as singular variables ranging over paths. ‘Γ grounds A via path p’ or ‘Γ grounds A in the manner p’ are English renderings of ‘Γ <p A’. By quantifying over paths, I treat them as reified features of the grounding hierarchy. A plausible ontological view—suggested by our talk of ways for a fact to ground another fact—is that paths are properties that facts of grounding instantiate. Admittedly, this view does not sit well with Fine's project of developing an ontologically neutral account of grounding, but I will not aim to provide an ontologically neutral account of paths here. I assume that ‘<’ and ‘<p’ are intelligible, and will make free use of both expressions in what follows (I use ‘facts/claims of grounding’, from now on, so as to include facts/claims of the form ‘Γ <p A’). ‘<’ probably comes closer to expressing a pre-theoretical concept of grounding than ‘<p’. But ‘<p’ is arguably conceptually prior to ‘<’, since ‘<’ can be recovered from ‘<p’ by existentially binding the path variable:   ${\rm Equiv}{:} \, \Gamma < {\rm A \,iff} \exists {\rm p}\,(\Gamma <_{\rm p} \;{\rm A})$ This says that Γ grounds A simpliciter iff Γ grounds A via some path or other, which seems hard to deny. On the other hand, it is hard to see how we might define ‘<p’ in terms of ‘<’. Given Equiv, we can see that there are certain interactions between ‘<’ and ‘<p’. In particular, the following thesis (which plays an important role in the next section) is plausible: Gen: If Γ < (Δ <p A) then Γ < (Δ < A) We can argue for Gen by appealing to the idea that Δ’s grounding A in a specific way grounds Δ’s grounding A in some way or another; that is, if Δ <p A then (Δ <p A) < (Δ < A). This is an instance of the plausible idea that existential generalizations are grounded by their instances. Gen then follows: any ground of a fact of the form Δ <p A will, given Trans, also ground Δ < A. We will allow paths to be constructed from other paths. In connection with our discussion of amalgamation and transitivity principles, it seems we can distinguish two ways of concatenating paths: we can combine them ‘end to end’ to make longer paths, and we can combine them ‘side by side’ to make wider ones. This intuitive contrast should be reflected in our theory of paths.17 To that end, I introduce two concatenation relations. The first—elongation—is involved in applications of transitivity, and I will express it with the symbol ‘∩’: if A <p1 B and B < p2 C, A grounds C via a path constructed from p1 and p2, where these are constructed sequentially. I express this idea by saying A <p1∩p2 C. When p is not the result of elongating other paths, I will say that p is L-Simple, and that L-simple paths lack L-parts. The notion of L-simplicity is closely tied to that of immediate grounding, as follows: Γ <IMM A = df ∃p(Γ <p A & p is L-simple) Intuitively, this says that Γ immediately grounds A just in case there is some path p that is not the result of elongating any other paths, and Γ grounds A via p. Mediate grounding is defined as grounding via a path that is not L-simple: Γ <MED A = df ∃p(Γ <p A & p is not L-Simple) Mediate and immediate grounding, so defined, are compatible: the claim that Γ <MED A is compatible with the claim that Γ <IMM A, since a fact can take distinct paths, of different structures, to ground another. This allows us to accommodate the plausible idea grounding connections can be seen as somehow both immediate and mediate—like the grounding connection between the fact that A and the fact that A or (A or B) (Fine 2012: 51). Note as well that defining immediate grounding in terms of ‘<p’ has the virtue of consistency with the possibility of dense grounding: we can allow there to be grounding between A and B in the absence of a series of immediate grounding connections between them. As we noted in Section III, a defect of Fine's (2012: 51) analysis of ‘<’ as the transitive closure of immediate grounding is that it rules out this possibility. One might think there ought to a way of making sense of the immediate vs mediate distinction on which these kinds of grounding are incompatible. This is naturally done in terms of paths. For we can say that Γ immediately grounds A via p if and only if Γ <p A and p is L-simple; and that Γ mediately grounds A via p if and only if p is not L-simple. Since a path cannot be both L-simple and not L-simple, these are incompatible. Consider now the path concatenation relation at work in instances of amalgamation. If A <p1 C and B <p2 C, when we apply amalgamation, A and B together will ground C by ‘straddling’ p1 and p2. To capture the idea, we can consider the plurality A, B as jointly grounding C via a path comprised of p1 and p2, combined ‘side by side’. To express this relation of concatenation—call it widening—I will use the symbol ‘+’, and say that A, B <p1+p2 C. When p is not the result of widening other paths, I will say that p is W-Simple, and that W-simple paths lack W-parts. The notion of W-simplicity is closely tied to that of minimal grounding, as follows: Γ <MIN A = df ∃p(Γ <p A & p is W-Simple) This says that Γ minimally ground A just in case there is some path p that is not the result of widening any other paths, and Γ grounds A via p. Non-minimal grounding is defined as grounding via a path that is the result of widening other paths: Γ <NON-MIN A = df ∃p(Γ <p A & p is not W-Simple) Minimal and non-minimal grounding, so defined, are compatible. Again, one might think there ought to a way of making sense of the minimal vs non-minimal distinction on which these kinds of grounding are incompatible. This is easily done: say that Γ minimally grounds A via p if and only if Γ <p A and p is W-simple; and that Γ non-minimally grounds A via p if and only if Γ <p A and p is not W-simple. Since a path cannot be both W-simple and not W-simple, these are incompatible. Let us consider how appealing to paths helps resolve the problem cases we discussed at the beginning of this section. There are distinct paths that the fact that x is scarlet takes to ground the fact that x is coloured. We can represent the case diagrammatically: Once we separate p1, p2 and p3, we can distinguish the following claims in SCA’s vicinity: SCA1: (x is scarlet <p1 x is red), (x is red <p2 x is coloured) < (x is scarlet <p1 ∩p2 x is coloured) SCA2: (x is scarlet <p1 x is red), (x is red <p2 x is coloured) < (x is scarlet <p3 x is coloured) SCA1 says that, relative to the path comprised of p1 and p2, that x is scarlet < x is coloured is grounded by the fact that x is scarlet < x is red and that x is red < x is coloured. I think that someone attracted to the intuitive idea that the transitivity principle tracks grounding connections should endorse this claim, as an expression of that idea. At least, we can accept it while offering a plausible diagnosis of what is implausible about SCA. For SCA is implausible if it is interpreted as SCA2: the immediate grounding connection via path p3 does not depend on the grounding connections mediated by p1 and p2. Once we take account of paths, we find that there is a falsehood (i.e., SCA2) and a truth (i.e., SCA1) in SCA’s vicinity. The moral is that we need to formulate minimalism in such a way that the truth, and not the falsehood, is implied by it. Consider now the second problem case. Supposing there to be paths corresponding to each disjunct of a disjunctive fact, the fact that A or B or (A & B) has three paths leading to it. The grounding connection that takes the path corresponding to the fact's third disjunct is an instance of the law that truths ground their conjunction, and this does not involve an application of Amalg. But if we consider how A and B ground A or B or (A and B) as disjuncts—rather than conjuncts of a disjunct—then A and B’s jointly grounding the disjunctive fact does plausibly depend on an application of Amalg, and on the ground-theoretic connections between the disjunctive fact and A and B individually. Now to reformulate minimalism in terms of ‘<p’, so that our two problem cases are resolved in these ways. We start with the analogue of Trans for ‘<p’, which is as follows. TransPATH: If Γ <p1 A and A, Δ <p2 B then Γ,  Δ <p1∩p2 B TransPATH tells us that if some facts Γ ground A via path p1, and if A and Δ ground B via p2, then Γ and Δ together ground B via a path comprised of p1 and p2. TransPATH goes beyond Trans by telling us that how Γ,  Δ < B is determined by the ways that Γ < A and A, Δ < B (in addition to telling us that that Γ,  Δ < B is determined by the facts that Γ < A and A, Δ < B). The analogue of Amalgamation is: AmalgPATH: If Γ <p1 A and Δ <p2 A then Γ,  Δ <p1+p2 A Minimalism then becomes—and, from now on, will be understood as—the conjunction of the following two theses: Trans-GroundPATH: If Γ <p1 A and A, Δ <p2 B then (Γ <p1 A), (A, Δ <p2 B) < (Γ,  Δ <p1 ∩ p2 B) Amalg-GroundPATH: If Γ <p1 A and Δ <p2 A then (Γ <p1 A), (Δ <p2 A) < (Γ,  Δ <p1+p2 A) We argued above for the principle Gen: that whatever grounds the fact that Γ <p A will also ground that Γ < A. The following theses then follow from minimalism: Trans-GroundGEN: If Γ <p1 A and A, Δ <p2 B then (Γ <p1 A), (A, Δ <p2 B) < (Γ, Δ < B) Amalg-GroundGEN: If Γ <p1 A and Δ <p2 A then (Γ <p1 A), (Δ <p2 A) < (Γ, Δ < A) This shows that, although minimalism is in the first instance an account of how certain facts of the form ‘Γ <p A’ are grounded, it also provides an account of how certain facts of the form ‘Γ < A’ are grounded. This potentially brings minimalism into competition with other such accounts, as we will now see. V. GENERAL THEORIES AND OVERDETERMINATION Many philosophers deny the existence of systematic causal overdetermination, a view that plays a prominent role in discussions of mental causation and material constitution.18 The topic of ground-theoretic overdetermination remains relatively unexplored but there are plausibly some cases of systematic ground-theoretic overdetermination. Any existential generalization with multiple instances and any disjunction with multiple true disjuncts will be ground-theoretically overdetermined. These cases of systematic overdetermination do not seem problematic. At least, the threat of overdetermination is not commonly regarded as a reason to reject the thesis that existential generalisations are grounded by their instances, or that disjunctions are grounded by their true disjuncts. I suspect that the reason these cases are not considered problematic is that, in each case, the overdetermination follows from a single and plausible idea. In the disjunctive case, the plausible idea is that disjunctions are grounded by their true disjuncts; in the existential case, the plausible idea is that existential generalizations are grounded by their instances. Widespread overdetermination just follows from these plausible ideas. Despite the multiplication of grounds, we have in these cases a unity of theory: there are not different competing theories of how disjunctive facts are grounded, or different metaphysical laws involved. Surely not all cases of ground-theoretic overdetermination are innocent, though. Suppose I claim that the vase's brittleness is fully grounded by some fact about the structure of its constituent atoms, but add that it is also fully grounded by facts about angels—angels always glare at brittle things, and this makes them brittle. And suppose I add that neither grounds the other, nor helps the other ground the vase's brittleness. This seems an implausible combination of views, because it involves an unparsimonious disunity of theory: I am running in tandem what should be two competing theories about how the vase's brittleness is grounded. The different grounds I posit for the vase's brittleness do not follow from a unified underlying theory; the metaphysical laws involved will inevitably be different.19 The claim that this sort of overdetermination is widespread is extremely suspect. To the extent that this attitude is warranted, the above defence of minimalism puts pressure on certain general theories of iterated grounding. One leading thesis about iterated grounding is the superinternality thesis, developed independently by Bennett (2011) and deRosset (2013): Superinternality: If Γ < A then Γ < (Γ < A) According to Superinternality, facts of grounding ‘bottom out’ by being grounded by the facts on their left-hand side. Even if some facts of grounding ground others, repeated applications of Superinternality will presumably deliver facts that are not facts of grounding, which ground the whole grounding hierarchy. This indeed is part of the motivation for Superinternality. Bennett and deRosset both seek to avoid fundamental facts of grounding, and cite the ability of Superinternality to deliver this result as a count in its favour. Superinternality applies to all facts of grounding, whereas minimalism provides grounds only for some facts of grounding—those that are the result of applying the transitivity and amalgamation principles. Consider any such fact of grounding, Γ < A. Minimalism implies that this fact is grounded by other facts of grounding. Superinternality predicts that it is grounded by Γ. These—the worry goes—are distinct, competing theories about how the fact in question gets grounded. We get a clearer view of the sort of overdetermination involved by considering an example. Consider a case involving Trans-GroundGEN. Suppose that A <p1 B, B <p2 C, and consider how the fact that A < C is grounded. Minimalism—specifically, its corollary Trans-GroundGEN—implies ATO: (A <p1 B), (B <p2 C) < (A < C) Superinternality implies SUP: A < (A < C) ATO and SUP seem to be distinct, competing theories about how the fact that A < C is grounded. Moreover, this kind of overdetermination will be systematic, arising for any instance of mediate grounding. Perhaps defenders of Superinternality should extend their view to accommodate paths. If so, it is in the spirit of Superinternality to supplement the thesis with the following one: SuperinternalityPATH: If Γ <p A then Γ < (Γ <p A) SuperinternalityPATH is of a piece with Superinternality: both are expressions of the idea that features of the grounding hierarchy are grounded by its bottom layer. According to SuperinternalityPATH Γ does not just determine that Γ < A; Γ determines how Γ < A. SuperinternalityPATH provides grounds for the facts on the left of ATO: A < (A <p1 B) and B < (B <p2 C). Given Trans, it follows that: A, B < (A < C)20 And the superinternalist might push farther still: since we have assumed that A < B, Trans lets us infer from A, B < (A < C) that A, A < (A < C). Assuming that if A, A < B then A < B, this gives us the thesis that A < (A < C). In sum, the superinternalist can plausibly resist the charge that she is committed to positing multiple fundamental grounds for the fact that A < C, for by her lights it is plausible to maintain that A is the only fundamental ground for the fact that A < C (or, if A is not itself fundamental, A’s fundamental grounds will play this role). These remarks are suggestive but there is reason to doubt that, on their own, they mitigate the worry that combining minimalism and Superinternality results in problematic overdetermination. This is because ATO and SUP are plausibly understood as identifying immediate grounds for the fact that A < C. If that is how we understand ATO and SUP, their conjunction will commit us to saying that A grounds the fact that A < C in two ways: both immediately (as per SUP) and mediately (by applying SuperinternalityPATH to ATO). So, although combining ATO and SUP arguably does not multiply fundamental grounds for the fact that A < C, it does multiply the paths we posit between A and A < C; it multiplies, as it were, the non-causal mechanisms connecting A to the grounded fact, that A < C. And this seems problematic, because it involves an objectionable disunity of theory. That A < C is grounded in two ways does not follow from a single plausible idea (thus this case contrasts with the unproblematic overdetermination we find when a disjunction has many true disjunctions). It follows from running together distinct theories about how the fact that A < C is immediately grounded. We are familiar with isolated cases in which distinct causal mechanisms connect a cause to one of its effects. For instance, suppose that Jimbo throws a ball at a window, Jane sees this happen and immediately throws another ball at the window, and both balls hit the window and break it simultaneously. But the ground-theoretic overdetermination we are considering would be systematic, arising for any instance of mediate grounding. To that extent, it seems problematic: we have two generally applicable theories about how facts of mediate grounding are immediately grounded, and this seems unparsimonious. So if ATO and SUP are both claims of immediate grounding—and if, more generally, minimalism and Superinternality are in the business of stating immediate grounds for the facts of grounding they concern—then the tension between these theories of iterated grounding seems to remain. Unfortunately, we lack clear and informative diagnostics for immediate grounding, although, as Fine (2012: 51) points out, we do seem to have strong intuitions in particular cases about when a given grounding connection is immediate.21 A plausible conjecture is that the fundamental metaphysical laws are laws of immediate grounding. We can make an initial case for this by considering examples. It is (we may suppose) a fundamental metaphysical law that the existence of any non-empty set is grounded by the existence of its members; and plausibly non-empty sets are immediately grounded by their members. It is (we may suppose) a fundamental metaphysical law that the existence of any composite object is grounded by the existence or arrangement of its constituents. This law again seems to track immediate grounding connections—composites seem to be immediately grounded by the existence and arrangement of their constituents. Contrast these cases with the non-fundamental metaphysical law that the existence of a non-empty set is grounded by the existence and arrangement of the constituents of its composite members: this law results from chaining two more fundamental laws (i.e., the law governing the grounding of sets and that governing the grounding of composite objects): its instances are all instances of mediate grounding. It is, then, initially plausible that the following thesis holds: Imm-Law: If L is a fundamental metaphysical law, then its instances are instances of immediate grounding.22 If we regard Superinternality as a fundamental metaphysical law then there is some pressure on us to accept the following thesis (recall, ‘<IMM’ expresses immediate full grounding): SuperinternalityIMM: If Γ < A then Γ <IMM (Γ < A) SuperinternalityIMM goes beyond Superinternality; to my knowledge, extant defences of Superinternality do not explicitly commit to SuperinternalityIMM. SuperinternalityIMM brings the Superinternality thesis into conflict with minimalism, since SuperinternalityIMM and minimalism are competing theories of how a large class of facts of grounding—the ones that both theories apply to—are immediately grounded. To avoid commitment to overdetermination, superinternalists who accept minimalism should reject SuperinternalityIMM and also the thesis that Superinternality is a fundamental metaphysical law. Instead, they should regard their theory as an account of how the most fundamental facts of grounding are immediately grounded—allowing minimalism to provide immediate grounds for the less fundamental facts of grounding. After all, the main motivation for the superinternality thesis is that it avoids commitment to fundamental facts of grounding, and to establish this we need not provide immediate grounds for facts of grounding that minimalism applies to. Superinternalists might then take the fundamental metaphysical law in which their thesis is articulated to be as follows: SuperinternalityWEAK: If Γ < A, and there is no Δ that only contains facts of grounding and is such that Δ < (Γ < A), then Γ < (Γ < A) The logical relationship between SuperinternalityWEAK and Superinternality is debatable, and depends on the background special theory of iterated grounding assumed. What matters for us is that the thesis that SuperinternalityWEAK is a fundamental metaphysical law does not entail that Superinternality is one—and SuperinternalityWEAK does not imply SuperinternalityIMM. SuperinternalityWEAK is a thesis about how the most fundamental layer of facts of grounding is grounded by facts that are not themselves facts of grounding. Once this layer of facts of grounding is accounted for, minimalism (or perhaps some other special theory of iterated grounding) is invoked to provide immediate grounds for the other facts of grounding. The resulting picture witnesses the division of labour described in Section II. Our account of the fundamental metaphysical laws by which facts grounding are grounded divides across the distinction between general and special theories of iterated grounding. Dividing the topic up like this is a promising strategy for avoiding the threat of overdetermination that arises from combining the superinternality thesis with minimalism. A problem with this strategy, however, is that it is speculative that all facts of grounding can be accounted for in this manner. Suppose that A <p B, and that p is the only path connecting A to B. Further, suppose that p is ‘gunky’: that p is not L-simple, and that none of p's L-parts is L-simple either. (This case illustrates a path-sensitive analogue to the notion of dense grounding that was introduced in Section II.) If we were to represent p's structure by listing its L-parts connected by ‘∩’, our representation of p would be isomorphic to the real number series. p mediates an infinity of ever more immediate grounding connections, but no immediate grounding connections. Divide p in half: p = p1 ∩ p2, where A <p1 C and C <p2 B. Minimalism implies that the fact that A <p B is grounded by these two facts of grounding. But not only these: p1 and p2 can be split up into L-parts which—according to minimalism—mediate still more fundamental grounding connections. On the minimalist's picture, that A <p B is grounded by an infinite series of ever longer sequences of ever more immediate grounding connections, without end: an infinitely descending chain of facts of grounding. SuperinternalityWEAK would not apply to any of the facts in this chain, because each such fact would be grounded by other facts of grounding. It would simply have nothing to say about this part of the grounding hierarchy. Similarly, suppose that Γ <p B, and that every proper W-part of p itself has a proper W-part. Intuitively, we could keep removing facts from Γ to yield a more minimal ground for B, never arriving at a minimal ground. There is, in fact, a non-absurd model for this scenario. That there are infinitely many numbers might be grounded by the fact that 1 is a number, 2 is a number… and so on for all the natural numbers—call this plurality of facts ‘N’. N does not minimally ground that there are infinitely many numbers, because the following proper sub-plurality of N would suffice as well: 2 is a number, 4 is a number… and so on for all the even numbers. Indeed, any proper sub-plurality of N that grounds the fact that there are infinitely many numbers will itself have a proper sub-plurality that does so as well. Consider, then, the following facts of grounding 1 is a number, 3 is a number, 5 is a number … <p1 There are infinitely many numbers. 2 is a number, 4 is a number, 6 is a number … <p2 There are infinitely many numbers. Applying AmagPATH to these facts of grounding yields 1 is a number, 2 is a number, 3 is a number…<p1+p2 There are infinitely many numbers. According to minimalism—specifically, Amalg-GroundPATH—this fact of grounding is less fundamental than those to which AmalgPATH was applied to. And we can repeat the reasoning—for example, p1 in turn can be split into W-parts, p3 and p4, such that: 1 is a number, 5 is a number, 9 is a number … <P3 There are infinitely many numbers. 3 is a number, 7 is a number, 11 is a number … <P4 There are infinitely many numbers. We can keep dividing N into subsets to arrive at ever more minimal grounds for the fact there are infinitely many numbers—grounds that, by the lights of minimalism, are ever more fundamental—without arriving at minimal grounding connections between subsets of N and the fact that there are infinitely many numbers. Commitment to infinitely descending chains of facts of grounding is not the same as commitment to fundamental facts of grounding. But, if we reject fundamental grounding, the former commitment hardly seems palatable. For, while each step in an infinitely descending chain might be accounted for, the chain itself would not be. There would seem to be an important sense in which this part of the grounding hierarchy has not been accounted for by metaphysically more fundamental matters. We have focussed on the superinternality thesis, but the discussion generalizes to other general theories of iterated grounding. Abstractly put, the threat of overdetermination arises whenever the special and general theories of iterated grounding imply—in a systematic way—distinct immediate grounds for facts of grounding. Restricting the general theory to the most basic layer of the facts of grounding is always an option. This sidesteps the threat of overdetermination, because the general and special theories will then be concerned with different facts of grounding—the special theory will only pick up where the general theory leaves off. But if our ambition is to show that the existence and features of the grounding hierarchy are explicable in terms of more fundamental matters, it will likely be frustrated by the existence of (perhaps even the mere possibility of) infinitely descending chains of facts of grounding, like those described in the preceding paragraphs. VI. CONCLUSION We have seen that transitivity and amalgamation principles plausibly yield facts of grounding that are grounded by others. In order to formulate minimalism, the view in which this intuitive idea finds expression, we found that it is important to pay attention to the notion of a grounding path. We have also discussed interactions between minimalism and Superinternality—and, by extension, other general theories of iterated grounding—finding that there is a threat of objectionable overdetermination that arises from combining minimalism with extant theories of iterated grounding. More broadly, we have seen that the theory of iterated grounding is not exhausted by the general theory, and that neglecting the special theory might indeed impoverish our general theory.23 Footnotes 1 Trogdon (2013: 16) cites unpublished work by Dasgupta defending this kind of grounding claim. 2 The transitivity principle seems more intuitively plausible than the amalgamation principle (although see Schaffer 2012 for scepticism about the former). Fine endorses the amalgamation principle for theoretical reasons: ‘I doubt that there is simple and natural account of the logic of ground that can do without it’ (2012: 57). I simply assume the amalgamation principle here. A good question—laid aside for reasons of space—concerns whether an analogue of the second claim would remain plausible if we replaced the amalgamation principle with another describing conditions under which we can add to the grounds of some fact A to yield another ground for A. Arguably, some such ‘augmentations’ of a fact's grounds are legitimate (see Clark 2015; Litland 2013: 21; Scott-Dixon 2016; but see Audi 2012: 699 for a dissenting voice). Thanks to an anonymous referee here. 3 Compare Armstrong (2004: 19–21) discussion of the related notion of minimal truthmaking. 4 I take the term ‘iterated grounding’ from Litland (forthcoming). 5 It is fairly plausible that the fact that A does not ground B might be grounded in the fact that B grounds A, given that grounding is asymmetrical. The cases we discuss are importantly different from this, since they concern positive facts about what grounds what, rather than negative facts. 6 See Bennett (2011), Dasgupta (2014), Litland (forthcoming), Rosen (2010)and deRosset (2013). For an overview, see Bliss and Trogdon (2014: sect. 7). Sider ms argues that we should not expect a fully general account of iterated grounding; as the next section indicates, I am sympathetic to this view. 7 Arguably, expressing grounding with a sentential connective allows us to avoid ontological commitment to a grounding relation and its relata (Correia 2010: 254; Fine 2012: 47). But to keep the discussion idiomatic, I will often write as though there is a grounding relation between facts. 8 The distinction between special and general theories is not often drawn but Trogdon (2013: 116) suggests it, citing unpublished work by Dasgupta. As I have drawn it, the distinction is not exhaustive, for it leaves room for mixed theories of iterated grounding that conjoin special and general theories; and for theories that entail claims of the form ‘Γ < (Δ < A)’, where some but not all members of Γ are claims of grounding. But the distinction is not arbitrary: extant theories of iterated grounding tend to be general in the sense defined, and the theory proposed in this paper is special in the defined sense. 9 Thanks to an anonymous referee for prompting me to comment on these issues. 10 My use of ‘metaphysical law’ is different to that of Wilsch (2016), on which the metaphysical laws are ‘general principles that govern construction-operations’ (Wilsch 2016: 4). Roughly, such principles describe how certain operations generate entities from other entities. See Wilsch (2016: 3–8) for further discussion. 11 Compare the discussion of propositional forms in Rosen (2010: 131). 12 Thanks to an anonymous referee for raising these problem cases. 13 The case might seem still more problematic if whenever F is a shade of G, there is a third shade of G that F is a more determinate shade of—for then there would be an infinite series of grounding connections between x's scarletness and its being coloured. 14 I use ‘dependence’ here and below to express the converse of grounding and not the concept of ontological dependence, as discussed by Fine (1995), Lowe (1994) and others. 15 The notion of a path, or one very similar, is invoked by Fine (2012: 51). 16 Schaffer (2016: 83–6) argues that his structural equations approach competes with Fine's framework for grounding, and that Fine's is a less illuminating setting for theorising about grounding. I do not see the present regimentation of paths—which extends Fine's framework—as competing with the structural equations framework. It is relatively straightforward, and adequate for the purpose of making some philosophical claims about iterated grounding, but these claims should survive being transplanted into a graph-theoretic setting. They are artefacts, not of regimentation, but of philosophical intuition. 17 Thanks to an anonymous referee here. 18 For overdetermination in the context of mental causation, see Kim (1989, 1993) and Malcolm (1968); an overview is provided by Heil and Robb (2014: § 6.2). For overdetermination in the context of material constitution see Merricks (2001) and Paul (2007, 2010). Sider (2003) questions the motivation of scepticism about systematic overdetermination. 19 It is sometimes said that problematic cases of causal overdetermination involve overdetermination by events of the same type; see Dretske (1988: 42ff) and Jaworski (2016: 280 ff). There seems to be no problem in assuming that the case described meets any such additional requirements for problematic ground-theoretic overdetermination. 20 The argument is as follows: (P1) (A <p1 B), (B <p2 C) < (A < C) [ATO]. (P2) A < (A <p1 B) [From SuperinternalityPATH]. (P3) A, (B <p2 C) < (A < C) [From Trans, P1, P2]. (P4) B < (B <p2 C) [From SuperinternalityPATH]. (C) A, B < (A < C) [From Trans, P3, P4]. 21 Poggiolesi (forthcoming) offers an analysis of the related notion of immediate formal grounding. 22 The existence of a set can be mediately grounded by the existence of its members, as in the case of {a {a}}, and the existence of a composite can be mediately grounded by features of its constituent matter, as where some object's constituents also constitute another of its constituents. But this is not in tension with the Imm-Law: we have seen, in Section 4 that immediate grounding does not rule out mediate grounding by another path. 23 This paper has benefitted from discussion at Hamburg's Forschungskolloquium seminar. I am especially grateful to Stephan Krämer, David Liggins, Giovanni Merlo, Stefan Roski, Kelly Trogdon and Nathan Wildman for providing very valuable comments on previous versions of this paper. This research was carried out as part of the Sinergia project Grounding: Metaphysics, Science, and Logic, which is funded by the Swiss National Sciences Foundation; I am very grateful for their support. REFERENCES Armstrong D. ( 2004) Truth and Truthmakers . Cambridge: CUP. Google Scholar CrossRef Search ADS   Audi P. ( 2012) ‘ Towards a Theory of the ‘in Virtue of’ Relation’, Journal of Philosophy , 109: 685– 711. Google Scholar CrossRef Search ADS   Bennett K. ( 2011) ‘ By Our Bootstraps’, Philosophical Perspectives , 25: 27– 41. 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Google Scholar CrossRef Search ADS   Sider T. ms. ‘ Ground grounded’, unpublished manuscript. 〈http://tedsider.org/papers/ground_grounded.pdf〉 accessed 6 June 2017. Trogdon K. ( 2013) ‘ An Introduction to Grounding’, in Hoeltje M., Schnieder B., Steinberg A. (eds) Varieties of Dependence. Basic Philosophical Concepts , 97– 122. Munich: Philosophia Verlag. Weslake B. (forthcoming) ‘ A Partial Theory of Actual Causation’, The British Journal for the Philosophy of Science . Wilsch T. ( 2016) ‘ The Deductive-nomological Account of Metaphysical Explanation’, Australasian Journal of Philosophy , 94: 1– 23. Google Scholar CrossRef Search ADS   © The Author 2017. Published by Oxford University Press on behalf of The Scots Philosophical Association and the University of St Andrews. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Philosophical Quarterly Oxford University Press

# What Grounds What Grounds What

, Volume 68 (270) – Jan 1, 2018
22 pages

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Oxford University Press
ISSN
0031-8094
eISSN
1467-9213
D.O.I.
10.1093/pq/pqx031
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### Abstract

Abstract If there are facts about what grounds what, are there any grounding relations between them? This paper suggests so, arguing that transitivity and amalgamation principles in the logic of grounding yield facts of grounding that are grounded by others. I develop and defend this view and note that combining it with extant accounts of iterated grounding commits us to seemingly problematic instances of ground-theoretic overdetermination. Taking the superinternality thesis as a case study, I discuss how defenders of this thesis should respond. It emerges that our discussion puts pressure on superinternalists to make an interesting qualification to their view: to only regard as a fundamental metaphysical law a version of the superinternality thesis that is restricted to minimal and immediate grounding. I. INTRODUCTION If there are facts about what grounds what, are there any grounding relations between them? Plausibly so. Suppose we want to explain why some portion of physical reality grounds the existence of a city. One good strategy is to trace the steps in the grounding chain between that portion of physical reality and the city: first, certain physical facts involving particles ground facts about larger physical objects (atoms, molecules…), which ground certain chemical facts, which ground facts about basic biological entities, which ground facts about organisms … (passing through the psychological and sociological realms) … and that is why this portion of physical reality gives rise to a city. We can conduct a more systematic discussion by connecting the question to issues in the logic of grounding. Two plausible claims then heave into view. Suppose that a fact A grounds another fact B and that B grounds a third fact C. Assuming that grounding is transitive, it follows that A grounds C. First claim: A grounds C in virtue of these mediating links in the grounding chain from A to C. It is because A grounds B and B grounds C that A grounds C.1 Suppose now that A grounds C and B grounds C. Assuming that the grounds of a fact always combine to yield another of its grounds—a principle that Fine calls amalgamation (Fine 2012: 57)—it follows that A and B together ground C.2 Second claim: A and B together ground C in virtue of the facts that A grounds C and B grounds C. It is because A and B individually ground C that they combine to yield a ‘bigger’ ground for C. These intuitively plausible claims suggest that the transitivity and amalgamation principles, assuming them to hold, yield facts of grounding that are grounded by others. Metaphorically speaking, the amalgamation principle allows us to ‘widen’ facts of grounding and the transitivity principle allows us to ‘lengthen’ them. The above claims suggest a picture on which the ‘narrower’ and ‘shorter’ facts of grounding are prior to the ‘wider’ and ‘longer’ facts of grounding that the transitivity and amalgamation principles yield when applied to them. I label this intuitive picture minimalism—a more precise formulation of minimalism is provided in Section III, and refined in Section IV. It is standard in the grounding debate to distinguish, on the one hand, immediate from mediate grounding connections (Fine 2012: 50–1) and, on the other, minimal from non-minimal grounding connections (Scott-Dixon 2016). These distinctions are firmly rooted in our intuitive thought about grounding—and explications of them will be offered in Section IV. Intuitively, an immediate grounding connection between two facts is direct; it does not need to be mediated by other grounding connections. That the apple is green directly grounds that the apple is coloured. But fundamental physical facts only indirectly, or mediately, ground the existence of cities. Where some facts minimally ground another, they do so without superfluity: the former facts suffice to ground the latter, and would not have grounded it if anything were removed from them (or, at least, not in the same way—see Section IV).3 The concepts of minimal and immediate grounding are absolute. Metaphorical talk of grounding connections being ‘longer’ and ‘wider’ expresses comparative cousins of these concepts: the longer grounding connections are less immediate than shorter ones, and wider grounding connections are less minimal. This paper aims to support minimalism and explore its bearing on other theories of iterated grounding—that is, theories that tell us when and how facts of grounding are grounded.4 A programmatic idea that underlies the discussion is that our theory of iterated grounding should be partly driven by our logic of grounding. This view would be strengthened if other logical principles, besides the transitivity and amalgamation principles, systematically yield derivative facts of grounding, but I leave discussion of this sort of extension to the project for another occasion.5 I focus on the transitivity and amalgamation principles because, firstly, we do well to start our inquiry focussing on the clearest cases we can find, and these strike me as the clearest cases in which logically derivative facts of grounding are grounded by more basic facts of grounding. Secondly, minimalism is, metaphorically speaking, the view that the big facts of grounding derive from the small. That we readily identify two dimensions along which facts of grounding can be big in this metaphorical sense suggests a degree of conceptual unity between them. The discussion proceeds as follows. Section II introduces some concepts and notational conventions. Section III tries to formulate minimalism with more precision and to motivate it. Section IV discusses a couple of problem cases for minimalism, and responds by incorporating the concept of a grounding path into minimalism's formulation. There are, in the literature, some general proposals about how facts of grounding—that is, all facts of grounding—are grounded by other kinds of fact.6 None of these entail either of the two claims we opened with so, if minimalism is correct, there are instances of iterated grounding that are not entailed by extant theories of it. It is not obvious that these general theories coexist peacefully with these instances. The worry is that combining any of the extant theories with minimalism commits us to problematic instances of ground-theoretic overdetermination. I explore this issue in Section V, using the superinternality thesis, defended by Bennett (2011) and deRosset (2013), as a case study. There are, indeed, several theories of iterated grounding on the market. I focus on the superinternality thesis because it is a prominent theory of iterated grounding and because discussing it does not require us to introduce controversial metaphysical concepts (such essentialist concepts), besides ground-theoretic concepts. This makes discussing the superinternality thesis comparatively straightforward, and I indicate the extent to which Section V’s discussion generalizes to competing theories of iterated grounding. Section VI provides concluding remarks. II. PRELIMINARIES Grounding is a concept of non-causal and explanatory determination, often expressed in metaphysical discussions with the phrase ‘in virtue of’. To say that a fact fully grounds another is to say that the former is wholly responsible for the latter, in the relevant non-causal sense. Where a fact partially grounds another, the former helps to bring it about that the latter obtains, but may not suffice to do so (for more by way of introduction to the topic of grounding, see Bliss and Trogdon 2014; Correia and Schnieder 2012; Rosen 2010: §1 and 2). I assume a broadly Finean background theory of grounding, although the discussion should generalize to other comparable frameworks. Adopting Fine's notation, I use the symbol ‘<’ as a sentential connective to express the concept of full grounding.7 ‘<’ takes variably long lists of sentences on its left and single sentences on its right. I use capitalized letters ‘A’, ‘B’… as sentential variables and capital Greek letters ‘Γ’, ‘Δ’ … as plural variables ranging over lists of sentences. Although listing sentences may not be the same as referring to a set of sentences, I will occasionally use familiar set-theoretic concepts in reasoning about lists. Like Fine, I assume that (plural) sentential quantification is intelligible. The general form of claims of full grounding is, then, as follows:   \begin{equation*}{\rm{\Gamma }} < {\rm{A}} \end{equation*} A claim of this form may be read: A wholly in virtue of the facts in Γ. The facts of grounding are just those facts, or truths, that have the above form (I use ‘fact’ interchangeably with ‘truth’). A fact of iterated grounding is a fact of the form ‘Γ < (Δ < A)’. A claim of iterated grounding is a claim to the effect that such a fact obtains. And a theory of iterated grounding is a theory that entails a claim of iterated grounding. We can contrast two kinds of theory of iterated grounding. What I will call the general theory investigates how and when facts of grounding are grounded by facts that are not facts of grounding. The central question for the general theory is whether grounding is a fundamental phenomenon and, if it is not, how facts of grounding are grounded by more fundamental matters. The general theory contrasts with the special theory, which investigates grounding relations only within the class of facts of grounding. The central question of the special theory is whether facts of grounding ever ground others. More precisely, say that a theory of iterated grounding is general just in case every claim it entails of the form ‘Γ < (Δ < A)’ is such that no member of Γ is a claim of grounding; and special just in case every claim it entails of the form ‘$${\rm{\Gamma}} < {\rm{}}({\rm{\Delta}} < {\rm{A}})$$’ is such that every member of Γ is a claim of grounding. The two claims that we began with constitute a rudimentary special theory. A main thesis of this paper is that, whatever our general theory, the logic of grounding should guide our special theory.8 Distinguishing the special and general theories allows for a division of labour in the theory of iterated grounding. Suppose G is the set of all and only the facts of grounding and we have a special theory according to which there is a certain set K meeting the following conditions: a) K⊂G b) $$\forall {\rm{A}}( {\rm{A}} \in {\rm{G\ }}\& {\rm{\ }}\sim ( {{\rm{A}} \in {\rm{K}}} ) ) \to \exists {\rm{\Gamma }}({\rm{\Gamma }} \subseteq {\rm{K\ }}\& {\rm{\ \Gamma }} < {\rm{A}})$$ c) $$\forall {\rm{A}}( {{\rm{A}} \in {\rm{K}}} ) \to \sim \exists {\rm{\Gamma }}({\rm{\Gamma }} \subseteq {\rm{G\ }}\& {\rm{\ \Gamma }} < {\rm{A}})$$ Condition (a) says that K contains only facts of grounding; (b) that every fact of grounding that is not in K is grounded by members of K; and (c) that no member of K is fully grounded by facts of grounding. Intuitively, the special theory we are considering claims that K forms a minimal grounding base for the facts of grounding. We then turn to the general theory. Suppose we aim to show that every fact of grounding is grounded by some other kind of fact (an essentialist fact, say). To establish this, we need a general theory that provides plausible grounds for the members of K, but we do not even need to discuss the other facts of grounding, because our special theory provides an account of how these are grounded by the members of K. Grounding is transitive, so whatever grounds the members of K will also ground the facts of grounding that are not in K. But our theory of what grounds the members of K need not apply directly to the facts of grounding not in K. For all we know at the outset, it might only provide grounds for these in conjunction with our special theory. Perhaps the best general theory directly applies to all facts of grounding, but that it does is a substantive thesis about iterated grounding, which the thesis that there are no fundamental facts of grounding does not obviously entail. It is, then, sensible to develop the special theory before the general theory; for only with a special theory in hand will we have a clear idea of which facts we need our general theory to directly apply to. III. MINIMALISM: FORMULATION AND MOTIVATION The two claims we opened with seem immediately plausible. Insofar as they suggest the general view that the facts of grounding to which the transitivity and amalgamation principles are applied are prior to those the principles yield, minimalism is a plausible default view. Still, it is instructive to trace minimalism's plausibility to that of some more general metaphysical ideas. The idea that grounding is transitive can be made precise in various ways. A strong transitivity thesis is as follows (outer universal quantifiers are omitted for readability, where it is obvious how to add them): Trans: If Γ < A and A,  Δ < B then Γ,  Δ < B I assume that grounding is indeed transitive in this strong sense. It might be suggested that Trans is trivial, since ‘<’ should be understood as the transitive closure of immediate grounding, which is trivially transitive (see Fine 2012: 51). In my view, this definition of ‘<’ is neither mandatory nor attractive. It is not mandatory because our intuitive concept of grounding is robust—we do not have to introduce ‘<’ as the transitive closure of immediate grounding. The definition is not attractive because it rules out the possibility of dense grounding, understood as follows: Γ densely grounds A iff $${\rm{\Gamma }} {< }{\rm{A }}\& {\rm{}}\forall {\rm{\Delta }}( {{\rm{\Delta }} {<} {\rm{A}} \to \exists {\rm{{\rm K}}}\exists {\rm{B}}( {( {{\rm{\Delta }} < {\rm{B}}} )\& ( {{\rm{{\rm K}}},{\rm{\ B}} {<} {\rm{A}}} )} )} )$$ If Γ densely grounds A, there are no links of immediate grounding between Γ and A, and so Γ does not bear the transitive closure of immediate grounding to A. So, if ‘<’ expresses the transitive closure of immediate grounding, the pair of conditions specified by the definition of dense grounding can never both obtain, and so dense grounding is impossible. But dense grounding seems possible, so we should resist the analysis of ‘<’ in terms of immediate grounding. In any case, it seems to me that the discussion below does not depend on any thesis about the epistemic status of Trans (or of the amalgamation principle, discussed below). I take the question of whether Trans is trivial to be orthogonal to the issues discussed here.9 The first component of minimalism says that Trans tracks grounding connections: it licenses the inference of a derivative fact from facts that ground it, or would do if they obtained. This idea is naturally expressed by the following thesis: Trans-Ground: If Γ < A and A,  Δ  < B then (Γ < A),  (A,  Δ < B) < (Γ,  Δ < B) It follows from Trans-Ground that if A < B and B < C then (A < B), (B < C) < (A < C), which is the first intuitive idea we started with; but Trans-Ground also applies in cases that are not one-one. Trans-Ground is analogous to principles connecting grounding to the logical constants—that, for instance, if A and B then A, B < (A and B) and that if Fa then Fa < Something is F. These principles tell us that certain classical inference rules (conjunction introduction, existential generalization) track grounding connections. Trans-Ground is similar, except the inference rule being claimed to track grounding connections is not classical but one in the logic of grounding. We can motivate Trans-Ground by appealing to the plausible idea that general facts are grounded by their more specific realisers. Consider the following plausible grounding claims: 1) Ball is scarlet < Ball is red 2) A < (A or B) 3) Fa < There are Fs These claims are aptly described using the locution ‘one way (among others)’. One way for the ball to be red is for it to be scarlet. One way for it to be true that A or B is for it to be true that A. And one way for there to be Fs is for a to be F. We tend to think that where some fact realises another, in this broad sense, the former grounds the latter; alternatively put, where some fact is a specific way for another fact to obtain, it is plausible that the latter's obtaining is grounded by the former's. Note that the locution ‘one way (among others)’ does not require, for its applicability, that the contrasting other ways be metaphysically possible; it can apply when there is only one metaphysically possible way for a given fact to obtain. On the standard account of the grounding of disjunctions, if A then A < (A or Not-A), whatever the modal status of the fact that A. If it is necessary that A then the fact that A or Not-A is necessarily not grounded by the fact that Not-A. Yet, the disjunction is still less specific than the disjunct, in the relevant sense. One way for A to ground C is for A to ‘go via’ B. Another (perhaps not metaphysically possible) way would be for A to ground C immediately. More generally, one way for it to be the case that Γ, Δ < B is for it to be the case that Γ < A and A, Δ < B and so—by Trans—Γ, Δ < B. The fact that Γ, Δ < B seems in the relevant sense less specific than the facts that Γ < A and A, Δ < B: we know more about the grounding hierarchy if we know that Γ < A and A, Δ < B than if we just know that Γ,  Δ  < B −  just as we know more about the apple if we know that it is scarlet than if we know merely that it is red. This asymmetry of informational content, I suspect, helps explain the plausibility of the first intuitive claim that we opened with. Trans-Ground, if it holds, is a general principle governing grounding connections; it is a generalization of the form ‘∀Γ∀A(φ → (Γ <  A))’, where ‘φ’ is a schematic sentence letter. I will use the label metaphysical law for all and only truths of this form, and their natural language equivalents.10 Examples 1–3 also seem law-like: all disjunctions are grounded by their true disjuncts; all facts of the form ‘… is red’ are grounded by facts about determinates of red; and all existentially general facts are grounded by their instances. But there is a notable difference between 1–3 and the grounding claims implied by Trans-Ground. The facts on the right of 1–3 each have a form, such that it is a metaphysical law that facts of that form are derivative. We can look just at the facts on the right of 1–3 and, as long as we know the relevant metaphysical laws, we will be in a position to know that these facts are derivative and how, broadly speaking, they are grounded.11 By contrast, it does not have to do with the form of the fact that A < C that it is grounded, as per Trans-Ground, by the facts that A < B and B < C. Not all facts of the form ‘Γ <  A’ are grounded in this manner, since some grounding connections are immediate. Does this disanalogy threaten my strategy of trying to motivate Trans-Ground by appealing to analogies between the facts of grounding it implies and examples 1–3? Not obviously. It is not obvious that whether and how a fact is grounded—and which metaphysical laws apply to it—can always be ‘read off’ from its form. What is more important, it seems to me, is whether the ‘one way (among others)’ locution applies and it seems evident that one way for A to ground C is for A to ground B and for B to ground C. That we cannot tell whether A grounds B and B grounds C if we only know that A grounds C is not obviously relevant. Turn now to minimalism's second component. The amalgamation rule is as follows (Fine 2012: 54–7): Amalg: If Γ < A and Δ < A … then Γ, Δ… < A Amalg tells us that collections of facts that fully ground A always combine to yield another full ground for A. The most obvious way to articulate the intuitive claim that Amalg yields derivative facts of grounding is as follows: Amalg-Ground: If Γ < A and Δ < A … then (Γ < A), (Δ < A)… < (Γ, Δ…  <  A) This captures the idea that Amalg, assuming it to be valid, licenses the inference of a derivative fact from facts that ground it, or would ground it if they obtained. Like Trans-Ground, Amalg-Ground is motivated by plausible general ideas. First, the same general-specific considerations apply here: we paint a richer picture of the grounding hierarchy if we say that A < C and B < C and hence—by Amalg—A, B < C than if we say merely that A, B < C. Amalg-Ground is also an instance of the plausible idea that macroscopic facts are grounded by microscopic ones. We tend to expect that facts about composite objects and pluralities are grounded by facts about their parts and sub-pluralities; as Kim puts it, this is ‘the metaphysical principle underwriting the research strategy of microreduction and the method of microreductive explanation’ (Kim 1994: 67). Assuming that Amalg itself holds, we naturally think that the ground-theoretic relations that a plurality of facts enters into can be determined by the ground-theoretic relations entered into by its subpluralities—in particular, that the facts of grounding yielded by Amalg are grounded by the those that Amalg is applied to. A further idea that motivates both Trans-Ground and Amalg-Ground (assuming Trans and Amalg themselves to hold) is that the fundamental facts in a given domain are those that suffice to minimally characterize that domain (Schaffer 2009: 377). Given Amalg and Trans, a host of facts of grounding (mediate and non-minimal ones) supervene on the class of immediate and minimal facts of grounding. More generally, Trans and Amalg fix the less immediate and minimal facts of grounding on the basis of the more immediate and minimal ones. It is then plausible to conjecture that the more minimal and immediate facts of grounding ground the less minimal and immediate ones. Supervenience is distinct from grounding, but it is often plausible that the presence of supervenience relations indicates an underlying grounding relation (Kim 1993: 167). And the supervenience in the case at hand is not mutual: we cannot recover the smaller facts of grounding from the bigger ones. We cannot, for instance, infer from the fact that Γ, Δ < A that either Γ < A or Δ < A. This has to do with the fact that there is nothing distinctive about the form of facts of grounding that determines whether they are derivative, by the lights of Trans-Ground or Amalg-Ground. IV. GROUNDING PATHS Our discussion so far has been friendly to minimalism. But minimalism, understood as the conjunction of Trans-Ground and Amalg-Ground, has plausible counterexamples.12 Begin with a problem for Trans-Ground. Given that x is scarlet < x is red and that x is red < x is coloured, Trans-Ground implies that SCA: (x is scarlet < x is red), (x is red < x is coloured) < (x is scarlet < x is coloured) SCA tells us that the fact that x is scarlet < x is coloured is grounded by the facts that x is scarlet < x is red and x is red < x is coloured. But intuitively, an object's being scarlet grounds its being coloured directly. Being scarlet is a determinate of both being coloured and being red. The relevant metaphysical law here is that if F is a determinate of G, then whenever something is F, its being F makes it G. No mention of intermediate grounding steps is made in stating this law and, intuitively speaking, it does not seem that x is F grounds x is G by virtue of grounding other determinates of G.13 Trans-Ground appears to yield dependencies when intuition says there are none.14 The second problem case targets Amalg-Ground. Consider the fact that A, B < (A or B or (A and B)). Given that A < (A or B or (A and B)) and B < (A or B or (A and B)), Amalg-Ground tells us that AMA: (A < (A or B or (A and B))), (B < (A or B or (A and B))) < (A, B < (A or B or (A and B))) AMA tells us that it is because A grounds the fact that A or B or (A and B), and B grounds the fact that A or B or (A and B), that A and B together ground A or B or (A and B). But this seems wrong. Rather, A, B jointly ground the disjunctive fact by virtue of grounding its third disjunct. To reinforce the point, consider a different disjunction: C or D or (A and B). We might reasonably think that the explanation for why A, B ground A or B or (A and B) should be the same as the explanation for why A, B ground C or D or (A and B). Both problem cases trade on the fact that one fact can ground another in different ways or, as I will say, via different paths. This notion is analogous to that of a causal path and can be illustrated with an example: A < (A or (A and B)) In a sense, A grounds the fact that A or (A and B) twice over: A fully grounds A or (A and B) ‘via’ the latter fact's left disjunct and partially grounds it (in combination with B) ‘via’ the right disjunct.15 Distinguishing paths between A and A or (A and B) enables us to make sense of this sort of idea. Grounding paths can be represented formally by structural equation models for grounding (Schaffer 2016), just as structural equation models for causation can represent causal paths. Representing such models graph-theoretically, causal paths appear as edges connecting nodes representing causally relevant variables in the model (see Weslake forthcoming: §2.). I will not try to provide a rigorous formal representation of paths here; nor will I provide anything like a complete metaphysics of paths, or answer every good question that arises about them. My aim is just to say something illuminating about how paths bear on the theory of iterated grounding and about how path-sensitive notions of grounding interact with more familiar ones.16 To that end, introduce a triadic locution ‘<p’. ‘<p’ is like ‘<’, except that it has a subscripted argument place for path-denoting terms. I will understand the subscripted argument as singular, using ‘p’, ‘p0’, ‘p1’… as singular variables ranging over paths. ‘Γ grounds A via path p’ or ‘Γ grounds A in the manner p’ are English renderings of ‘Γ <p A’. By quantifying over paths, I treat them as reified features of the grounding hierarchy. A plausible ontological view—suggested by our talk of ways for a fact to ground another fact—is that paths are properties that facts of grounding instantiate. Admittedly, this view does not sit well with Fine's project of developing an ontologically neutral account of grounding, but I will not aim to provide an ontologically neutral account of paths here. I assume that ‘<’ and ‘<p’ are intelligible, and will make free use of both expressions in what follows (I use ‘facts/claims of grounding’, from now on, so as to include facts/claims of the form ‘Γ <p A’). ‘<’ probably comes closer to expressing a pre-theoretical concept of grounding than ‘<p’. But ‘<p’ is arguably conceptually prior to ‘<’, since ‘<’ can be recovered from ‘<p’ by existentially binding the path variable:   ${\rm Equiv}{:} \, \Gamma < {\rm A \,iff} \exists {\rm p}\,(\Gamma <_{\rm p} \;{\rm A})$ This says that Γ grounds A simpliciter iff Γ grounds A via some path or other, which seems hard to deny. On the other hand, it is hard to see how we might define ‘<p’ in terms of ‘<’. Given Equiv, we can see that there are certain interactions between ‘<’ and ‘<p’. In particular, the following thesis (which plays an important role in the next section) is plausible: Gen: If Γ < (Δ <p A) then Γ < (Δ < A) We can argue for Gen by appealing to the idea that Δ’s grounding A in a specific way grounds Δ’s grounding A in some way or another; that is, if Δ <p A then (Δ <p A) < (Δ < A). This is an instance of the plausible idea that existential generalizations are grounded by their instances. Gen then follows: any ground of a fact of the form Δ <p A will, given Trans, also ground Δ < A. We will allow paths to be constructed from other paths. In connection with our discussion of amalgamation and transitivity principles, it seems we can distinguish two ways of concatenating paths: we can combine them ‘end to end’ to make longer paths, and we can combine them ‘side by side’ to make wider ones. This intuitive contrast should be reflected in our theory of paths.17 To that end, I introduce two concatenation relations. The first—elongation—is involved in applications of transitivity, and I will express it with the symbol ‘∩’: if A <p1 B and B < p2 C, A grounds C via a path constructed from p1 and p2, where these are constructed sequentially. I express this idea by saying A <p1∩p2 C. When p is not the result of elongating other paths, I will say that p is L-Simple, and that L-simple paths lack L-parts. The notion of L-simplicity is closely tied to that of immediate grounding, as follows: Γ <IMM A = df ∃p(Γ <p A & p is L-simple) Intuitively, this says that Γ immediately grounds A just in case there is some path p that is not the result of elongating any other paths, and Γ grounds A via p. Mediate grounding is defined as grounding via a path that is not L-simple: Γ <MED A = df ∃p(Γ <p A & p is not L-Simple) Mediate and immediate grounding, so defined, are compatible: the claim that Γ <MED A is compatible with the claim that Γ <IMM A, since a fact can take distinct paths, of different structures, to ground another. This allows us to accommodate the plausible idea grounding connections can be seen as somehow both immediate and mediate—like the grounding connection between the fact that A and the fact that A or (A or B) (Fine 2012: 51). Note as well that defining immediate grounding in terms of ‘<p’ has the virtue of consistency with the possibility of dense grounding: we can allow there to be grounding between A and B in the absence of a series of immediate grounding connections between them. As we noted in Section III, a defect of Fine's (2012: 51) analysis of ‘<’ as the transitive closure of immediate grounding is that it rules out this possibility. One might think there ought to a way of making sense of the immediate vs mediate distinction on which these kinds of grounding are incompatible. This is naturally done in terms of paths. For we can say that Γ immediately grounds A via p if and only if Γ <p A and p is L-simple; and that Γ mediately grounds A via p if and only if p is not L-simple. Since a path cannot be both L-simple and not L-simple, these are incompatible. Consider now the path concatenation relation at work in instances of amalgamation. If A <p1 C and B <p2 C, when we apply amalgamation, A and B together will ground C by ‘straddling’ p1 and p2. To capture the idea, we can consider the plurality A, B as jointly grounding C via a path comprised of p1 and p2, combined ‘side by side’. To express this relation of concatenation—call it widening—I will use the symbol ‘+’, and say that A, B <p1+p2 C. When p is not the result of widening other paths, I will say that p is W-Simple, and that W-simple paths lack W-parts. The notion of W-simplicity is closely tied to that of minimal grounding, as follows: Γ <MIN A = df ∃p(Γ <p A & p is W-Simple) This says that Γ minimally ground A just in case there is some path p that is not the result of widening any other paths, and Γ grounds A via p. Non-minimal grounding is defined as grounding via a path that is the result of widening other paths: Γ <NON-MIN A = df ∃p(Γ <p A & p is not W-Simple) Minimal and non-minimal grounding, so defined, are compatible. Again, one might think there ought to a way of making sense of the minimal vs non-minimal distinction on which these kinds of grounding are incompatible. This is easily done: say that Γ minimally grounds A via p if and only if Γ <p A and p is W-simple; and that Γ non-minimally grounds A via p if and only if Γ <p A and p is not W-simple. Since a path cannot be both W-simple and not W-simple, these are incompatible. Let us consider how appealing to paths helps resolve the problem cases we discussed at the beginning of this section. There are distinct paths that the fact that x is scarlet takes to ground the fact that x is coloured. We can represent the case diagrammatically: Once we separate p1, p2 and p3, we can distinguish the following claims in SCA’s vicinity: SCA1: (x is scarlet <p1 x is red), (x is red <p2 x is coloured) < (x is scarlet <p1 ∩p2 x is coloured) SCA2: (x is scarlet <p1 x is red), (x is red <p2 x is coloured) < (x is scarlet <p3 x is coloured) SCA1 says that, relative to the path comprised of p1 and p2, that x is scarlet < x is coloured is grounded by the fact that x is scarlet < x is red and that x is red < x is coloured. I think that someone attracted to the intuitive idea that the transitivity principle tracks grounding connections should endorse this claim, as an expression of that idea. At least, we can accept it while offering a plausible diagnosis of what is implausible about SCA. For SCA is implausible if it is interpreted as SCA2: the immediate grounding connection via path p3 does not depend on the grounding connections mediated by p1 and p2. Once we take account of paths, we find that there is a falsehood (i.e., SCA2) and a truth (i.e., SCA1) in SCA’s vicinity. The moral is that we need to formulate minimalism in such a way that the truth, and not the falsehood, is implied by it. Consider now the second problem case. Supposing there to be paths corresponding to each disjunct of a disjunctive fact, the fact that A or B or (A & B) has three paths leading to it. The grounding connection that takes the path corresponding to the fact's third disjunct is an instance of the law that truths ground their conjunction, and this does not involve an application of Amalg. But if we consider how A and B ground A or B or (A and B) as disjuncts—rather than conjuncts of a disjunct—then A and B’s jointly grounding the disjunctive fact does plausibly depend on an application of Amalg, and on the ground-theoretic connections between the disjunctive fact and A and B individually. Now to reformulate minimalism in terms of ‘<p’, so that our two problem cases are resolved in these ways. We start with the analogue of Trans for ‘<p’, which is as follows. TransPATH: If Γ <p1 A and A, Δ <p2 B then Γ,  Δ <p1∩p2 B TransPATH tells us that if some facts Γ ground A via path p1, and if A and Δ ground B via p2, then Γ and Δ together ground B via a path comprised of p1 and p2. TransPATH goes beyond Trans by telling us that how Γ,  Δ < B is determined by the ways that Γ < A and A, Δ < B (in addition to telling us that that Γ,  Δ < B is determined by the facts that Γ < A and A, Δ < B). The analogue of Amalgamation is: AmalgPATH: If Γ <p1 A and Δ <p2 A then Γ,  Δ <p1+p2 A Minimalism then becomes—and, from now on, will be understood as—the conjunction of the following two theses: Trans-GroundPATH: If Γ <p1 A and A, Δ <p2 B then (Γ <p1 A), (A, Δ <p2 B) < (Γ,  Δ <p1 ∩ p2 B) Amalg-GroundPATH: If Γ <p1 A and Δ <p2 A then (Γ <p1 A), (Δ <p2 A) < (Γ,  Δ <p1+p2 A) We argued above for the principle Gen: that whatever grounds the fact that Γ <p A will also ground that Γ < A. The following theses then follow from minimalism: Trans-GroundGEN: If Γ <p1 A and A, Δ <p2 B then (Γ <p1 A), (A, Δ <p2 B) < (Γ, Δ < B) Amalg-GroundGEN: If Γ <p1 A and Δ <p2 A then (Γ <p1 A), (Δ <p2 A) < (Γ, Δ < A) This shows that, although minimalism is in the first instance an account of how certain facts of the form ‘Γ <p A’ are grounded, it also provides an account of how certain facts of the form ‘Γ < A’ are grounded. This potentially brings minimalism into competition with other such accounts, as we will now see. V. GENERAL THEORIES AND OVERDETERMINATION Many philosophers deny the existence of systematic causal overdetermination, a view that plays a prominent role in discussions of mental causation and material constitution.18 The topic of ground-theoretic overdetermination remains relatively unexplored but there are plausibly some cases of systematic ground-theoretic overdetermination. Any existential generalization with multiple instances and any disjunction with multiple true disjuncts will be ground-theoretically overdetermined. These cases of systematic overdetermination do not seem problematic. At least, the threat of overdetermination is not commonly regarded as a reason to reject the thesis that existential generalisations are grounded by their instances, or that disjunctions are grounded by their true disjuncts. I suspect that the reason these cases are not considered problematic is that, in each case, the overdetermination follows from a single and plausible idea. In the disjunctive case, the plausible idea is that disjunctions are grounded by their true disjuncts; in the existential case, the plausible idea is that existential generalizations are grounded by their instances. Widespread overdetermination just follows from these plausible ideas. Despite the multiplication of grounds, we have in these cases a unity of theory: there are not different competing theories of how disjunctive facts are grounded, or different metaphysical laws involved. Surely not all cases of ground-theoretic overdetermination are innocent, though. Suppose I claim that the vase's brittleness is fully grounded by some fact about the structure of its constituent atoms, but add that it is also fully grounded by facts about angels—angels always glare at brittle things, and this makes them brittle. And suppose I add that neither grounds the other, nor helps the other ground the vase's brittleness. This seems an implausible combination of views, because it involves an unparsimonious disunity of theory: I am running in tandem what should be two competing theories about how the vase's brittleness is grounded. The different grounds I posit for the vase's brittleness do not follow from a unified underlying theory; the metaphysical laws involved will inevitably be different.19 The claim that this sort of overdetermination is widespread is extremely suspect. To the extent that this attitude is warranted, the above defence of minimalism puts pressure on certain general theories of iterated grounding. One leading thesis about iterated grounding is the superinternality thesis, developed independently by Bennett (2011) and deRosset (2013): Superinternality: If Γ < A then Γ < (Γ < A) According to Superinternality, facts of grounding ‘bottom out’ by being grounded by the facts on their left-hand side. Even if some facts of grounding ground others, repeated applications of Superinternality will presumably deliver facts that are not facts of grounding, which ground the whole grounding hierarchy. This indeed is part of the motivation for Superinternality. Bennett and deRosset both seek to avoid fundamental facts of grounding, and cite the ability of Superinternality to deliver this result as a count in its favour. Superinternality applies to all facts of grounding, whereas minimalism provides grounds only for some facts of grounding—those that are the result of applying the transitivity and amalgamation principles. Consider any such fact of grounding, Γ < A. Minimalism implies that this fact is grounded by other facts of grounding. Superinternality predicts that it is grounded by Γ. These—the worry goes—are distinct, competing theories about how the fact in question gets grounded. We get a clearer view of the sort of overdetermination involved by considering an example. Consider a case involving Trans-GroundGEN. Suppose that A <p1 B, B <p2 C, and consider how the fact that A < C is grounded. Minimalism—specifically, its corollary Trans-GroundGEN—implies ATO: (A <p1 B), (B <p2 C) < (A < C) Superinternality implies SUP: A < (A < C) ATO and SUP seem to be distinct, competing theories about how the fact that A < C is grounded. Moreover, this kind of overdetermination will be systematic, arising for any instance of mediate grounding. Perhaps defenders of Superinternality should extend their view to accommodate paths. If so, it is in the spirit of Superinternality to supplement the thesis with the following one: SuperinternalityPATH: If Γ <p A then Γ < (Γ <p A) SuperinternalityPATH is of a piece with Superinternality: both are expressions of the idea that features of the grounding hierarchy are grounded by its bottom layer. According to SuperinternalityPATH Γ does not just determine that Γ < A; Γ determines how Γ < A. SuperinternalityPATH provides grounds for the facts on the left of ATO: A < (A <p1 B) and B < (B <p2 C). Given Trans, it follows that: A, B < (A < C)20 And the superinternalist might push farther still: since we have assumed that A < B, Trans lets us infer from A, B < (A < C) that A, A < (A < C). Assuming that if A, A < B then A < B, this gives us the thesis that A < (A < C). In sum, the superinternalist can plausibly resist the charge that she is committed to positing multiple fundamental grounds for the fact that A < C, for by her lights it is plausible to maintain that A is the only fundamental ground for the fact that A < C (or, if A is not itself fundamental, A’s fundamental grounds will play this role). These remarks are suggestive but there is reason to doubt that, on their own, they mitigate the worry that combining minimalism and Superinternality results in problematic overdetermination. This is because ATO and SUP are plausibly understood as identifying immediate grounds for the fact that A < C. If that is how we understand ATO and SUP, their conjunction will commit us to saying that A grounds the fact that A < C in two ways: both immediately (as per SUP) and mediately (by applying SuperinternalityPATH to ATO). So, although combining ATO and SUP arguably does not multiply fundamental grounds for the fact that A < C, it does multiply the paths we posit between A and A < C; it multiplies, as it were, the non-causal mechanisms connecting A to the grounded fact, that A < C. And this seems problematic, because it involves an objectionable disunity of theory. That A < C is grounded in two ways does not follow from a single plausible idea (thus this case contrasts with the unproblematic overdetermination we find when a disjunction has many true disjunctions). It follows from running together distinct theories about how the fact that A < C is immediately grounded. We are familiar with isolated cases in which distinct causal mechanisms connect a cause to one of its effects. For instance, suppose that Jimbo throws a ball at a window, Jane sees this happen and immediately throws another ball at the window, and both balls hit the window and break it simultaneously. But the ground-theoretic overdetermination we are considering would be systematic, arising for any instance of mediate grounding. To that extent, it seems problematic: we have two generally applicable theories about how facts of mediate grounding are immediately grounded, and this seems unparsimonious. So if ATO and SUP are both claims of immediate grounding—and if, more generally, minimalism and Superinternality are in the business of stating immediate grounds for the facts of grounding they concern—then the tension between these theories of iterated grounding seems to remain. Unfortunately, we lack clear and informative diagnostics for immediate grounding, although, as Fine (2012: 51) points out, we do seem to have strong intuitions in particular cases about when a given grounding connection is immediate.21 A plausible conjecture is that the fundamental metaphysical laws are laws of immediate grounding. We can make an initial case for this by considering examples. It is (we may suppose) a fundamental metaphysical law that the existence of any non-empty set is grounded by the existence of its members; and plausibly non-empty sets are immediately grounded by their members. It is (we may suppose) a fundamental metaphysical law that the existence of any composite object is grounded by the existence or arrangement of its constituents. This law again seems to track immediate grounding connections—composites seem to be immediately grounded by the existence and arrangement of their constituents. Contrast these cases with the non-fundamental metaphysical law that the existence of a non-empty set is grounded by the existence and arrangement of the constituents of its composite members: this law results from chaining two more fundamental laws (i.e., the law governing the grounding of sets and that governing the grounding of composite objects): its instances are all instances of mediate grounding. It is, then, initially plausible that the following thesis holds: Imm-Law: If L is a fundamental metaphysical law, then its instances are instances of immediate grounding.22 If we regard Superinternality as a fundamental metaphysical law then there is some pressure on us to accept the following thesis (recall, ‘<IMM’ expresses immediate full grounding): SuperinternalityIMM: If Γ < A then Γ <IMM (Γ < A) SuperinternalityIMM goes beyond Superinternality; to my knowledge, extant defences of Superinternality do not explicitly commit to SuperinternalityIMM. SuperinternalityIMM brings the Superinternality thesis into conflict with minimalism, since SuperinternalityIMM and minimalism are competing theories of how a large class of facts of grounding—the ones that both theories apply to—are immediately grounded. To avoid commitment to overdetermination, superinternalists who accept minimalism should reject SuperinternalityIMM and also the thesis that Superinternality is a fundamental metaphysical law. Instead, they should regard their theory as an account of how the most fundamental facts of grounding are immediately grounded—allowing minimalism to provide immediate grounds for the less fundamental facts of grounding. After all, the main motivation for the superinternality thesis is that it avoids commitment to fundamental facts of grounding, and to establish this we need not provide immediate grounds for facts of grounding that minimalism applies to. Superinternalists might then take the fundamental metaphysical law in which their thesis is articulated to be as follows: SuperinternalityWEAK: If Γ < A, and there is no Δ that only contains facts of grounding and is such that Δ < (Γ < A), then Γ < (Γ < A) The logical relationship between SuperinternalityWEAK and Superinternality is debatable, and depends on the background special theory of iterated grounding assumed. What matters for us is that the thesis that SuperinternalityWEAK is a fundamental metaphysical law does not entail that Superinternality is one—and SuperinternalityWEAK does not imply SuperinternalityIMM. SuperinternalityWEAK is a thesis about how the most fundamental layer of facts of grounding is grounded by facts that are not themselves facts of grounding. Once this layer of facts of grounding is accounted for, minimalism (or perhaps some other special theory of iterated grounding) is invoked to provide immediate grounds for the other facts of grounding. The resulting picture witnesses the division of labour described in Section II. Our account of the fundamental metaphysical laws by which facts grounding are grounded divides across the distinction between general and special theories of iterated grounding. Dividing the topic up like this is a promising strategy for avoiding the threat of overdetermination that arises from combining the superinternality thesis with minimalism. A problem with this strategy, however, is that it is speculative that all facts of grounding can be accounted for in this manner. Suppose that A <p B, and that p is the only path connecting A to B. Further, suppose that p is ‘gunky’: that p is not L-simple, and that none of p's L-parts is L-simple either. (This case illustrates a path-sensitive analogue to the notion of dense grounding that was introduced in Section II.) If we were to represent p's structure by listing its L-parts connected by ‘∩’, our representation of p would be isomorphic to the real number series. p mediates an infinity of ever more immediate grounding connections, but no immediate grounding connections. Divide p in half: p = p1 ∩ p2, where A <p1 C and C <p2 B. Minimalism implies that the fact that A <p B is grounded by these two facts of grounding. But not only these: p1 and p2 can be split up into L-parts which—according to minimalism—mediate still more fundamental grounding connections. On the minimalist's picture, that A <p B is grounded by an infinite series of ever longer sequences of ever more immediate grounding connections, without end: an infinitely descending chain of facts of grounding. SuperinternalityWEAK would not apply to any of the facts in this chain, because each such fact would be grounded by other facts of grounding. It would simply have nothing to say about this part of the grounding hierarchy. Similarly, suppose that Γ <p B, and that every proper W-part of p itself has a proper W-part. Intuitively, we could keep removing facts from Γ to yield a more minimal ground for B, never arriving at a minimal ground. There is, in fact, a non-absurd model for this scenario. That there are infinitely many numbers might be grounded by the fact that 1 is a number, 2 is a number… and so on for all the natural numbers—call this plurality of facts ‘N’. N does not minimally ground that there are infinitely many numbers, because the following proper sub-plurality of N would suffice as well: 2 is a number, 4 is a number… and so on for all the even numbers. Indeed, any proper sub-plurality of N that grounds the fact that there are infinitely many numbers will itself have a proper sub-plurality that does so as well. Consider, then, the following facts of grounding 1 is a number, 3 is a number, 5 is a number … <p1 There are infinitely many numbers. 2 is a number, 4 is a number, 6 is a number … <p2 There are infinitely many numbers. Applying AmagPATH to these facts of grounding yields 1 is a number, 2 is a number, 3 is a number…<p1+p2 There are infinitely many numbers. According to minimalism—specifically, Amalg-GroundPATH—this fact of grounding is less fundamental than those to which AmalgPATH was applied to. And we can repeat the reasoning—for example, p1 in turn can be split into W-parts, p3 and p4, such that: 1 is a number, 5 is a number, 9 is a number … <P3 There are infinitely many numbers. 3 is a number, 7 is a number, 11 is a number … <P4 There are infinitely many numbers. We can keep dividing N into subsets to arrive at ever more minimal grounds for the fact there are infinitely many numbers—grounds that, by the lights of minimalism, are ever more fundamental—without arriving at minimal grounding connections between subsets of N and the fact that there are infinitely many numbers. Commitment to infinitely descending chains of facts of grounding is not the same as commitment to fundamental facts of grounding. But, if we reject fundamental grounding, the former commitment hardly seems palatable. For, while each step in an infinitely descending chain might be accounted for, the chain itself would not be. There would seem to be an important sense in which this part of the grounding hierarchy has not been accounted for by metaphysically more fundamental matters. We have focussed on the superinternality thesis, but the discussion generalizes to other general theories of iterated grounding. Abstractly put, the threat of overdetermination arises whenever the special and general theories of iterated grounding imply—in a systematic way—distinct immediate grounds for facts of grounding. Restricting the general theory to the most basic layer of the facts of grounding is always an option. This sidesteps the threat of overdetermination, because the general and special theories will then be concerned with different facts of grounding—the special theory will only pick up where the general theory leaves off. But if our ambition is to show that the existence and features of the grounding hierarchy are explicable in terms of more fundamental matters, it will likely be frustrated by the existence of (perhaps even the mere possibility of) infinitely descending chains of facts of grounding, like those described in the preceding paragraphs. VI. CONCLUSION We have seen that transitivity and amalgamation principles plausibly yield facts of grounding that are grounded by others. In order to formulate minimalism, the view in which this intuitive idea finds expression, we found that it is important to pay attention to the notion of a grounding path. We have also discussed interactions between minimalism and Superinternality—and, by extension, other general theories of iterated grounding—finding that there is a threat of objectionable overdetermination that arises from combining minimalism with extant theories of iterated grounding. More broadly, we have seen that the theory of iterated grounding is not exhausted by the general theory, and that neglecting the special theory might indeed impoverish our general theory.23 Footnotes 1 Trogdon (2013: 16) cites unpublished work by Dasgupta defending this kind of grounding claim. 2 The transitivity principle seems more intuitively plausible than the amalgamation principle (although see Schaffer 2012 for scepticism about the former). Fine endorses the amalgamation principle for theoretical reasons: ‘I doubt that there is simple and natural account of the logic of ground that can do without it’ (2012: 57). I simply assume the amalgamation principle here. A good question—laid aside for reasons of space—concerns whether an analogue of the second claim would remain plausible if we replaced the amalgamation principle with another describing conditions under which we can add to the grounds of some fact A to yield another ground for A. Arguably, some such ‘augmentations’ of a fact's grounds are legitimate (see Clark 2015; Litland 2013: 21; Scott-Dixon 2016; but see Audi 2012: 699 for a dissenting voice). Thanks to an anonymous referee here. 3 Compare Armstrong (2004: 19–21) discussion of the related notion of minimal truthmaking. 4 I take the term ‘iterated grounding’ from Litland (forthcoming). 5 It is fairly plausible that the fact that A does not ground B might be grounded in the fact that B grounds A, given that grounding is asymmetrical. The cases we discuss are importantly different from this, since they concern positive facts about what grounds what, rather than negative facts. 6 See Bennett (2011), Dasgupta (2014), Litland (forthcoming), Rosen (2010)and deRosset (2013). For an overview, see Bliss and Trogdon (2014: sect. 7). Sider ms argues that we should not expect a fully general account of iterated grounding; as the next section indicates, I am sympathetic to this view. 7 Arguably, expressing grounding with a sentential connective allows us to avoid ontological commitment to a grounding relation and its relata (Correia 2010: 254; Fine 2012: 47). But to keep the discussion idiomatic, I will often write as though there is a grounding relation between facts. 8 The distinction between special and general theories is not often drawn but Trogdon (2013: 116) suggests it, citing unpublished work by Dasgupta. As I have drawn it, the distinction is not exhaustive, for it leaves room for mixed theories of iterated grounding that conjoin special and general theories; and for theories that entail claims of the form ‘Γ < (Δ < A)’, where some but not all members of Γ are claims of grounding. But the distinction is not arbitrary: extant theories of iterated grounding tend to be general in the sense defined, and the theory proposed in this paper is special in the defined sense. 9 Thanks to an anonymous referee for prompting me to comment on these issues. 10 My use of ‘metaphysical law’ is different to that of Wilsch (2016), on which the metaphysical laws are ‘general principles that govern construction-operations’ (Wilsch 2016: 4). Roughly, such principles describe how certain operations generate entities from other entities. See Wilsch (2016: 3–8) for further discussion. 11 Compare the discussion of propositional forms in Rosen (2010: 131). 12 Thanks to an anonymous referee for raising these problem cases. 13 The case might seem still more problematic if whenever F is a shade of G, there is a third shade of G that F is a more determinate shade of—for then there would be an infinite series of grounding connections between x's scarletness and its being coloured. 14 I use ‘dependence’ here and below to express the converse of grounding and not the concept of ontological dependence, as discussed by Fine (1995), Lowe (1994) and others. 15 The notion of a path, or one very similar, is invoked by Fine (2012: 51). 16 Schaffer (2016: 83–6) argues that his structural equations approach competes with Fine's framework for grounding, and that Fine's is a less illuminating setting for theorising about grounding. I do not see the present regimentation of paths—which extends Fine's framework—as competing with the structural equations framework. It is relatively straightforward, and adequate for the purpose of making some philosophical claims about iterated grounding, but these claims should survive being transplanted into a graph-theoretic setting. They are artefacts, not of regimentation, but of philosophical intuition. 17 Thanks to an anonymous referee here. 18 For overdetermination in the context of mental causation, see Kim (1989, 1993) and Malcolm (1968); an overview is provided by Heil and Robb (2014: § 6.2). For overdetermination in the context of material constitution see Merricks (2001) and Paul (2007, 2010). Sider (2003) questions the motivation of scepticism about systematic overdetermination. 19 It is sometimes said that problematic cases of causal overdetermination involve overdetermination by events of the same type; see Dretske (1988: 42ff) and Jaworski (2016: 280 ff). There seems to be no problem in assuming that the case described meets any such additional requirements for problematic ground-theoretic overdetermination. 20 The argument is as follows: (P1) (A <p1 B), (B <p2 C) < (A < C) [ATO]. (P2) A < (A <p1 B) [From SuperinternalityPATH]. (P3) A, (B <p2 C) < (A < C) [From Trans, P1, P2]. (P4) B < (B <p2 C) [From SuperinternalityPATH]. (C) A, B < (A < C) [From Trans, P3, P4]. 21 Poggiolesi (forthcoming) offers an analysis of the related notion of immediate formal grounding. 22 The existence of a set can be mediately grounded by the existence of its members, as in the case of {a {a}}, and the existence of a composite can be mediately grounded by features of its constituent matter, as where some object's constituents also constitute another of its constituents. But this is not in tension with the Imm-Law: we have seen, in Section 4 that immediate grounding does not rule out mediate grounding by another path. 23 This paper has benefitted from discussion at Hamburg's Forschungskolloquium seminar. I am especially grateful to Stephan Krämer, David Liggins, Giovanni Merlo, Stefan Roski, Kelly Trogdon and Nathan Wildman for providing very valuable comments on previous versions of this paper. This research was carried out as part of the Sinergia project Grounding: Metaphysics, Science, and Logic, which is funded by the Swiss National Sciences Foundation; I am very grateful for their support. REFERENCES Armstrong D. ( 2004) Truth and Truthmakers . Cambridge: CUP. Google Scholar CrossRef Search ADS   Audi P. ( 2012) ‘ Towards a Theory of the ‘in Virtue of’ Relation’, Journal of Philosophy , 109: 685– 711. Google Scholar CrossRef Search ADS   Bennett K. ( 2011) ‘ By Our Bootstraps’, Philosophical Perspectives , 25: 27– 41. 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