Abstract
I propose a way to accommodate plausible examples of cyclic grounding chains that have been proposed in the literature while preserving a good measure of the metaphysical foundationalist’s grounding hierarchy with a foundational level. This precludes the need to adopt a strong form of coherentism, such as the view that everything grounds everything, or even the view that everything grounds everything other than itself. I do this by developing axiomatizations of grounding which allow for localized cycles of ground at the non-foundational and foundational levels, but which still place certain foundationalism-like constraints on the grounds that grounded facts must have. In addition, I point to considerations, previously put forth in favor of foundationalism and a strong form of coherentism, that also help support the views I develop.
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Notes
Gabriel Rabin (2018) argues that conceptions of grounding which deny one or more of irreflexivity, antisymmetry, transitivity, and the aforementioned foundationalism axiom are consistent with reality being layered hierarchically. But he captures the latter idea in terms of relative fundamentality, not in terms of grounding, and is ultimately out to show only that the status of relative fundamentality as a strict partial order can be preserved even if grounding fails to count as such. In contrast, I’m out to investigate how much of a ground-imposed hierarchy (with a foundational layer) can be preserved while denying that grounding obeys all four of these axioms. In an interesting investigation into several forms of metaphysical coherentism, Jan Swiderski (2022) develops a view, which he calls ‘rebarism’, that is similar to the view I develop here. (His paper was published while this one was undergoing the review process.) Rebarism differs from metaphysical foundherentism in three substantive ways. First, rebarism is stated in terms of partial ground only, and so does not provide conditions for the full grounds of facts, as does foundherentism. Second, it is stated in terms of grounding, rather than the transitive closure of grounding, and so is incompatible with versions of coherentism on which transitivity is false. Third and most substantively, everything in a rebarist world, as Swiderski says, must be “either (i) part of a web of mutual grounding or (ii) grounded by such a web (but not both)” (2022). Rebarism is therefore incompatible with non-foundational cyclic grounding chains. It is also incompatible with the existence of ungrounded facts, and with the existence of disjoint webs of grounding that together ground all other facts. Each of these things is possible according to metaphysical foundherentism.
I therefore take the view that the fundamental notion of grounding is expressed by a predicate rather than an operator. Despite this, what is said below can be recast in operationalist terms. For further discussion of the predicate-operator debate, see Correia (2010, pp. 245–254, 2020, pp. 239–240).
While many authors restrict the grounding relation to facts, others [notably, Jonathan Schaffer (2009, pp. 375–376)] are more permissive about the sorts of things that may stand in the grounding relation. I frame the following discussion in a way that is neutral between the two sorts of view.
See Rosen (2010, p. 115), Audi (2012, p. 698), Fine (2012a, p. 50), and Raven (2013, p. 194) for endorsements of this definition. Despite its wide acceptance, this definition is not completely free from controversy. Some argue that a partial ground of something need not be among some things which fully ground it. See, for example, Leuenberger (2020) and Trogdon and Witmer (Trodgdon and Witmer (2021)). I ignore this issue in what follows. Nothing I argue for depends upon it as far as I know.
Regard the name of this principle as an abbreviation for ‘Irreflexivity of Full Grounding’. Mutatis mutandis for the Full and Partial versions of the principles that follow.
Jenkins actually describes the case she has in mind rather generally, as a truth that is its own truthmaker. But I suspect she has an example like this in mind. See also Pleitz (2020, pp. 194–195).
Wilson (2014, pp. 571–572) also mentions this general sort of case, though she doesn’t mention Leibniz specifically.
Martin Pleitz (2020, sec. 3) provides a detailed discussion of this sort of case.
Wilson (2012) argues that determinable properties should be considered to be among the fundamental entities in the world, despite (instantiations of them) being grounded by (instantiations of) appropriate determinate properties. But she does not argue that (instantiations of them) ground (instantiations of) determinate properties. So her argument there does not, on its own, support the idea that cyclic grounding chains exist.
The same result would follow if it is supposed instead that, in general, if p is true, then the proposition that p is true is grounded in p. See Thompson (2018, pp. 110–111) and example J2 above.
‘Smallest relation’ should be understood extensionally, i.e., its extension contains all the ordered pairs that the extension of partial grounding contains, plus any more ordered pairs that are needed to ensure the truth of Partial Transitivity, and no other ordered pairs.
One might have noticed that the above definition allows for “pinched” cyclic grounding chains. That is, it allows for grounding structures like the following to count as cyclic chains: a grounds b, b grounds c, c grounds b, b grounds a, and no other grounding claims hold. Represented diagramatically, this structure has the look of a cyclic grounding chain that has been “pinched” together at b. While one might, intuitively, think that such a structure ought not to count as a single cyclic grounding chain, it will according to the definition given. Ultimately, however, it doesn’t matter which choice is made; the points I make below hold of the category I call ‘cyclic grounding chains’, even if that name is ultimately reserved for only a species of that genus.
I ignore the following singular variants of weak and strong coherentism.
Strong Singular Full Coherentism. Everything is fully grounded by everything. Weak Singular Full Coherentism. Everything is fully grounded by everything other than it.
These are entailed by strong full coherentism and weak full coherentism, respectively. But denying one of these more general theses while espousing its singular variant clashes with the extremely plausible and widely accepted view that some entities, e.g., conjunctive facts, whose conjuncts are distinct and each true) are fully grounded (collectively) only by more than one entity.
Strong weak partial coherentism does not imply strong/weak full coherentism without further assumptions about the behavior of grounding. Of course, the right-to-left direction of the definition of partial grounding must be true. See fn. 5 above. But suppose that there exist just three entities a, b, and c, and that a is fully grounded by b and partially but not fully grounded by c. Then a is partially grounded by everything other than it, so weak partial coherentism is true, but it does not automatically follow, given just partial-orderism, that a is fully grounded by b, c. Indeed, if Paul Audi’s (2012: 699) principle Minimality is true, then it follows that a is not fully grounded by b, c.
Minimality. If x is fully grounded by \(\Gamma \), then, for any y not among \(\Gamma \), it is not the case that x is fully grounded by \(y, \Gamma \).
Elsewhere (Dixon 2016a: sec. 5) I have argued that Minimality is false. But Minimality is stronger than it needs to be to secure the result that partial coherentism doesn’t imply full coherentism. For the implication to hold, it would be enough that
Amalgamation \((<, \prec )\). If x is fully grounded by \(\Gamma \) and partially grounded by y, then x is fully grounded by \(y, \Gamma \).
But my (ibid. 387) replacement for Minimality,
Full Non-Monotonicity. If something is partially grounded by something, then it is not the case that, for any x, y, and \(\Gamma \), if x is fully grounded by \(\Gamma \), then x is fully grounded by \(y, \Gamma \).
which, I argue, is better motivated than Minimality by the consideration Audi provides in support of Minimality, isn’t this strong, and doesn’t support the implication. Further, Amalgamation \((<, \prec )\) does not appear to be derivable in Fine’s (2012b) system of the pure logic of ground. A model in which b strictly fully grounds a and c strictly partially grounds a but c does not strictly fully ground a, and in which b, c strictly fully ground a, seems to be consistent with, but not forced by, the rules Fine gives on p. 5.
At least if we are considering all possible grounding structures and not just actual ones. If, in the actual world, no entity grounds itself, then grounding would be irreflexive if its domain is only actual entities, even if it were possible for there to be cyclic grounding chains. Similarly, if there were no failures of transitivity in the actual world, then it would be transitive. But I take each of the axiomatizations of grounding that I discuss (partial-orderism, transitive grounding, and irreflexive grounding) to hold of metaphysical necessity if it holds at all.
I ignore the option of rejecting both Full Irreflexivity and Full Cut. It does not impose any constraints on the behavior of grounding, but even each of Strong Full and Strong Partial Coherentism ensures that one of Full Cut and Partial Transitivity is true. Thanks are due to an anonymous referee who suggested a particularly interesting version of this view: a hybrid view according to which the domain of grounding is partitioned into subsets on some of which only Full Cut is true and on the others of which only Full Irreflexivity is true. Interesting as this view is, I must leave it aside in this paper in the interests of space. But what I say below about each of Transitive Grounding and Irreflexive Grounding will apply to each of the relevant subsets of the domain of grounding in such a partitioned world, so some light will be shed on the view in what follows.
See Dixon (2016b, p. 446) and Rabin and Rabern (2016, pp. 363–364) for equivalent statements of this foundationalism axiom. Other foundationalism axioms that have been considered in the literature include:
-
(F1)
Every grounding structure \(\Gamma \) is well-founded, i.e., it contains a ground-minimal element, i.e., an element x such that no y among \(\Gamma \) even partially grounds x. [See Trogdon (2013, p. 108), Tahko (2014, p. 260), Dixon (2016b, sec. 5), and Rabin and Rabern (2016, p. 358).]
-
(F2)
Every grounding chain \(\Gamma \) terminates, i.e., there is an x among \(\Gamma \) that grounds every other y among \(\Gamma \). [See Schaffer (2010, p. 37), Bennett (2011, p. 30), Dixon (2016b, sec. 4) and Rabin and Rabern (2016, sec. 3.3).]
-
(F3)
Every maximal grounding chain \(\Gamma \) terminates, i.e., if \(\Gamma \) is a grounding chain such that nothing even partially grounds everything in it, then something in it grounds everything else in it. [See Dixon (2016b, sec. 6). For equivalent axioms, see Trogdon (2013, p. 108) and Rabin and Rabern (2016, sec. 4.2).]
-
(F4)
Every weakly maximal grounding chain \(\Gamma \) terminates, i.e., if \(\Gamma \) is a grounding chain such that no thing or things fully ground everything in some final segment of it, then something in it grounds everything else in it. [See Dixon (2016b, sec. 7).]
-
(F5)
Every non-collapsible grounding structure is well-founded, where a structure \(\Gamma \) is non-collapsible \(=_{df}\) every grounding chain x grounds y grounds z among \(\Gamma \) is such that x’s grounding z is grounded by x’s grounding y together with y’s grounding z. [See Pleitz (2020, sec. 10).]
These definitions will, like Full Foundations (see discussion in the main text below), need to be revised for contexts in which partial-orderism is not taken for granted. (F1) may need to be revised in such a way that allows for the ground-minimal element to be grounded by itself, though nothing else among \(\Gamma \). (F2), (F3), and (F4) will need to be revised so that the thing in it that grounds everything else in it needs merely to ground\(^{+}\) everything else in it. (F2), (F3), (F4) may also need to be revised to ensure that the thing that ground\(^{+}\)s everything else in the chain is not ground\(^{+}\)ed by anything else in the chain. The definition of maximal chains in (F3) will need to be revised so that nothing partially ground\(^{+}\)s everything in a maximal chain. The notion of a final segment of a chain, which appears in (F4), will need to be defined in terms of the transitive closure of partial grounding\(^{+}\), rather than in terms of partial grounding itself, as it is in Dixon (2016b, sec. 7). And (F5) will need to be revised to apply only to chains x grounds y grounds z in \(\Gamma \) such that x also grounds z. The axiom I will make use of in my formulation of metaphysical foundherentism is an analog of Full Foundations. It may be that systems slightly stronger and weaker than the version of metaphysical foundherentism I develop result from pairing one of Full Cut and Full Irreflexivity (or neither) with a foundherentist analog of the appropriately revised version of one of these five alternative foundationalism axioms. Due to considerations of space, however, I must leave the exploration of any such alternative versions of metaphysical foundherentism for another occasion.
-
(F1)
Some take foundationality and fundamentality to be the same thing [see, for example, Dixon (2016b, p. 442) and Rabin and Rabern (2016, p. 354, fn. 9)]. Others do not, thinking instead that fundamentality can’t be characterized (solely) in terms of foundationality [see, for example, Raven (2016)]. I do not take a stand on this issue in what follows, and speak only of foundational entities.
To ensure that sense can be made of the phrase ‘the smallest relation’, each of the relations full grounding and R can be understood as a binary operation which holds between an entity and a (non-empty) set. To accomplish this, the plural variables in the above definition can be interpreted as set variables in the present context, ranging over sets of entities in the domain of the full grounding relation. Then the phrase can be interpreted in the usual way, i.e., as the relation identical to the intersection of every set which meets conditions (i) and (ii).
See the appendix for a proof of the claim that every cyclic chain will be a subchain of a unique maximal cyclic chain.
See the appendix for a proof that the definiens is equivalent to the claim that x is either foundational or an element of a foundational cyclic chain.
Bennett (2017, p. 136) articulates something very close to the notion of foundherentality, as just defined, with Independence* (“x is independent* just in case for all y such that y builds x, x builds y”), though she does not make allowance for non-transitive building relations. Her alternative characterization of the notion in fn. 33 (“x is independent* just in case it belongs to a set S none of whose members are partially built by anything outside S”) is reminiscent of my preliminary characterization of foundherentality in terms of foundational cyclic chains in the main text—as something that is either foundational or a member of a foundational cyclic chain. But her characterization is too permissive. Any entity x in the domain D of grounding (suppose for simplicity that grounding is the only building relation) counts as Independent*. No member of D is partially grounded by anything outside D, so x belongs to a set none of whose members are partially grounded by anything outside that set. The notion of foundherentality is also similar to Raven’s (2016) notion of ineliminability, though his characterization of this notion is sensitive to the repetition, as one moves groundward, not of entities that stand in the grounding relation to one another, but of their constituents. Still, it is clear that foundherentality has important ties to conceptions of fundamentality in terms of ineliminability. Compare also to Leuenberger’s (2020) notion of all-grounding. See fn. 29 below.
Structurally, metaphysical foundherentism differs substantially from the epistemological version developed by Susan Haack (1983, 1993). A notable difference is that, on Haack’s view, coherence plays a role throughout the entire set of one’s beliefs. While one will have foundational beliefs, the coherence of all other of one’s beliefs helps confer justification on those beliefs. It also differs from the weak foundationalism that has been attributed to Lewis (1946), e.g., in Olsson (2021, sects. 1 and 5), on which there is some amount of foundational justification present amongst the most basic of our beliefs, which are regarded as being beliefs that are close to our experiences, e.g., observational beliefs. Metaphysical foundherentism allows for the coherence amongst the elements of proper subsets of the domain of the grounding relation to confer satisfactory metaphysical explanation only on themselves and those things grounded by them. Still, metaphysical foundherentism is certainly not a version of foundationalism. It is less clear whether it should be regarded as a form of coherentism. Thompson (2018, p. 110) supposes that metaphysical interdependence (which can be plausibly regarded as a form of metaphysical coherentism) requires at least one violation of both asymmetry and Full Foundations\(^{+}\). Thus the presence of just non-foundationalist cyclic grounding chains (and not foundationalist ones) in an otherwise foundationalist hierarchy would not count as interdependence on her view. This is still consistent with foundherentism being a form of coherentism. And perhaps it is. Not a lot rests on this choice. The matter is only one of terminology. At the least, ‘foundherentism’ is a useful label to use for the view, as it emphasizes the important feature of the view that it allows for there to be, in a sense, a bottom layer of entities which ground all other entities in the domain of the grounding relation. See further discussion of this in the main text below.
The two versions of foundherentism also allow for the structures that result from deleting c from each of \(S_{7}\)–\(S_{10}\). But the structures I have depicted better illustrate that foundational cyclic chains can play the role in foundherentism that foundational entities play in foundationalism, viz., the grounds of all the other facts.
Each of the entities in this layer, if such a layer exists, are all-grounding or complete, though not necessarily ungrounded or independent [see Leuenberger (2020) and Bennett (2017, ch. 5)]. Leuenberger (2020, p. 2651) defines an all-grounding entity as one which belongs to every grounding base, where
Grounding Bases. \(\Gamma \) form a grounding base \(=_{df}\) for every entity x not among \(\Gamma \), there are \(\Delta \) among \(\Gamma \) such that x is fully grounded by \(\Delta \).
Even once full grounding is replaced with full grounding\(^{+}\), the resulting definition of grounding bases, along with Leuenberger’s definition of an all-grounding entity, won’t count elements of foundational cyclic chains as all-grounding. Consider the grounding structure \(S_{10}\), depicted in Fig. 4. Neither a nor b belongs to every grounding base. a, b is a grounding base. But so is a and so is b. And b isn’t among a nor is a among b. But there is a nearby alternative to Leuenberger’s definition of grounding bases.
Proper Grounding Bases. \(\Gamma \) form a proper grounding base \(=_{df}\) for every entity x not among \(\Gamma \), there are \(\Delta \) among \(\Gamma \) such that x is fully ground\(^{+}\)ed by \(\Delta \) and x does not partially ground\(^{+}\) any y among \(\Gamma \).
Because b ground\(^{+}\)s a, a doesn’t count as a proper grounding base. Mutatis mutandis for b. a, b does, however, since c doesn’t ground\(^{+}\) either a or b. Then an all-grounding entity can be defined as one which belongs to every proper grounding base. It is worth noting that these definitions aren’t unmotivated. With the notion of an all-grounding entity, Leuenberger is attempting to pin down a sense of ‘fundamental’, but only within a foundationalist framework. He notes that the notion of an all-grounding entity is not well-defined in an infinitist framework (ibid. 2653). In an infinitist framework, consisting of just non-cyclic infinitely descending chains of ground, there is no principled point below which one might say everything is “basic” and above which would be that which depends upon the basic. And granted, in coherentist frameworks, which technically count as foundherentist frameworks, little sense is to be made of the notion of an all-grounding entity also. But in other foundherentist frameworks, a hierarchy, or approximate hierarchy, of ground exists. And in such frameworks, there will be a principled line: that below which things are either ungrounded or grounded only in one another, and above which everything is fully ground\(^{+}\)ed by things below that line. It would be ideal to characterize the notion of an all-grounding entity, in broadest terms, in a way that counts entities below such a line as all-grounding, as I have just done.
See Bliss (2011, 226 ff.) for further considerations in favor of local cyclic grounding chains.
See Bliss (2011, ch. 10) for a discussion of the permissibility of circular definitions more generally, and for a discussion of the permissibility of circularity in philosophical analyses.
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Acknowledgements
Thanks to Roberto Loss, Yash Agarwal, and three anonymous referees for helpful comments on drafts of this article, and to Martin Pleitz for helpful discussions about multiple issues that arise in this paper. Thanks also to an audience in the philosophy department at Lawrence University for helpful questions and comments.
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Appendix: proofs
Appendix: proofs
Proposition 1
Every cyclic chain is a subchain of a unique maximal cyclic chain.
Proof
Suppose \(\Gamma \) form a subchain of maximal cyclic chains \(\Delta \) and \(\text {E}\). Since \(\Gamma \) form a subchain of each of \(\Delta \) and \(\text {E}\), everything among \(\Gamma \) is among each of \(\Delta \) and \(\text {E}\). Suppose for reductio that \(\Delta \ne \text {E}\). Then either \(\Delta \) contains something, a, not among \(\text {E}\) or \(\text {E}\) contains something not among \(\Delta \). Suppose the former. Since \(\Gamma \) are among \(\Delta \), something among \(\Gamma \), b, ground\(^{+}\)s and is ground\(^{+}\)ed by a. Since \(\Gamma \) are among \(\text {E}\), there is something, viz., b, among \(\text {E}\) that ground\(^{+}\)s and is ground\(^{+}\)ed by a. But since a is not among \(\text {E}\), \(\text {E}\) would not be maximal, contrary to the initial hypothesis. In a similar way, we reach a contradiction when we suppose that \(\text {E}\) contains something not among \(\Delta \). Hence \(\Delta = \text {E}\). \(\square \)
Proposition 2
The claim that for any x and y, if x is partially ground\(^{+}\)ed by y, then y is partially ground\(^{+}\)ed by x is equivalent to the claim that x is either foundational or an element of a foundational cyclic chain.
Proof
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(i)
First, consider an arbitrary entity a and suppose that, for any y, if a is ground\(^{+}\)ed by y, then y is ground\(^{+}\)ed by a. Now either a is ungrounded or it is grounded. If it’s ungrounded, then it’s foundational, and so it’s either foundational or an element of a foundational cyclic chain. Now suppose it’s grounded, and consider the entities \(\Gamma \) such that, for every y, y is among \(\Gamma \) iff a is ground\(^{+}\)ed by y. Since a is grounded by something, we know \(\Gamma \) are non-empty. By hypothesis, every y among \(\Gamma \) is also ground\(^{+}\)ed by a. And since partial grounding\(^{+}\) is transitive, we know a is among \(\Gamma \). It can now be shown that \(\Gamma \) form a foundational cyclic chain. I show first that \(\Gamma \) form a grounding chain. Consider arbitrary b, c among \(\Gamma \). We know by the characterization of \(\Gamma \) that a is ground\(^{+}\)ed by b, and we know that c is ground\(^{+}\)ed by a by that characterization along with our initial hypothesis. So by the transitivity of partial grounding\(^{+}\), c is ground\(^{+}\)ed by b. This chain is cyclic. By the last result and the initial hypothesis, b is ground\(^{+}\)ed by c, and so c is ground\(^{+}\)ed by b iff b is ground\(^{+}\)ed by c. But b and c are arbitrary. And this cyclic chain is maximal. Consider an arbitrary b and suppose that b both ground\(^{+}\)s something c among \(\Gamma \) and is ground\(^{+}\)ed by something d among \(\Gamma \). (The second conjunct will not be needed.) Since c is among \(\Gamma \), a is ground\(^{+}\)ed by c. But since c is ground\(^{+}\)ed by b, the transitivity of partial grounding\(^{+}\) guarantees that a is ground\(^{+}\)ed by b, which ensures that b is among \(\Gamma \). By Proposition 1, which was proved above, \(\Gamma \) is the only maximal cyclic chain that \(\Gamma \) are among. And nothing among \(\Gamma \) has a ground that is not among \(\Gamma \). If something, b, did have such a ground, c, then b would be ground\(^{+}\)ed by c, and so a would be ground\(^{+}\)ed by c, ensuring that c is among \(\Gamma \), contrary to the hypothesis. So a is an element of a foundational cyclic chain, and so it is either foundational or such an element.
-
(ii)
Now suppose that a is either foundational or an element of a foundational cyclic chain. If a is foundational, then it is ungrounded, and so it is vacuously true that, for all y, if a is partially ground\(^{+}\)ed by y, then y is partially ground\(^{+}\)ed by a. Now suppose that a is an element of a foundational cyclic chain \(\Gamma \), consider an arbitrary b, and suppose that a is ground\(^{+}\)ed by b. Since \(\Gamma \) form a foundational cyclic chain, the maximal cyclic chain \(\Delta \) that includes \(\Gamma \) must be foundational as well. (On the uniqueness of \(\Delta \), see Proposition 1, which was proved above.) If it were not, the chain formed by \(\Gamma \) would not be foundational, contrary to the hypothesis, since \(\Gamma \) would be among some things that form a cyclic maximal chain which include something that has a ground not included in them (the things that form the maximal cyclic chain). But suppose now for reductio that b is not ground\(^{+}\)ed by a. Then b wouldn’t be among \(\Delta \), since \(\Delta \) is cyclic, and so something among \(\Delta \), viz., a, would have a ground not among \(\Delta \), contrary to the above result that \(\Delta \) is foundational. \(\square \)
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Dixon, S. Metaphysical foundherentism. Synthese 201, 86 (2023). https://doi.org/10.1007/s11229-023-04053-1
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DOI: https://doi.org/10.1007/s11229-023-04053-1