Grounding, Well-Foundedness, and Terminating Chains

It has recently been argued that foundationalists, those who take grounding to be well-founded, should not understand the well-foundedness of grounding as the condition that every grounding chain terminates in the downward direction, because this interpretation of well-foundedness fails to correctly classify certain complex grounding structures. Some structures that plausibly would be acceptable to the foundationalist are classified as not well-founded and others that plausibly would not be acceptable to the foundationalist are classified as well-founded. In this paper I provide a characterisation of well-foundedness in terms of termination that correctly characterises all these difficult cases. This acts as a response to these recent arguments. Furthermore, it allows us to better evaluate their importance: these arguments have not shown that foundationalists are wrong to harbour the intuition that grounding chains must terminate in the downward direction, rather, they have shown that foundationalists need to be more clear about this intuition and how it is born out in more complex grounding structures.


Introduction
Foundationalism can be understood as the view that all grounding structures are well-founded. In this context well-foundedness has been interpreted as the condition that all chains of grounding terminate in the downward direction. However, recently this has been challenged with a number of writers arguing that the termination of chains doesn't capture well-foundedness in the context of grounding and 1 3 foundationalism. 1 In this paper I will respond to these arguments. In particular, I will show that we can understand foundationalism as the view that grounding is well-founded, where this is understood as the condition that every inclusive full grounding chain terminates in the downward direction.
There are various phenomena in the world that appear to occur in virtue of other phenomena, but without being caused by them. For example, the fact that a certain mental state occurs appears to be non-causally explained by the fact that a certain brain state occurs, the fact that certain action is good appears to be noncausally explained by the fact that the action has various natural properties, and the fact that a complex whole exists seems to be non-causally explained by the fact that all of its parts exist. The world thus appears to involve at least one form of non-causal explanation.
Grounding is an example of such non-causal explanation. I am not here concerned to argue that any particular case ought to be understood as an instance of grounding. However, the variety of phenomena just listed hopefully indicates the potential importance of the notion. 2 I will take grounding to be a transitive, asymmetric, and irreflexive relation that holds between facts. 3 Relations of grounding give the world structure and this structure involves a form of ontological priority. Grounds appear to have ontological priority over those facts that occur in virtue of them.
According to foundationalism, this structure must be well-founded. This point has been taken to mean that relations of grounding must terminate. It is important to note that termination here specifically concerns termination in the downward direction, that is, in the direction towards the grounds. (From here on I will simply speak of termination for ease of presentation, though I intend this to specifically pick out termination in the downward direction.) So, for example, we might think that some specific mental fact is grounded in some specific biological facts about a brain, and we might in turn think that these biological facts are themselves grounded in some further chemical facts, but at some point in this series of grounding relations we must reach facts that are not themselves grounded. Perhaps in this case these will be micro physical facts: ultimately the mental facts occur in virtue of the micro physical ones.
Foundationalism is a general view about the structure of the world with potential implications for a variety of topics. It is a view, however, that has been met with a recent wave of opposition. 4 Clarifying the notion of well-foundedness at the heart of foundationalism will hence help us to both clarify and evaluate foundationalism.
1 3 Philosophia (2023Philosophia ( ) 51:1539Philosophia ( -1554 The term 'well-founded' has an established use in mathematics and set theory, and this use perhaps inspired the term's use in discussions of grounding. However, relations of grounding encompass more than mathematics and set theory. 5 In this paper I am concerned to try to better understand what the foundationalist's notion of well-foundedness is. It will not be assumed that this is the same as the established notion from mathematics and set theory, and, in fact, it will be indirectly argued it is distinct from it (whilst the words are the same, the notions differ).

Inclusive Chains
A grounded entity is metaphysically explained by what grounds it, as the conjunctive fact that Jo is happy and Sue is happy is metaphysically explained by the facts that Jo is happy and that Sue is happy. (From here on I will often adopt the convention of using square brackets to pick out facts, so '[p]' means the fact that p.) We can distinguish full and partial grounding. Like some of the authors that I am discussing, I will take the notion of full ground to be primitive. 6 By way of an example, we might say the facts [Jo is happy] and [Sue is happy] together provide a full ground for the fact [Jo is happy and Sue is happy].
The fact f or the facts F are a partial ground of g, if H is a full ground of g and f or the Fs are amongst the members of H.
The fact f or the facts F are a merely partial ground of g, if H is a full ground of g and f or the Fs are amongst the members of H but f or the Fs are not a full ground of g.
For example, the fact [Jo is happy] provides a merely partial ground for the fact [Jo is happy and Sue is happy].
We can also distinguish mere groups of facts from grounding structures and in turn chains of grounding.
A grounding structure is a collection of facts such that some of those facts are in grounding relations with others.
A chain of grounding is a collection of facts each of which stand in grounding relations to each other member.
For example, the fact [Jo exists] partially grounds the fact [Jo is happy], which in turn partially grounds the fact [Jo is happy and Sue is happy], so these facts together form a chain of grounding. These ideas are common to the debate and the authors I am addressing. 7 Foundationalists are concerned with the ultimate nature of reality. They are not merely concerned with the local grounding relations of a fact, but with all of the grounding relations for a fact. This is made clear by recognising that part of what a foundationalist is denying is that a grounded fact can be grounded by other facts all of which are grounded facts.
An independent fact is a fact that is not grounded and doesn't require grounds in order to obtain. 8 Consider the following chain of grounding, f is fully grounded in g, which is fully grounded in h, where h is an independent, or f < g < h. This chain includes the sub-chain f < g. This sub-chain includes only grounded entities. However, a foundationalist is not concerned to suggest that this sub-chain is impossible. Rather, the foundationalist would be happy with this sub-chain precisely because they are happy that the whole chain of which it is a part is possible. Given this, it is appropriate to express the foundationalist's concern as a concern with inclusive chains.
An inclusive grounding chain is a chain of grounding such that it is not the case that each member of the chain is grounded by a fact or facts that are not members of the chain.
In my example the chain f < g is not inclusive and the chain f < g < h is inclusive. Foundationalism involves the idea that no inclusive grounding chains contain only grounded entities.
The foundationalist's dissatisfaction with chains only made up of grounded entities is reflected in Schaffer's (2010: 62) argument that if every entity existed merely in virtue of another, then being would only ever be deferred and never established. 9 Schaffer (2010: 37), ruling out such cases, proposes that the foundationalist takes grounding structures to be well-founded, where this is understood in terms of termination as follows.
Following Dixon, we can understand termination as follows.
A grounding chain terminates if a member of that chain grounds all of the other members of that chain. Dixon (2016) and Rabin and Rabern (2016) argue that T1 is not sufficient to capture a notion of well-foundedness adequate for foundationalism. Dixon's argument is essentially that a foundationalist will plausibly be happy with the existence of fully pedestalled chains, but T1 rules these out. The following would be an example of a fully pedestalled chain: g1 is fully grounded in h and h is an independent; g1 is also fully grounded in g2, where g2 is fully grounded in h and fully grounded in g3; in turn, g3 is fully grounded in h and fully grounded in g4; and so on. Pictorially we have the following (Fig. 1) 10 .
More specifically, the following case adapted from Dixon would constitute a fully pedestalled chain. Take h to be the fact [Joe is happy] and take this to be an independent fact. Then take the gs to be a series of (infinite) disjunctive facts, where each has [Joe is happy] as a disjunct and the disjunctive fact that is gn + 1 is formed by removing an instance of [Joe is happy] from the disjunctive fact that is gn. So, for example, g1 is [Joe is happy or ((Sue is sad or Joe is happy) or (((Sue is sad or Sue is sad) or Joe is happy) or ((((Sue is sad or Sue is sad) or Sue is sad) or Joe is happy) or …)))] and g2 is [(Sue is sad or Joe is happy) or (((Sue is sad or Sue is sad) or Joe is happy) or ((((Sue is sad or Sue is sad) or Sue is sad) or Joe is happy) or …))].
This case assumes that a disjunctive fact will be fully grounded by a fact that is a disjunct of it, so, for example, [Joe is happy] fully grounds [Joe is happy or Sue is sad]. Thus, h will fully ground each of the gs because it is a disjunct of each of them and gn will be fully grounded by gn + 1 as the latter is a disjunct of the former.
Plausibly a foundationalist would be comfortable with such a structure because h is an independent and adequately accounts for g1, and h adequately accounts for g2, and so on. Assuming the fact [Joe is happy] is an independent, it adequately accounts for any disjunctive fact of the form [Joe is happy or …].
However, in this fully pedestalled chain, the chain g1 < g2 < g3 < … does not terminate. Thus, if the foundationalist is comfortable with this case and not comfortable with non-well-founded grounding structures, then T1 cannot capture what it is for a structure to be well-founded.
Nevertheless, we can accommodate the foundationalist's concerns and the fully pedestalled case if we replace T1 with T2_inc.
T2_inc accommodates fully pedestalled chains. The chain g1 < g2 < g3 < … is not an inclusive chain. The only inclusive chains in the structure are g1 < h, g1 < g2 < h, etc. and these all terminate. Furthermore, as already indicated, it is natural for the foundationalist to focus on inclusive chains. It is of note that in all finite and many infinite grounding structures it will only be the case that every inclusive grounding chain terminates if it is also the case that every grounding chain terminates. A foundationalist might therefore naturally think of T1 instead of T2_inc. However, what fully pedestalled chains reveal is that in some grounding structures every inclusive grounding chain will terminate even if not every grounding chain does. Dixon (2016: 453) considers the foundationalist adopting a principle like T2_inc, which I refer to as T2_max. T2_max differs from T2_inc because it utilises the notion of a maximal chain, rather than of an inclusive chain.

Inclusive Versus Maximal Chains
A maximal chain of grounding is a chain of grounding which is such that it is not the case that there is an entity that is not a member of the chain and that partly grounds every member of the chain.
T2_max, like T2_inc, accommodates fully pedestalled chains. Dixon, however, raises a problem for a foundationalist that relies on the notion of maximal chains. 11 T2_max rules out fully crutched chains, though plausibly a foundationalist would be happy to accept such structures. The following is an example of a fully crutched chain: g1 is fully grounded in h1 and h1 is an independent; g1 is also fully grounded in g2, which is in turn fully grounded in the independent h2; g2 is also fully grounded in g3, which is in turn fully grounded in the independent h3; and so on. Pictorially we have the following (Fig. 2).
The following case adapted from Dixon would constitute a fully crutched chain. Assume that there are an infinite number of spacetime points, s1, s2, s3 etc. and related to these an infinite number of facts of the form [s1 exists], [s2 exists], etc., where each of these facts is an independent. Take h1 to be the fact [s1 exists] and h2 to be the fact [s2 exists] etc. Now take g1 to be an infinite disjunctive fact of the form [s1 exists or (s2 exists or (s3 exists or …))] and take g2 to be an infinite disjunctive fact of the form [s2 exists or (s3 exists or …)], etc. Now, hn will be a disjunct of gn, and so will fully ground it, gn + 1 will also be a disjunct of gn and so will also fully ground it.
In this fully crutched chain, the chain g1 < g2 < g3 < … qualifies as a maximal chain and one that doesn't terminate. Therefore, T2_max would rule out such a fully crutched chain. However, plausibly a foundationalist would be happy with the case, as h1 is an independent and g1 is adequately accounted for by h1, h2 is an independent and g2 is adequately accounted for by h2, and so on.
Essentially the same argument is also given by Rabin and Rabern (2016) who use a fully crutched chain to criticise interpreting well-foundedness as being bounded from below, where a chain is bounded from below if it terminates (2016: 366). Their argument is also given focusing on "maximal" chains which are described as chains which are not substructures, where being a substructure appears akin to being non-maximal (2016: 364).
T2_inc, however, accommodates fully crutched chains. The chain g1 < g2 < g3 < … does not qualify as inclusive, because in the fully crutched case each member of the chain of gs is grounded in something outside that chain (even though there is no one thing outside the chain that they are all grounded in). I suggest that what this case highlights is that the foundationalist ought to be concerned with inclusive chains not maximal chains. This is because the case clarifies that a chain might be maximal without providing the full picture of how any of the members of that chain are grounded and, as noted above, it is such a full picture that the foundationalist is concerned with. However, it is of note that in all finite structures and many infinite structures inclusive and maximal chains will coincide. So, one can understand why the foundationalist might accidentally focus on maximal rather than inclusive chains. 12 12 Litland (2015) has raised concerns with the cases of infinite disjunction used to constitute the fully pedestalled and fully crutched chains above. Specifically, Litland argues they involve assumptions which, if universalised, are wrong. However, Litland (2015Litland ( : 1368 admits that these assumptions may still stand in the particular cases at issue and hence his arguments do not necessarily undermine them. Litland does offer a different grounding structure including an infinite chain of grounding relations that will plausibly be acceptable to the foundationalist. What is of note as far as our discussion is concerned, is that T2_inc (and T4) holds in Litland's structure, whilst neither T1 nor T2_max do. (This is because Litland's structure contains an infinite chain of grounds (contra T1) that would qualify as a maximal chain (contra T2_max) but that would not qualify as an inclusive chain (or an inclusive full grounding chain). The foundationalist may be content with Litland's infinite chain, because each member of it is fully grounded by a fact or facts that may be independents or fully grounded by independents, in a way similar to the fully crutched chain.) Litland's structure thus brings us to the same point as our consideration of fully pedestalled and crutched chains.

An Infinitely Branching Chain
However, T2_inc faces a problem. It is possible to have a grounding structure that does not contain any inclusive grounding chains or that contains entities that do not lie on any inclusive grounding chains. This would occur if every entity was grounded by more than one further entity. I will call such a structure an infinitely branching chain. Pictorially this would look as follows (Fig. 3). An infinitely branching chain could be constituted by the following case, were it to occur. Let us assume that the world is gunky, so every entity has parts. So, we have the entity e1 and its two halves e2 and e3, and their halves e4, e5, e6, and e7, and so on. Next, let us assume that the fact that an entity exists is grounded in facts concerning the existence of its parts, so [e1 exists] is grounded in [e2 exists] and [e3 exists]. We can then construct an infinitely branching chain as follows. Take g to be the fact [e1 exists]. Take h1 to be the fact [e2 exists] and take h2 to be the fact [e3 exists], and so on for facts concerning their halves (i1, i2, i3, and i4), etc.
An infinitely branching tree would not satisfy the foundationalist. Our inability to locate a single inclusive grounding chain in the structure appears to reflect an inability to fully establish the being of any of the entities in the structure. Being would appear to be forever deferred. Therefore, T2_inc should be replaced by T3.
T3. Every grounded entity is a member of at least one inclusive grounding chain and every inclusive grounding chain terminates.
T3 rules out infinitely branching chains and accommodates fully pedestalled chains and fully crutched chains, as the foundationalist would want.

Full Grounding Chains
Dixon, however, raises a further case that causes trouble for T2_max and also T2_inc and T3. Crucially, T3, T2_inc and T2_max appear to allow for partially pedestalled chains, though plausibly the foundationalist will not want to. The following would be an example of a partially pedestalled chain: g1 is merely partially grounded in h and h is an independent; g1 is also grounded in g2, where g2 and h together provide a full ground for g1; g2 is in turn merely partially grounded in h and grounded in g3, where g3 and h together provide a full ground for g2; and so on. Pictorially we have the following (Fig. 4). The following case could provide us with a partially pedestalled chain, were it to occur. Firstly, assume a regress of instantiation, so given the fact [a instantiates Fness] occurs, the fact [a and Fness instantiate instantiation] occurs, and the fact [a and Fness and instantiation instantiate instantiation] occurs, and so on. Next, assume that this regress involves grounding, so the first of these facts is grounded in the second, which is grounded in the third, and so on. Next assume that everything that occurs does so in part because of an omnipresent god, Gee. The fact [Gee exists] is an independent and grounds every other fact. However, note Gee is not omnipotent. Gee cannot create things ex nihilo, rather Gee requires materials to work with. So, the fact [Gee exists] merely partially grounds every other fact. Finally, let us assume that in our regress of instantiation the grounding is merely partial, because of Gee's crucial role: each fact is created out of the fact that grounds it by Gee. So, for example, [a instantiates Fness] is merely partially grounded in [a and Fness instantiate instantiation] and merely partially grounded in [Gee exists], and these last two facts together fully ground the first fact. 13 Now we can construct our partially pedestalled chain as follows. Take h to be the fact [Gee exists] and take the gs to give our regress of instantiation, so g1 is the fact [a instantiates Fness], and g2 is the fact [a and Fness instantiate instantiation], and so on.
This whole structure of a partially pedestalled chain qualifies as an inclusive chain that terminates and as a maximal chain that terminates. However, the foundationalist might be concerned about the possibility of such a chain. Going back to Schaffer's argument, for example, the being of g1 seems to be forever deferred and never established.
We can capture the foundationalist's worry if we focus on the relations of full grounding in this structure. Specifically, there appears to be a worrying non-terminating run of full grounds in the series given by g1 < {h and g2} < {h and g3} < …. Thus the being of g1 might seem to be forever deferred and never 13 If one does not like Gee, one might replace the fact [Gee exists] with the independent fact [grounding exists] and take this to be a merely partial ground of every grounded fact and take the other grounds of any grounded fact to therefore also only be merely partial grounds of it. g1 g2 g3 h Fig. 4 A partially pedestalled chain established (at least to the foundationalist). It is the termination of full grounding that the foundationalist is centrally concerned with, because full grounding is what is required for being and the foundationalist's concern, as expressed by Schaffer, is that the being of the grounded entities is established.
This suggests that we need to modify T3 further to focus specifically on series of full grounding.
We can consider such series of full grounding if we introduce the notion of a node. A node is a group of facts, its members, that together form a full ground. However, a fact might have more than one full ground in the sense of having more than one complete explanation (as a disjunction might be fully grounded by each disjunct). We want a notion that can consider each such complete explanation, so we need to understand a node in a way that accommodates these complexities.
A node is a group of facts that together provide a full ground for a fact, unless that group has a subgroup that fully grounds that fact.
We then need to distinguish chains of such nodes from mere groups of nodes.
A chain of nodes is a group of nodes which are each in full grounding relations with one another.
A node is in a full grounding relation with another node if (i) it is a full ground for that node or full ground for a member of that node, or (ii) it is fully grounded by that node or has a member fully grounded by that node. (Note, these grounding relations are transitive, so if g1 is a node thus fully grounded by g2 and g2 is a node thus fully grounded by g3, then g1 and g3 will be nodes that stand in grounding relations in the relevant sense.) A grounded entity might not itself be a node, so such entities must also be added.
A full grounding chain is a fact and a chain of nodes all of which are in grounding relations with that fact and at least one of which provides a full ground for that fact.
We can then make sense of inclusive such chains and their termination as follows.
An inclusive full grounding chain is a full grounding chain which is such that it is not the case that each node of the chain is grounded by some entity or entities which are not members of that chain.
An inclusive full grounding chain terminates if there is a node in the chain which grounds all of the other nodes.
The notion of a node allows us to pick out groups such as {h and g2} that play the full grounding roles in our partially pedestalled chain. The notion of a chain of nodes allows us to capture the full grounding series of g1 < {h and g2} < {h and g3}. In turn, the notion of an inclusive full grounding chain then allows us single out a grounding series such as the series g1 < {h and g2} < {h and g3} < …. In this way we are able to accommodate the foundationalist's concern with the full picture of how an entity is fully grounded even in complex structures. I therefore suggest that we replace T3 with T4.
T4. Every grounded entity is a member of at least one inclusive full grounding chain and every inclusive full grounding chain terminates.
As a foundationalist would want, T4 does not rule out fully pedestalled chains or fully crutched chains. In our fully pedestalled case, although h is a full ground of g1 and g2 is a full ground of g1, the group {h and g2} is not a node grounding g1 and does not play a part in a full grounding chain. The only inclusive full grounding chains in the fully pedestalled case are therefore the chains g1 < h, g1 < g2 < h, g1 < g2 < g3 < h, etc. and all of these chains terminate.
In the fully crutched case, the only inclusive full grounding chains are the chains g1 < h1, g1 < g2 < h2, g1 < g2 < g3 < h3, etc. and again each of these terminate. As with the fully pedestalled case, h1 and g2 do not together form a node grounding g1.
T4 does rule out our partially pedestalled chain, as the foundationalist would want. In the partially pedestalled chain g1 does lie on an inclusive full grounding chain, the chain g1 < {h and g2} < {h and g3} < …. However, this chain does not terminate.
It is of note that Dixon describes partially pedestalled chains in a way that allows them to come in two forms. Specifically, he describes the relations between the gs as relations of partial grounding, allowing that they therefore could be either relations of merely partial grounding or relations of full grounding. So far we have only considered a structure of the former kind, which we might call a merely partially pedestalled chain. A structure of the latter kind, which we might call an improperly partially pedestalled chain, would look as follows (Fig. 5).
Dixon provides us with the following case that would constitute an improperly partially pedestalled chain. Firstly, take the gs to be constituted by a regress of instantiation, so g1 is the fact [a instantiates Fness], and g2 is the fact [a and Fness instantiate instantiation], and so on. Next, suppose that the relations of grounding g1 g2 g3 h Fig. 5 An improperly partially pedestalled chain here are relations of full grounding, so the first fact is fully grounded by the second, and the second by the third, and so on. (Gee's work is not required here.) Now suppose that h is the conjunctive fact that [a, Fness, and instantiation exist], that this is an independent fact, and that any other fact involving a, Fness, and instantiation, is merely partially grounded in it. This gives us a structure in which gn is partially grounded in h, fully grounded in gn + 1, and also fully grounded in h and gn + 1 together.
This improperly partially pedestalled chain, like our first partially pedestalled chain, looks to be unacceptable to a foundationalist. Specifically, there appear to be worrying non-terminating runs of full grounds in the series given by g1 < g2 < g3 < … and in the series given by g1 < {h and g2} < {h and g3} < …. Thus the being of g1 might seem to be forever wanting and never established.
However, neither of these series of full grounds are inclusive full grounding chains. The former is not, because every member of the series is grounded by something outside the series. The latter is not because {h and g2} and {h and g3} etc. do not count as nodes. 14 In fact, g1 does not lie on any inclusive full grounding chains and as such it, and our improperly partially pedestalled chain, are ruled out by T4 as the foundationalist would want. Pleitz (2020) has recently provided an argument that might imply we should not take the well-foundedness of grounding to be given by T4 (although, Pleitz did not give the argument with T4 in mind). Pleitz argues that many paradoxes concern situations that can be understood as involving poorly formed grounding structures and, further, that these infelicities of grounding can help us understand and perhaps solve these paradoxes. For Pleitz there is hence an advantage to a theory of grounding if it can enable us to identify infelicities of grounding in paradoxical situations.

Pleitz
On the basis of this advantage, Pleitz considers and rejects an account of the well-foundedness of grounding that is akin to T2_max, because it does not render non-well-founded a situation that is paradoxical. 15 1 3 Philosophia (2023Philosophia ( ) 51:1539Philosophia ( -1554 The paradoxical situation at issue concerns an infinite series of sentences that appears paradoxical because there is no consistent way to attribute truth-values to them all. The series of sentences runs as follows: s1, 'for every n greater than 1, sn is false'; s2, 'for every n greater than 2, sn is false'; s3, 'for every n greater than 3, sn is false'; … sω, '0 = 1'. (To recognise the difficulty here, suppose that s1 is true, then all the other sentences are false. However, if that is the case then s2 is true, as all the sentences that follow it are false. Alternatively, suppose s1 is false. If that is the case then some other sentence sn must be true. However, if that is the case then we arrive at a problem for sn parallel to the one we just arrived at supposing s1 is true: if sn is true, sn + 1 is both false and true.) Such a series of sentences does not constitute a grounding structure if grounding holds between facts. Nevertheless, Pleitz's discussion suggests two ways we might see the series as involving/necessitating grounding structures, one relating to reference and one relating to truth-value.
Pleitz suggests that relations of reference involve grounding. So, the series of sentences gives rise to the following infinite chain of grounding [s1 exists] < [s2 exists] < [s3 exists] … [sω exists]. 16 Pleitz also suggests that relations of truthmaking and truth denying involve grounding. The former point implies the fact [p exists] grounds the fact [the sentence 'p' is true]. 17 Extrapolating from here, though Pleitz does not himself say as much, we might imagine the series of sentences involving/necessitating a grounding chain of the following form [s1 has the truth-value it does] < [s2 has the truth-value it does] < [s3 has the truth-value it does] < … [sω has the truthvalue that it does].
Pleitz's point with the grounding structures that arise from the series of sentences is that they involve infinite maximal chains that terminate. Pleitz takes this point to show that taking T2_max to give the well-foundedness of grounding does not account for there being an infelicity of grounding in the paradoxical situation of the series of sentences. Given Pleitz has argued that there is value to a theory of grounding that does reveal infelicities of grounding in paradoxical situations, Pleitz hence says we have reason not to accept T2_max as the foundationalists' criterion of well-foundedness. 18 The point of note for our discussion is that if this argument is right, it might also appear to raise trouble for T4 too, if the chains of grounding are inclusive full grounding chains. However, I will now argue that the chains at issue are not inclusive full grounding chains.
Concerning the chain of reference, I would firstly challenge Pleitz's claim that reference involves grounding. Crucially, this claim involves the idea that 16 Pleitz (2020: 199) specifically talks of a chain of grounding between the sentences themselves occurring due to reference, but given I am here taking grounding to be a relation between facts, I adjust the example in accord with his footnote 17. 17 See Pleitz's (2020: 197) discussion of truth grounding. By truthmaking I mean his claim that p grounds that the sentence 'p' is true, and by truth denying I mean his claim that not-p grounds that it is not the case that the sentence 'p' is true. 18 See especially Pleitz (2020: Sects. 8 and 9). the existence of the referent of an expression partially explains the being of that expression. However, the converse of this sounds worryingly close to the idea that an expression cannot be an expression referring to an entity unless that entity exists. Down this path lie all sorts of Meinongian worries that are better avoided if possible. Thus, I am hesitant to accept reference involves grounding. Secondly, even if reference does involve grounding, plausibly it is merely partial grounding. 19 This is because in order for a sentence to refer to an entity, certain other facts fixing the meaning of that sentence must obtain. For example, let us suppose the sentence 'John is happy' refers to the fact [John is happy], plausibly it does so only given various other facts about our linguistic community. Therefore, the grounding chain [s1 exists] < [s2 exists] < … [sω exists], is not an inclusive full grounding chain. Therefore, this chain alone qualifies as non-well-founded by the standards of T4 and, prima facie, there is an infelicity of grounding here as Pleitz would want. (Though, plausibly this chain is in fact merely part of a larger grounding structure which has not been expanded. Without such an expansion Pleitz's comments have failed to show that T4 or even T2_max would not rule out such a case.) Concerning the chain of truth-value, a similar concern to the last arises. Even if relations of truth involve relations of grounding, these are plausibly merely partial relations of grounding because they require facts concerning conventions of meaning. 20 Therefore, the grounding chain [s1 has the truth-value it does] < … [sω has the truth-value that it does], is not an inclusive full grounding chain. Therefore, this chain alone qualifies as a non-well-founded structure by the standards of T4 and whilst it is plausible that it is in fact part of a wider grounding structure, this has not been shown to be well-founded by the standards of T4.
Therefore, as it stands Pleitz's arguments are not yet sufficient to undermine T4 (or T2_max) as capturing the well-foundedness of grounding. 21

Conclusion
We might present the discussion thus far in the following terms. The foundationalist is initially concerned that all chains of grounding must terminate and they are concerned about this because they believe that being must be established and not left wanting. However, when we focus our attention on just parts of a grounding chain it becomes clear that such parts ought to be able to leave being wanting, so long as the wholes of which they are parts do not. (We do not expect a part of an explanation to do the work of the whole explanation.) This enables us to say that it is really inclusive chains, as potentially complete explanations, that ought not to leave being wanting and hence that intuitively ought to terminate. We next shift our attention to chains that are only supposed to merely partially ground something. Again, it becomes clear that such chains clearly ought to be able to leave being wanting, so long as the full grounds formed from these do not. (We do not expect a partial explanation to do the work of a whole explanation.) This enables us to say that it is really relations of full grounding that ought not to leave being wanting and, hence, that intuitively ought to terminate. Thus, we are ultimately led to T4.
The foundationalists' concern that the metaphysical explanation of a fact must be complete, must not leave being wanting, might initially appear as the intuition that grounding chains ought to terminate. However, a consideration of the complexities of grounding structures shows that this concern is better understood as the intuition that every fact lies on an inclusive full grounding chain and that all such chains terminate. The arguments of Dixon and Rabin and Rabern do not undermine the role of termination, in fact they help to clarify it.
Pleitz has argued that our notion of grounding well-foundedness should be such that when certain paradoxical situations are understood as grounding structures, these grounding structures are not rendered as well-founded. I have argued that understanding well-foundedness in terms of T4 does not so render the structures Pleitz mentions and hence well-foundedness can be understood in terms of termination whilst acknowledging Pleitz's concern.
Dixon argues that the foundationalist ought to understand the well-foundedness of grounding in terms of FS.
FS. Every grounded fact x is fully grounded by some independent facts Y. 22 Dixon's argument for this is that unlike other interpretations of well-foundedness, FS accommodates fully pedestalled chains and fully crutched chains, whilst ruling out partially pedestalled chains.
Rabin and Rabern argue that the core notion of grounding well-foundedness is given by a principle akin to FS which insists that every grounding structure has a foundation, where a foundation is a substructure of independent entities that ground all the other entities (2016: 363/4, 369). Their argument for this is that other interpretations of well-foundedness fail to accommodate fully crutched chains.
I have just shown that understanding well-foundedness in terms of T4 would share the positive results of Dixon's and Rabin and Rabern's proposal. It accommodates fully pedestalled chains and fully crutched chains, whilst ruling out partially pedestalled chains. To this extent T4 and FS are equally well motivated. I have thus shown that foundationalists have more than one option for how they define their core notion of well-foundedness. Furthermore, T4 also achieves something that