1 Introduction

In the past decade, the notion of grounding has gained increased attention in the metaphysical literature. Some discussions about it focus on general, topic-neutral issues like whether grounds necessitate what they ground (e.g. Skiles, 2015) and whether grounding talk should be regimented with a relational predicate or a sentential connective (e.g. Correia, 2010). On the other hand, metaphysicians with first-order concerns use it to define physicalism (e.g. Dasgupta, 2014), characterize the notion(s) of fundamentality (e.g. Bennett, 2017: ch. 5), and formulate new versions of the cosmological argument for the existence of God (e.g. Deng, 2020), to name but a few of the uses to which grounding has been put. Also, there are grounding sceptics who raise questions regarding the intelligibility of the notion itself (e.g. Daly, 2012; Wilson, 2014).

Among the varieties of discussions about grounding, there is a small but growing portion of the literature paying attention to a somewhat peculiar idea about grounding, namely the notion of ‘zero-grounding’, first introduced by Fine (2012: 47). Proponents of zero-grounding distinguish ungrounded truths, which are not grounded in any truths at all, from zero-grounded truths—which are grounded, yet the number of truths that they are grounded in is zero.Footnote 1 Or as Fine put it:

“The case in which a given statement is zero-grounded, i.e. grounded in zero antecedents, must be sharply distinguished from the case in which it is ungrounded, i.e. in which there is no number of statements – not even a zero number – by which it is grounded” (Fine, 2012: 47).

Proponents of this distinction will typically acknowledge that zero-grounding “sounds odd at first” (Muñoz, 2020: 211) and that it may seem like an “exotic possibility” at best (Fine, 2012: 48). Yet on the face of it, what they assert sounds not only odd and exotic. It also seems plainly unintelligible, just like the claim that someone is a daughter but has zero parents, or the claim that something is composite but composed of zero parts. Surprisingly, although it is common to hear others report, in conversation, their failure to make sense of Fine’s idea, to date we know of no published attempt to articulate and defend scepticism about zero-grounding in detail.Footnote 2

In what follows, we will argue that zero-grounding is indeed unintelligible. But it is important to begin by explaining why zero-grounding is an interesting notion which, if intelligible, has metaphysical potential, and is not merely an academic curiosity.

First, a number of metaphysicians have attempted to answer certain notorious metaphysical questions by drawing on the notion of zero-grounding. These include (i) the suggestion by Fine (2012, 2016) and Litland (2023) that identity and distinctness truths are zero-grounded, (ii) Litland’s (2017) account of iterated grounding on which (non-factive) iterated grounding truths, e.g. the truth that P (non-factively) grounds Q, are zero-grounded, (iii) Thomas Donaldson’s (2017) and Louis deRossett and Øystein Linnebo’s (2023) accounts of arithmetic on which some arithmetical truths featuring in the foundations of arithmetic are zero-grounded, (iv) Daniel Muñoz’s (2020) account of negative existentials on which true negative existentials, e.g. the truth that there are no titans, are zero-grounded, and (v) Julio De Rizzo’s (2020) account of necessity on which necessary truths are necessary because they are zero-grounded.

Proponents of these applications of zero-grounding claim that we can evade a host of problems that emerge were we to take these truths to be either grounded in other truths, or not grounded at all, by taking them to be zero-grounded instead. For instance, take distinctness truths: i.e. truths of the form a ≠ b, with “a” and “b” proper names for numerically different objects. On the one hand, finding a systematic account of the grounds of such truths has proven difficult. On the other hand, several authors have argued that if an object is a constituent of an ungrounded truth, then that object is fundamental (e.g. Sider, 2011). Yet since every object is a constituent of at least one distinctness truth, it would then follow that every object is fundamental, assuming that distinctness truths are ungrounded. To navigate between these two unpalatable outcomes, one might instead take distinctness truths to be zero-grounded (cf. Litland, 2023: § 3 for more on this style of argument). Others have likewise argued that the option of zero-grounding a truth allows one to avoid problems with either grounding it in other truths, or taking it to be ungrounded instead. This shows that zero-grounding, if intelligible, can be used to do important metaphysical work, and its allure shows no sign of retreating (e.g. Donaldson, 2020; Kappes, 2021b, 2021b, 2022, 2023; Miller, 2022 for further applications).

Second, at an abstract level, the notion of zero-grounding has an interesting, though little discussed, consequence: it gives rise to the possibility of a novel sort of grounding structure. The relevant discussions in the recent literature have mainly focused on three kinds of grounding chains, each of which obeys different formal restrictions on grounding, e.g. transitivity or asymmetry.Footnote 3 Some grounding chains are well-founded ones, which end in ungrounded elements. This is the standard grounding structure which most grounding theorists have in mind. Other grounding chains are non-terminating ones, which come in two types: (i) infinitely descending grounding chains and (ii) cyclic grounding chains.Footnote 4 Now, in addition to the foregoing three kinds of grounding chains, zero-grounding gives rise to the possibility of grounding chains which are neither non-terminating ones, since they terminate in some elements, nor are they our familiar well-founded chains, since they do not end in ungrounded elements but rather in (zero-)grounded ones.Footnote 5

The foregoing two reasons indicate the metaphysical potential of the notion of zero-grounding, if intelligible. However, we shall argue that zero-grounding is actually an unintelligible notion. Specifically, we will first give two arguments against the intelligibility of zero-grounding (§§ 2–4) and then reject several attempts to make zero-grounding intelligible that have been suggested by its proponents (§§ 5–9).

2 The first argument against zero-grounding

As we noted in the previous section, the notion of zero-grounding has been used to account for negative existential, necessary truths, arithmetic truths, identity and distinctness truths, etc. But, in and of themselves, there seems to be little in common among these sorts of truths. The only thing that seems to be common to them is that there are reasons for not taking them as ungrounded and yet at the same time there is no clarity about what their grounds could be. Thus, the disunity and variegated character of the truths which zero-grounding has been used to account for seriously suggests that the enterprise has an ad hoc nature. What independent reason is there to believe that such different facts have exactly the same ultimate ground? Although we are sympathetic to this worry, this is not the point we want to press against zero-grounding, since we believe that we have arguments that show zero-grounding to be unintelligible.

Here is our first argument. The basis of the idea of zero-grounding is an unintelligible distinction: the distinction between there being no truths that ground a truth (this is the case of ungrounded truths) and there being zero truths grounding a truth (this is the case of zero-grounded truths). But this does not make sense unless there is a distinction between there being no truths satisfying a certain condition C and there being zero truths satisfying condition C. But, clearly, what does not make sense is this distinction. To put it another way: there is no such distinction, that is, the number of distinctions of this sort is zero. There is no distinction, for instance, between a bachelor having no spouses on the one hand, and a bachelor having zero spouses on the other. Since the distinction that the notion of zero-grounding relies on—at least as Fine characterized it in the quote above—is unintelligible, the notion of zero-grounding itself is also unintelligible. This is our first argument against the intelligibility of zero-grounding.

Could it be that the distinction between having no grounds and having zero grounds is difficult to perceive because they are necessarily co-instantiated concepts? That would not help the proponents of zero-grounding, for in that case, every instance of zero-grounding would also be an instance of ungroundedness, when the entire point behind taking a given truth to be zero-grounded was to avoid the problems that emerge from taking it to not be grounded at all.

3 The second argument against zero-grounding

Perhaps there is some other way to characterize what zero-grounding is, one that does not rely on the distinction between a truth being grounded in no truths vs. zero truths. Or perhaps, this distinction can be made sense of after all. Even so, to claim that there are grounded truths that are grounded in zero truths remains as prima facie unintelligible as claiming that there are daughters who have zero parents. The analogy is an apt one. Whatever grounding is, it seems to be a truism that the truth of a grounded truth is in some sense inherited from the truths that ground it. Some have argued, on this basis, for the stronger claim that there is some ultimate source from which all truth is inherited (cf. Trogdon, 2018 for discussion of various ‘inheritance arguments’ for the view that all grounding chains are well-founded). However, the truism we have in mind is weaker and, we believe, relatively uncontroversial. Grounding, whatever it is, involves the transfer of truth from certain truths to others, regardless of whether there are truths whose truth is inherited from nothing else. But just as it is unintelligible to claim that there is a daughter that inherits her genetic material, yet there are zero parents that she inherits this genetic material from, it is also unintelligible to claim that there is a truth that inherits its truth, yet there are zero truths that it inherits its truth from.

Thus, grounding is inherently relational in our view. There are two standard ways of capturing the logical form of truths about grounding, one is to represent grounding by means of a relational predicate, the other is to represent grounding by means of a sentential connective. As we shall see, our second argument against the intelligibility of zero-grounding is independent of how grounding is represented. We shall discuss the view that grounding is represented by a sentential connective in Sect. 4. For the time being we shall take grounding to be represented by a relational predicate. Now, relations link relata, by which we simply mean that a relational statement of the form “R(a, b, …)” is true only if the relata terms “a”, “b”, … refer to entities that are related in the way the statement says they are related (we leave open whether the relational predicate “R” itself refers to some entity as well). But in (alleged) cases of zero-grounding there are grounded truths but zero grounds, and so at least one of the relata of the grounding relation is missing. Since a statement of the form “Such-and-such truth is zero-grounded” attempts to ascribe a relation without enough relata, the statement is not true.

The idea that relations link entities is as plausible as the extremely plausible idea that instantiated properties are instantiated by entities (which is not to be confused with the idea that properties, if they exist, must be instantiated). Given that grounding has a genuinely relational and generative nature—for it is destined, as noted above, to help transfer the status of truth from the ground to what is grounded—the foregoing ‘relations-link-relata’ principle requires that all grounded truths, including zero-grounded ones if any, must be linked via the grounding relation to at least one ground, as opposed to zero.

Thus, if our explanation about the relational and generative nature of grounding is the correct one, then zero-grounding is unintelligible. This is our second argument against the intelligibility of zero-grounding.

4 Objections

Let us discuss two types of objections to our argument. One is to question the principle that relations link entities. For instance, it might be alleged that the relation of logical consequence sometimes obtains between a logically true proposition and no proposition. Indeed, the objector might claim that this is exactly what logicians mean when they write “⊢ P” (rather than “Q1, Q2, … ⊢ P”) to indicate that there are no propositions from which P follows.

Our response to this alleged counterexample (and others) is that even if there are relations that do not link relata, there are some that clearly do; in particular, relations that involve transfer, or inheritance, clearly link relata. Nothing can inherit anything unless it inherits it from something and, similarly, nothing can be transferred to anything unless it is transferred by something. The relation of being a daughter is a case in point. A daughter, or an offspring more generally, is someone who inherits her genetic material, and so it is absurd to think that there are no beings from which she inherits her genetic material. Since grounding is an inheritance relation, it must link relata, whether or not the principle that relations link relata holds in full generality. But since grounding must link relata, zero-grounding is not possible.

Furthermore, we believe that there are independent reasons for thinking that if logical consequence is taken to be a relation between propositions, it is nonsense to claim that certain propositions are logical consequences of no propositions. Of course, there is a more technical notion of logical consequence, according to which it is a relation between a proposition and a set of propositions. And it is when thinking about this sense of logical consequence that sometimes logicians say that certain propositions follow from no propositions or premises, while what they mean is that such propositions follow from the empty set (or, redundantly, from the empty set of premises).

Of course, this is a technicality, and the objector could reply that even if a proposition follows from the empty set, it is still the case that there is no proposition from which it follows. But this is incorrect. There is no proposition that follows from no proposition: every proposition follows from itself, and from the conjunction of itself and any other arbitrary proposition, and so on (at least in a classical setting, although this simplification will not affect our main point). Thus, there is a disanalogy between logical truths and zero-grounded truths: while zero-grounded truths are supposed to be grounded and have no grounds at all, logical truths are not supposed to follow from no propositions at all. In fact, the disanalogy is even clearer if we draw our attention to the fact that logical truths follow from all propositions. Indeed, the point of writing “⊢ P” is to exploit this distinctive fact about logical theoremhood: not that P follows from no propositions, but that for any propositions Q1, Q2, … we place to the left of the single turnstile (as it were), it remains true that Q1, Q2, … ⊢ P. The point is even clearer still when logicians write “⊨ P”: the point of this notation is not to note that P is true in no interpretations of the language, but rather that P is true in any (classical) interpretation you care to choose. There is no such thing as a proposition following from no propositions at all, and the use of the “⊢ P” notation does not show otherwise.

It has been suggested to us that the alleged disanalogy between logical truths and zero-grounded truths can be resisted since, the thought goes, logical truths would still be logical truths even if they did not follow from themselves or any other propositions at all. Thus, the idea is that zero-grounded truths are analogous to logical truths since, after all, they can be understood as those truths that would be grounded even if nothing grounded them. But this way of restoring the analogy between logical truths and zero-grounded truths actually undermines the notion of zero-grounded truths. For this way of understanding logical truths assumes that logical truths need not be consequences in order to be logical truths. Thus, if logical truths did not follow from any propositions at all, they would be logical truths but they would not be consequences, since consequences require something from which they follow. So, if logical truths and zero-grounded truth were analogous in this respect, we would have to say that, if nothing grounded them, zero-grounded truths would not be grounded. But zero-grounded truths are not supposed to be ungrounded.Footnote 6

So much for attempts to reject the idea that relations link relata by appealing to (alleged) counterexamples involving logical consequence. Nevertheless, even granting that inheritance relations link relata, one might object that zero-grounding does not violate this idea. There are at least four ways that one might mount this type of objection.

Here is the first. As is well known, some grounding theorists argue that grounding is to be expressed by means of a sentential connective. In this case, the standard view is that truths of grounding are to be expressed in terms of truths of the form “P because Q1, Q2, …” where “P” states what is grounded and “Q1”, “Q2”, … state what grounds it (cf. Correia, 2010; Fine, 2012; Schnieder, 2011).

But the idea that truths about grounding are to be formalized by means of a sentential connective is not at all in conflict with the idea that grounding involves the transfer of truth from one truth to another. Indeed, the ‘because’ is precisely signalling that one truth inherits its truth from another. Indeed, the relational and generative nature of grounding noted above supports a higher-order analogue of the relations-link-relata principle, i.e. that operations on propositions (or whatever is expressed by sentences) link propositions (or whatever is expressed by sentences). Again, this principle requires that all grounded truths, including zero-grounded ones if any, must be linked via the grounding operation to at least one ground, as opposed to zero.

Here is the second way to implement the objection. It is a standard view among grounding theorists that in at least some cases, a plurality of truths together ground some further truth, and in such cases they express grounding with statements of the form “P is grounded in Γ”, where “Γ” is a plural term for some number of truths and “P” refers to some single truth. Grounding theorists of this sort will sometimes explicitly stipulate that there be at least one truth among Γ for such a statement to be true (e.g. Rosen, 2010: 115). But why not allow for the number of truths in Γ to sometimes be zero? If one allows for the existence of at least one empty plurality of truths, one might maintain that truths of the form “P is grounded in Γ” express truths of zero-grounding whenever “Γ” refers to an empty plurality (cf. De Rizzo, 2020; Kappes, 2021b; Litland, 2023). Yet with Γ now serving as a relatum for the grounding predicate to relate, proponents of zero-grounding might claim to respect the principle that grounding links relata.

However, this view faces at least three problems. First, the nature of grounding requires that it transfers the status of truth from the ground to what is grounded. Nevertheless, since there are no things (and thus no truths) in an empty plurality, there is no entity in an empty plurality with the status of truth, and so no status of truth can be transferred from an empty plurality to what is grounded, even if empty pluralities exist.

Second, it is of course not at all obvious that empty pluralities exist. Traditional plural logic forbids their existence by positing the axiom of non-emptiness, expressible as the statement that ∀xx∃y y ≺ xx, with “∀xx” ranging over pluralities of things and the binary predicate “≺” standing for plural membership (Florio & Linnebo, 2021: 19). What we are saying here is not that this axiom cannot be jettisoned. Instead, our point is that if this is the only way to make sense of zero-grounding claims, then the extra theoretical costs that come with this should be made explicit, which may well outweigh whatever theoretical benefits may be gained from countenancing zero-grounding.

Third, suppose one rejects the axiom of non-emptiness, and thereby allows for the consistency of its negation. That is, suppose one allows that ∃xx∀y~(y ≺ xx), i.e. that there is some plurality such that everything is not among it. Even so, this does not help to make sense of zero-grounding. Suppose it were true that P is grounded in Γ, with Γ some empty plurality. Then P, the grounded truth in question, would turn out to be grounded in something: namely, whatever witnesses the negation of the axiom of non-emptiness—whatever plurality satisfies the open formula “∀y~(y ≺ xx)”. If so, then zero-grounded truths would be no different in nature from other typical grounded truths. A zero-grounded truth would be grounded in something—albeit a logically suspicious, non-alethic something—when the entire point was to avoid various notorious metaphysical problems by taking it to be grounded in nothing.Footnote 7

The final two replies we will consider also deny that zero-grounding is in conflict with the idea that grounding, in so far as it is a relation, links relata, but attack our argument in a subtly different way. We have presupposed that grounding must be expressed by a relational predicate, or even by a sentential connective. But why couldn’t it be expressed by a non-relational monadic predicate? If so, then a truth can be zero-grounded by instantiating the non-relational property expressed by this predicate, without being grounded in any other truth.

One problem with this suggestion is that it leads to a disjunctive view about grounding. On the one hand, non-zero-grounded truths are grounded because they stand in the relation of grounding to some truths. On the other hand, zero-grounded truths are grounded because they instantiate the non-relational property of being grounded. However, the idea of zero-grounding is not that there is a different kind of grounding, but that in the same sense of grounding there are grounded truths that are not grounded in anything and grounded truths that are grounded in other truths.

True, this only shows that the idea of zero-grounding as conceived by Fine and his followers cannot be taken to be a non-relational property. But couldn’t grounding have a dual, disjunctive nature, and obtain sometimes relationally and sometimes non-relationally? The proposer of such a line would face the ungrateful task of having to explain why such a non-relational property merits the title of grounding: in virtue of what has such property earned the right to be called grounding? In what sense could this non-relational property involve the transfer or inheritance of truth from the more fundamental truths to the less, perhaps the most distinctive feature of the relation of grounding? We think that no good answer to these questions is forthcoming, any more than if one were to introduce a non-relational predicate, “is a daughter”, and insisted that the property it expressed earns the right to be called daughterhood, despite the lack of genetic inheritance.

Alternatively, one might claim that grounding is a multigrade (or variably polyadic) property or operation, one that holds in a relational fashion on some occasions and in a non-relational fashion on others. In reply, note that even if this proposal deals with the previous worry about disjunctivity, it merely pushes the previous worry about entitlement one step back. For one could similarly say that “is a daughter of” expresses something that holds in a relational fashion on some occasions, and in a non-relational fashion on others; but in virtue of what does this thing, whatever it is, merit the title of daughterhood? It is unintelligible to claim that daughterhood could hold in a non-relational fashion, since on those occasions there would be no inheritance of genetic material from parent to daughter. And for the same reason, it seems unintelligible to claim that grounding could hold in a non-relational fashion, since on those occasions there would be no inheritance of truth from the more fundamental to the less. To claim that grounding is multigrade does nothing to address this point. Indeed, we believe that this points to an important but largely neglected lesson in the literature: the real issue is not whether we can formalize a predicate/operator with nothing on the left-hand side, like ‘⊢ P’, and call it ‘grounding’, but whether the notion expressed by that term deserves to be so-called.

5 Fine’s analogy

If the idea of zero-grounding, as these two arguments show, is unintelligible, there can be no proper elucidation of such a notion. This is what we shall confirm in what follows, where we shall discuss the attempted elucidations of zero-grounding we are aware of. We will argue that none of these attempts succeeds.

In addition to the characterization above, Fine also tries to bring out the difference between ungrounded truths and zero-grounded truths by means of an analogy with sets. For him any non-empty set is generated, via what he calls “the set-builder”, from its members; the empty set is generated from its members, but, in this case, Fine says, “there is a zero number of members from which it is generated” (2012: 47); and ur-elements are ungenerated: “there is no number of objects – not even a zero number – from which [they] may be generated” (2012: 47). This analogy, however, presupposes the unintelligible distinction between an entity having no members and an entity having zero members, and therefore the analogy cannot support anything.Footnote 8

But suppose someone says that generation is the operation of set formation, and that the relevant difference is that while the empty set is the result of applying the set formation operation to nothing, ur-elements are not generated at all. This is not any more intelligible than Fine’s own attempt. For an operation needs something to operate on to deliver a result. Indeed, the operation of set formation is, presumably, a function, but functions need arguments to deliver a value. Thus the empty set is not generated, at least not in the way that non-empty sets are generated from their members. Although both the empty set and ur-elements are ungenerated, the difference between them is that one of them is a set and the others are not.Footnote 9

Fine also imagines a machine that manufactures sets. The machine is such that if one feeds some objects into it, the set of those objects emerges from the machine; the empty set is the object that emerges from the machine when no objects are fed into it, and the ur-elements are the objects that never emerge from the machine (2012: 47–48).

This fantasy is meant to be an illustration of the analogy that appeals to the distinction between there being no number of objects from which an object is generated and there being zero number of objects from which an object is generated. But, interestingly, the fantasy does not require the distinction between no objects and zero number of objects. It only requires the distinction between emerging from a machine when nothing is fed into it and never emerging from the machine.

Does this make the idea of zero-grounding any more intelligible? Not at all. One can also imagine that the machine is such that if one feeds too many objects into it to form a set—such as the entire set-theoretic hierarchy—then the machine jams, and nothing emerges from it. But that alone should not convince us that there is an intelligible distinction between grounding no truths versus grounding zero truths. As a matter of symmetry, if the case involving the machine’s jamming does not make the distinction between grounding no truths and grounding zero truths intelligible, then Fine’s distinction between being grounded in no truths and being grounded in zero truths—which is supposed to be illuminated by an analogy with the very same machine—is not made intelligible by this analogy either.

Moreover, as we pointed out above, the empty set is ungenerated because in its case there is nothing for the operation of set formation to operate on, and therefore the fantasy of a set machine that generates the empty set by operating on nothing obscures rather than clarifies the distinction between the empty set and ur-elements. A fortiori, this fantasy does not clarify the distinction between ungrounded and zero-grounded truths, since such a distinction was supposed to be illuminated through the distinction between the empty set and ur-elements.

Another motivating case, like the empty set, given by Fine (2012: 48) is the so-called ‘null conjunction’. Fine holds that every conjunction is grounded in all of its conjuncts. As the null conjunction has no conjuncts, it is grounded in no truth at all. However, this case merely presupposes rather than clarifies the notion of zero-grounding, as the view that the null conjunction is grounded but grounded in nothing conflicts with the relation-link-relata principle mentioned in Sect. 3.Footnote 10

6 Ex nihilo

Thus, Fine provides no elements to understand what is prima facie an unintelligible idea. But there are other attempts at clarifying it. Muñoz, for instance, mentions that Jonathan Schaffer, in a presentation, proposed to elucidate the distinction between ungrounded and zero-grounded truths by reference to the distinction between uncaused initial conditions and ex nihilo effects. On this analogy the uncaused initial conditions are like ungrounded truths and the ex nihilo outcomes are like zero-grounded truths (Muñoz, 2020: 220).

The problem with this analogy is that it is premised on a false understanding of ex nihilo effects. Indeed, Muñoz glosses them as outcomes that are caused but not by anything. But ex nihilo effects are effects lacking a material cause, not an efficient one – indeed, in theological cosmology God is the efficient cause that causes ex nihilo, that is, from no previously existent material. In any case, for the analogy to work there must be a type of causation on which some things can exist uncaused and others are caused but not by anything – this distinction is no more intelligible than the distinction between ungrounded truths and truths that are grounded but not in anything. Thus this causal analogy does not help to make the idea of zero-grounding intelligible.

7 Explanatory arguments

Litland tries to elucidate the notion of zero-grounded truths by taking zero-grounded truths to be those that are the conclusion of an explanatory argument from the empty collection of premises (Litland, 2017: 280). To understand this, one needs to understand Litland’s notion of an explanatory argument. Litland does not attempt to give a reductive account of what makes something an explanatory argument but instead he takes the notion as primitive. But he says that the notion of an explanatory argument can be elucidated, and he elucidates it via the notion of basic explanatory inferences: explanatory arguments are composed from basic explanatory inferences (Litland, 2017: 289). And basic explanatory inferences are explained in terms of immediate grounding: in general, when Γ immediately grounds P, there is a basic explanatory inference from Γ to P (Litland, 2017: 289–290). But then, if there is an explanatory argument from the empty set of premises to P, either there is a basic explanatory inference from the empty set of premises to a conclusion P, or there is a basic explanatory inference from the empty set of premises to some intermediate conclusion Q (or intermediate conclusions Q1, Q2,…) from which one infers, immediately or mediately, P. Either way, in the case of zero-grounded truths, there has to be some basic explanatory inference with no premise. But this elucidation then just seems to pass the buck. It simply replaces the issue about the intelligibility of zero-grounding with the issue about the intelligibility of the idea of a basic explanatory inference with no premise, which also seems to be in need of explanation and thereby is not in better standing than the notion of zero-grounding.Footnote 11

8 Zero-premise arguments?

Litland might reply that the foregoing scenario does not exhaust all available ways for there to be an explanatory argument with no premise. For just like we may have a conditional proof in first-order logic from P to Q that leads to an argument with no premise for the conclusion that P → Q, there might be some explanatory argument with no premise that is not a result of chaining basic explanatory inferences, but a result of conditionalization. In fact, this is how Litland (2017: 298–302) argues for the contention that iterated grounding truths are zero-grounded.

This move, however, seems no less buck-passing than the previous elucidation. We agree that there are explanatory arguments, and (of course) we agree that there are valid arguments from zero premises (in the sense mentioned above). But from that alone it does not follow that there is any valid explanatory argument from zero premises—that is, an argument that explains its conclusion, with each step an explanatory inference, in terms of its zero premises. Perhaps they exist, but we are aware of no uncontroversial examples—by which we mean examples that either do not already presuppose the intelligibility of zero-grounding (as in Kappes, 2021b), or presuppose that there are basic inferences from zero premises that are explanatory, which are the only sort of basic inferences that an explanatory argument can be composed from. So the idea of basic explanatory inferences with no premise is not well-understood, nor is the idea of explanatory arguments with no premise, and regardless of whether the elucidation of the latter relies on the intelligibility the former.

Moreover, there is reason to believe that explanatory zero-premise arguments are impossible. Kovacs (2018) has recently argued against the possibility of a truth, P, that explains itself, and his reasoning can be easily extended to the case at hand. Kovacs argues that if P has an explanation, and Q is the true answer to the question ‘Why P?’, then there must be at least some possible situation, for at least some possible cognitive agent, in which learning the information conveyed by Q increases one’s overall understanding of why P is true, either by conveying information not already conveyed by P, or by presenting the information conveyed by P in a different way. But clearly, citing P itself in response to ‘Why P?’ would not possibly provide any new information, or any new way of presenting old information, that answers the question at hand. But just as clearly, not answering ‘Why P?’ does no better. Thus, elucidating the idea of zero-grounding in terms of the idea of explanatory zero-premise arguments appears to be a non-starter.

We anticipate two objections. One is to deny that the explanation-to-understanding link mentioned above applies in cases of zero-explanation. But once again the question arises as to why then it would merit being called a type of explanation. The idea of a type of explanation that could not increase any possible cognitive agent’s overall understanding in any possible circumstance seems unintelligible (cf. Kim, 1994: 54; Kovacs, 2020: 1661; Maurin, 2019: 1580; Thompson, 2016: 397). In what way could an explanatory inference, in Litland’s sense, be genuinely explanatory if it does not meet even this minimal constraint?

A second objection is to insist that not answering the question ‘Why P?’, or perhaps instead, answering it with ‘P is true just because’ (cf. Kappes, 2022: 441), can provide information relevant to the question at hand. We agree. Not answering the question ‘Why P?’ might inform the questioner that one never had an answer to give in the first place, or that one believes that the question falsely presupposes that P is true, or falsely presupposes that it has an explanation. Those pieces of information might increase the questioner’s understanding in all sorts of ways, but they would do nothing to help the questioner understand why P is true, which is what the explanation-to-understanding link above concerns. Moreover, when one says ‘P is true just because’, the information one provides is that P has no explanation, not that it has a zero-explanation. Compare it with the information one conveys with “I did it just because” when asked for the (normative) reasons behind some action. One is not saying that the action is supported or justified, yet there are no reasons that support or justify it. That is unintelligible. Rather, this colloquialism is used to either reject that the action required a reason behind it in the first place, or to communicate that there was no reason behind it at all. So although phrases like ‘…just because’ can provide information, it is not the sort of information that could help to zero-explain a truth, any more than utterances of ‘…just because’ could help to ‘zero-rationalize’ an action.

9 The mediator view of explanation

Mike Townsen Hicks and Al Wilson (2023a, 2023b) have defended what they call the mediator view of explanation.Footnote 12 On this view explanations are mediated by explanatory principles that take the form of laws of nature in the case of causal explanations and metaphysical laws in the case of grounding. These explanatory principles are not extra explanantia of the explanandum in question – instead they are higher-order explanantia that explain why the first-order explanans explains the first-order explanandum.Footnote 13 Their explanandum is thus not the original one, but the first-order explanation.

Interestingly, Hicks and Wilson have argued that the mediator view of explanation can also help to explain, and therefore clarify, the idea of zero-explanation, where zero-explanation is a generalization of zero-grounding to other ontic explanations (causal explanation in particular). If they are right, zero-explanation and zero-grounding should be intelligible.

How do Wilson and Hicks use the mediator view to explain zero-explanation? In cases of zero-explanation there is an explained explanandum but no explanans, and the explanatory principles explain why the explanandum in question is explained despite the absence of an explanans. In contrast, when there is an explanans what the explanatory principles explain is why the explanans explains the explanandum in question. (See also Kappes, 2021a: 2588–2589; 2021b: 1274; 2022: 442–443; 2023: Ch.1–2.)Footnote 14

Does this proposal provide a clear account of zero-explanation and the difference between zero- and non-zero-explanation? Here are two reasons for believing that it does not. The first is that it assumes precisely what needs to be shown. Remember, the challenge for the proponent of zero-grounding is to make sense of the idea of an explanandum being explained despite the absence of an explanans. However, all that the mediator proposal has done is to postulate the existence of ‘explanatory principles’ that would explain particular instances of this phenomenon if this phenomenon were possible. But that is exactly what is at issue: why believe that an explanandum can be explained despite the lack of an explanans, and how do the explanatory principles invoked by the proponents of this view explain what appears to be unintelligible? To these questions, they have no answer, and so no clear account of zero-explanation has been given.

Even putting that problem to the side, a further problem comes with how the mediator proposal distinguishes zero- and non-zero-explanation. Note that the explanatory principles are the explanantia of particular explanations. But if zero-explanations are possible, then there seems to be no reason why there could not be a zero-explanation that is zero-explained, that is, a zero-explanation which is explained by zero explanatory principles. Of course, there might be third-order explanatory principles that explain why the first-order zero-explanation is explained by zero second-order explanatory principles. But there seems to be no reason why there could not be a first-order explanation that is zero-explained at the second-order level and such that its zero-explanation is also zero-explained at the third-order level. In general, there seems to be no reason, if zero-explanations are possible, why there could not be an infinite hierarchy of zero-explanations, each one explaining the previous one. This is a case the mediator view cannot account for, since there are no explanatory principles explaining any explanations in such a hierarchy. So if zero-explanation is possible, the mediator view seems to be unable to account for zero-grounding in general since the infinite hierarchy of zero-explanations could be a hierarchy of zero-groundings.

Note that the real problem is not that the hierarchy of explanations is infinite. For suppose there were a highest explanation stage. If all the explanations up to and including that stage are cases of zero-explanation, then the mediator view cannot account for any of the explanations in that hierarchy, since there is nothing in that hierarchy doing the explanatory mediation. Indeed, the mediator view requires that explanations are mediated by explanantia explaining explanations, and so such explanations of explanations cannot be zero-explanations. But if zero-explanations are possible, there seems to be no reason why zero-explanations of explanations should not be possible too.

Of course, the proponents of the mediator view can postulate a principle according to which any zero explanations must be “capped off” by non-zero explanations at some order. But this does not provide a reason why zero-explanations at any order are not possible; it simply stipulates that there is some order at which zero-explanations are not possible in order to avoid the problem. Thus this is an explanatorily impotent ad hoc manoeuvre. It should be emphasised that we are not saying that such a stipulation would be incoherent; nor do we have an argument that proves that zero-explanations are possible at every order of the hierarchy. All we are saying is that, if a hierarchy of zero-explanations is possible, the mediator view cannot account for zero-explanation in general; and if a hierarchy of zero-explanations is not possible, then we are owed an explanation of why they are impossible, and simply stipulating that at some level zero-explanations must be non-zero explained is no explanation at all but simply an ad hoc manoeuvre.

10 Conclusion

We have seen that there are good reasons to take the idea of zero-grounding as unintelligible. And we have also seen that the different attempts to explain it, elucidate it, or clarify it, do not really succeed. We conclude that the idea of zero-grounding is unintelligible. We would like to thank Damian Aleksiev, Jose Tomas Alvarado, Paul Audi, Ricki Bliss, Andrew Brenner, Sebastian Briceñño, Rafael De Clercq, Julio De Rizzo, Loius deRosset, Sam Elgin, Kit Fine, Alex Geddes, NickJones, Yannic Kappes, Jessica Leech, Stephan Leuen berger, Jon Erling Litland, Annina Loets, Fraser MacBride, Dan Marshall, Farid Masrour, Susanna Melkonian-Altshuler, Alex Roberts, Howard Robinson, Bernhard Salow, Deniz Sarac, Carolina Sartorio, Jonathan Schaffer, Benjamin Schnieder, Richard Swinburne, Naomi Thompson, Kelly Trogdon, Bruno Whittle, Ezequiel Zerbudis, Hoshea Zhang, audiences at the University of Santiago, the University of Oxford, the University of Vienna, Rutgers University, the Chinese University of Hong Kong, the City University of New York, the University of Bern, and the referees for their comments and discussion. Lo’s research was funded by an Early Career Scheme grant from the Research Grants Council of Hong Kong (Project Number: CUHK 24601924). All three authors equally contributed to the writing of this article; the order is alphabetical.