ideas that first principles, God's existence, or certain grounding propositions themselves are self-explanatory. As it turns out, the notion can help make sense of Nozick's (1981, 119ff.) idea of "explanatory self-subsumption" and capture some strongly rationalist theses related to necessitarianism and the PSR. Section 6.6 concludes by showing that certain historical ideas about the explanation of God's existence give rise to a proposal for a self-explanation in the developed sense.

#### 6.1 Approaching self-explanation

Let us approach the topic of self-explanation by observing what I take to be a conceptual platitude: For a proposition to be self-explanatory is for it to explain itself. Or, schematically:

For a proposition x to be self-explanatory is for x to explain x.

Here, of course, 'explains' has to be used in the relational sense in which it expresses a relation that relates propositions or facts, viz. the kind of entities that constitute explananda and explanantia.

Before we disambiguate 'explains' further, note that the platitude already helps to distinguish self-explanation from related notions like our own emptybase explanation and Dasgupta's (2014b, 2016) explanatory autonomy, which might play a similar theoretical role or provide similar explanatory benefits as self-explanation proper. For example, one purpose of all three notions is to help satisfactorily end explanatory inquiry or avoid it all together.

Nevertheless, neither the notion of explanatory autonomy nor the notion of an empty-base explanation capture the idea expressed by the above platitude, namely that of a proposition explaining itself: Firstly, an explanatorily autonomous proposition is not explained, rather it is such that qua being autonomous it does not require an explanation.204 Therefore, it is not self-explanatory in the proper sense. Secondly, empty-base explanations are (to foreshadow a little: at least in general) not instances of a proposition explaining itself, as is for example witnessed by the zero-grounding explanation of non-factive grounding claims (à la Litland 2018): Non-factive grounding claims are (empty-base) explained, but they do not explain themselves.205

Perhaps it is possible that a proposition does not require an explanation and nevertheless has an explanation, but even this case does not capture the idea of a proposition explaining itself. Granted, a more relaxed sense of a self-explanatory proposition might exist in which for example merely empty-base explainable propositions count as self-explanatory. Perhaps such a sense functions similarly to 'self-evident': A self-evident proposition arguably need not be evidence for

Eventually, a comparison of the advantages and disadvantages of the three related notions will be desirable, but here my primary task is to investigate selfexplanation proper, as captured by the platitude (although see section 6.4 for some comparison of empty-base explanation and self-explanation with respect to the idea of ultimate explanation). Indeed, the possibility of self-explanation in the platitudinous sense is heavily contested. While this is often based on raw intuition, here I focus on the following three arguments from the asymmetry of related notions:

#### 'From "because"':

#### 'From explanatory dependence':

#### 'From reasonhood':

These arguments are similar in form: The first premise establishes a link between explanation and a further notion, the second premise establishes the asymmetry of that notion, and from this the asymmetry of explanation follows. The arguments are valid, so the proponent of self-explanation has to address the premises.

Admittedly, the arguments may perhaps be of somewhat limited dialectical value: A staunch defender of self-explanation might rather take them as counting against one of their premises than be convinced by them. In particular, the

An argument like this is suggested in Schnieder (2015).

itself. Rather, no further proposition is required for it to be evident. Here, though, I want to focus on the idea captured by the platitude above.

For an argument like this see Oppy (2006, 277f.). Let us ignore complications that might arise from quantifying into the contexts of 'explains' and 'because': My purpose here is to present a notion of self-explanation that can avoid these arguments independently of such concerns.

premises (P2), (P5) and (P7) that establish the asymmetry of the respective notion related to explanation seem to come quite close to the conclusion that nothing explains itself. Nevertheless, these premises enjoy considerable intuitive appeal and are widely endorsed.208

Therefore, I consider denial of either (P2), (P5) or (P7) to be a significant cost that would require serious argument.209 So instead of taking this route in defense of the possibility self-explanation, I will now employ the distinction between the restrictive and the inclusive sense of 'explains' introduced in chapter 1: While we can maintain premises (P1), (P4) and (P6) given the restrictive sense, these premises are doubtful given the inclusive sense (of course, we are still free to endorse the three simple arguments if we choose to understand 'explains' in the restrictive sense throughout).

#### 6.2 Two notions of (self-)explanation

Recall that in chapter 1 I argued that there is an inclusive sense of 'explains' in which not only the reasons (i.e. elements of the base) involved in an explanation (at least partially) explaininclusive the explanandum, but also the link of an explanation (partially) explainsinclusive its explanandum. As explained there, this sense of 'explains' stands in contrast to a more restrictive sense which corresponds more closely to because-statements and in which only the elements of the explanatory base (i.e. the reasons why), but not the link of an explanation (partially) explainrestrictive its result.

To make this explicit, we can define the two senses as follows:

But, of course, not universally: For example, one reason to deny causal irreflexivity may stem from the possibility of time travel and corresponding causal loops, cf. Smith (2019). For a critical discussion of the irreflexivity of metaphysical dependence see Jenkins (2011), and for the irreflexivity of grounding see Kovacs (2018) and the references therein.

For the case of grounding explanations, the start of such an argument might be provided by the puzzles of ground given in Fine (2010) and Krämer (2013). For some further discussion concerning the irreflexivity of grounding explanation see Bliss and Trogdon (2016, sec. 6.2).

210 This should be understood as a definition of immediate explanation to avoid ruling out here that x may in an inclusive sense explain y by in the restrictive sense explaining a link z of an explanation of y (and assuming a principle of transitivity, cf. the next section).

Correspondingly, we can distinguish self-explanation in the inclusive sense from selfexplanation in the restrictive sense: Proposals for self-causing or self-grounding facts concern self-explanation in the restrictive sense, while we will encounter candidates for self-explanations in the inclusive (but not restrictive) sense below.

Given this distinction, we can try to defend one type of self-explanation by arguing that the arguments against self-explanation only apply to the other type of self-explanation. Indeed, it can be argued that the first premise of each of the three arguments above is false given the inclusive sense of 'explains': For example, so understood, (P1) is false because if a proposition that P explainsinclusive a proposition that Q, then it is not in general the case that Q because P. The two sentential arguments of a 'because'-statement correspond to the base and result of an explanation and it is normally not the case that the explanatory link of an explanation is also in the base of the relevant explanation and thereby occurs in the corresponding 'because'-statement in this capacity. Rather, explanatory links correspond in a different way to 'because'-statements, for example by being tracked by the latter.211

Analogous considerations arise for (P4) and (P6) of the other arguments: If x explainsinclusive y, then it is not in general the case that y suitably depends on x: For example, the explanandum of a causal explanation does not causally depend on the causal connection or law of nature connecting it and its cause. Likewise, the grounding connection between a ground and a groundee does not normally also ground the groundee.212 Explanatory links involve the explanatory priority relation between an explanation's sources and its result, but in general do not themselves stand in such a relation to the result. Similarly, (P6) is false because if x explainsinclusive y (viz. by being the link of an explanation of y), then it is not in general the case that x is a reason for y. The base of an explanation consists of reasons for the explanation's result, but links normally do not play this role; instead links connect the reasons that constitute the explanation's base with its result.213

There is a more general lesson here: 'explainsinclusive' does not necessarily share the structural features of 'explainsrestrictive'. On the tripartite view of explanation and 'because' introduced in chapter 1, structural features often ascribed to explanation (e.g. asymmetry and transitivity) are captured by ascribing corresponding structural features to the link component. Additional analogous constraints on, e.g., the relation between explanatory links and results are unmotivated on this view: According to it,

Cf. Schnieder (2010).

Cf. Bolzano (1837, secs. 199, 344f.) and Litland (2018).

See chapter 1 and the discussion in Skow (2016).

what the relevant structural features of explanation come down to are the structural features of explanatory links. But normally, no additional explanatory links hold between the link and result of an explanation, so there appears to be no reason to assume corresponding structural features to govern the relation between link and result. In fact, stipulating corresponding constraints in addition to the structural features of the links would result in a disjoint account.

These considerations allow (but do not compel) us to maintain that selfexplanationrestrictive falls prey to versions of the three arguments in which each occurrence of 'explains' is understood in the restrictive sense while maintaining the intelligibility of self-explanationinclusive:

#### 'From "because" – revised':

#### 'From explanatory dependence – revised':

#### 'From reasonhood – revised':

Thus, we are free to deny the intelligibility of self-explanationrestrictive while maintaining the intelligibility of self-explanationinclusive, candidates for which we will look at in what follows.

### 6.3 On a circularity problem for Humeanism about laws of nature

Before we combine the notions of self-explanationinclusive and empty-base explanation to investigate the possibility of fully self-explanatoryinclusive propositions, I want to show how the previous result applies to matters that the reader might consider a bit more grounded. According to Humeanism about laws of nature (as I will understand them here), laws of nature are universal generalizations (or at least partially grounded in such). This idea is confronted with the following circularity problem that the distinction from the previous section can help solve:

Consider an explanation of [Ga] whose explanatory link is identical to or grounded in the universal generalization [∀x Fx ð Þ ! Gx ], and whose explanatory base contains [Fa].214 Together, the link and the base explain the result, so in particular:

(1) [∀x Fx ð Þ ! Gx ] partially explains [Ga].

But it is a widely accepted grounding principle about (true) universal generalizations that they are (partially) grounded in their instances, so [Fa ! Ga] partially explains [∀x Fx ð Þ ! Gx ]. Equally, it is widely accepted that if a material conditional has a true consequent, the former is grounded in the latter. So [Ga] explains [Fa ! Ga], and an application of transitivity for grounding yields:

(2) [Ga] partially explains [∀x Fx ð Þ ! Gx ].215

But (1) and (2) constitute an instance of symmetric (partial) explanation and an application of transitivity would even yield an instance of (partial) self-explanation.216

While several solutions to this problem have been discussed in the literature, the observations from the previous section afford a particularly straightforward solution: The derivation of a symmetric instance of 'explains' can only succeed given the inclusive sense of 'explains': (1) is true only in this sense. But as we have seen, there is reason to believe that structural features of explanation such as asymmetry only apply to the restrictive ('because'-corresponding) sense of 'explains', so the problem is avoided.217

As always, I use '[. . .]' to refer to the proposition or fact expressed by the sentence within the brackets.

For proponents of the relevant grounding principles see for example Fine (2012, 59ff.), Schnieder (2011, 406f.), Correia (2013a, 44f.), and for discussion in the present context Roski (2018). Note that for the problem to arise, all the Humean has to postulate is that laws are sometimes partially grounded in what they explain. This arguably already follows from the idea of Humean supervenience, championed by David Lewis, according to which nomic facts arise from a 'mosaic' of particular, non-nomic facts (cf. Weatherson 2016, sec. 5).

For discussion of this problem see, e.g., Loewer (2012), Lange (2013b), and Roski (2018), as well as the latter's bibliography.

Note that, alternatively, the application of transitivity in deriving a (partial) self-explanation from (1) and (2) could also be blocked like this.

### 6.4 Empty-base self-explanation

Self-explanations promise to be ultimate explanations, i.e. explanations that end explanatory regresses and do not give rise to further why-questions. Explanations by status (and thus empty-base explanations) may play a similar role: They explain without involving reasons why that could give rise to further why-questions. Nevertheless, empty-base explanations are (generally) not self-explanations in the platitudinous sense. Still, the notion of an empty-base explanation can be used to characterize a particular kind of full self-explanationinclusive that is not a self-explanationrestrictive, namely that of an empty-base explanation whose explanatory link is identical to its explanatory result.218 Schematically, such an 'empty-base self-explanation' has this form:

Base: ∅ Link: P Result: P

In such an explanation, the result explainsinclusive itself by being the link of its own empty-base explanation. Note that since there are no explanations without a link, self-explanations in the restrictive sense will likely involve a proposition that is distinct from its result, i.e. the explanatory link.219 In contrast, an empty-base selfexplanation would only involve one proposition, namely its explanatory result and link. Thus, in a sense, only an empty-base self-explanatory proposition would be fully self-explanatory in the sense of having an explanation with just it as a constituent, and only such explanations could be truly ultimate in that they do not involve any propositions that are unexplained or only explained by further explanations.

Before we consider candidates for empty-base self-explanations, let me address two arguments by Kovacs (2018, sec. 4) against the possibility of self-explanation that do not follow the pattern from section 6.1. In his first argument, Kovacs argues that just like circular ordinary arguments, circular explanatory arguments are objectionable, because just like ordinary arguments, explanatory arguments are supposed to provide reasons for their conclusions, but circular (ordinary as well as explanatory) arguments do not provide such reasons. Since Kovacs further assumes that every case of self-explanation corresponds to a circular explanatory argument, he concludes that self-explanation is objectionable.

We could in principle also consider explanations whose link and result are identical, but whose base contains different propositions, but these would not be full self-explanations.

'Likely' since we could in principle consider explanations whose reason, link, and result are identical. I will set aside this issue for what follows.

In response note first that an explanation whose result and link are identical is structurally related to the notion of rule-circular justification: In such an explanation, an explanatory link (partially) explains itself. Therefore, the corresponding explanatory argument has a conclusion that corresponds to the explanatory rule that governs the argument.220 Similarly, a rule-circular justification of an inference principle is provided by an argument to the conclusion that the principle in question holds (or perhaps to a conditional that corresponds to the inference principle), but which uses the inference principle in question to establish this.221

While some (e.g. Boghossian 2001) have endorsed the idea that rule-circular arguments may provide justification for their conclusions, their epistemic value is doubtful (for a recent criticism see Carter and Pritchard 2017). But note that even if the possibility of rule-circular justification is denied, the impossibility of empty-base self-explanation does not obviously follow: From the impossibility of rule-circular justification it would prima facie merely follow that if empty-base self-explanation is possible, then there are possible explanatory arguments that do not justify their conclusion, but they might still explain it.

Moreover, pace Kovacs, the premises of a good ordinary (or epistemic) argument justify its conclusion, viz. they are epistemic reasons for its conclusion, but the premises of a good explanatory argument explain its conclusion, they are reasons why the conclusion obtains. Kovacs appears to conflate these two notions of reasons and assumes that good explanatory arguments must justify (i.e. provide epistemic reasons for) their conclusions, but in many cases (e.g. many instances of inference to the best explanation), it is rather the case that a conclusion of an explanatory argument (i.e. an explanandum) justifies a premise of said argument (i.e. part of a corresponding explanans).

Kovacs' (2018, 1170) second argument turns on considerations about the relation between explanation and understanding:

[For] a statement such as 'p explains q' to express a genuine explanation, there should be a possible cognitive state of non-understanding, best expressed by the question 'Why q?', and an answer, 'p', learning of which replaces this state of non-understanding with a state of understanding. To achieve this goal, explanations have to be informative in the sense that the explanans clause conveys information not provided by the explanandum clause, or at least conveys information in a way not provided by the explanandum clause. Note that this requirement doesn't mean

Cf. Litland's (2017) calculus for explanatory arguments.

The analogy is not perfect: The result of an empty-base self-explanation is a proposition that is identical to its link. In contrast, the conclusion of a rule-circular argument is a proposition stating that a certain inference principle (that, moreover, arguably is not a proposition) holds. Thanks to an anonymous commenter on the paper on which this chapter is based.

#### 152 6 Self-Explanation

that the explanans clause conveys information to every audience that the explanandum clause doesn't convey in the same way, only that it's capable of doing so in the right circumstances.

In the above paper, Kovacs wants to argue that self-grounding is impossible and he does so by first arguing that self-grounding would give rise to self-explanation and then providing arguments against self-explanation. Thus, the intended targets of this argument are, in our terminology, self-explanations in the restrictive sense. But the argument is not convincing:

First, recall chapter 1, the tripartite account of explanation and its connection to why-questions and because-sentences: The explanans of an explanation (properly understood) has two components: The base component which is comprised of reasons why the explanandum obtains and which can be used to answer corresponding why-questions, and the link component which connects base and explanandum. Given these assumptions, the proponent of self-explanation in the restrictive sense can grant that the explanans needs to convey information not provided by the explanandum clause 'P', while maintaining that 'P' (or rather 'P because P') is a possibly correct answer to 'Why P?': The additional information conveyed by the explanans is then located in its link component, e.g. a proposition to the effect that [P] grounds [P].

Second, and supporting this point, observation of cases reveals that the step from a lack of understanding why towards understanding why often does not consist in coming to know the base-elements of the corresponding explanation why, but rather in coming to know (or to grasp) its link. For instance, many situations involving inference to the best explanation are like this: Sherlock may already know that the window is broken and that both Watson and Moriarty threw balls at the window, but coming to understand why the window broke involves grasping the causal (or law-like) link between Moriarty's throwing his ball and the window's breaking (see chapter 5 for more discussion).

While I am skeptical of self-explanation in the restrictive sense, Skiles (manuscript) has pointed out that Kovacs' argument might apply to explanations involving zero-grounding (and, we can add, empty-base explanations generally): Since the base of such explanations is empty, it does not contain any information that might lead to understanding. It seems then that Kovacs' argument would have us conclude that empty-base explanations are not possible. But again, empty-base explanations do involve another component, namely the link, grasping which can amount to understanding why the corresponding explanandum obtains. As argued in chapter 2, if we answer 'Why P?' with 'P just because' or 'P because' in the senses proposed there, we can communicate the relevant link.

Now, Kovacs' information constraint on explanation is more problematic for the notion of empty-base self-explanation, with which we are concerned here, because such explanations have an empty base (so there is no information to be found there), while the explanandum and the link are identical. Thus, neither base nor link provide information beyond the explanandum. In response, recall that in chapter 1 I have argued that mere knowledge of an explanatory link (plus base) need not be sufficient for understanding why the corresponding explanandum obtains. Rather, a mental state of grasping the link plus some associated cognitive control over the relationship is required. If this is correct, then there can be a possible cognitive state of non-understanding why P that is compatible with knowledge of [P] and [P] being the link and result of an empty-base self-explanation. For now, let us proceed to develop this notion further and look at candidates for empty-base self-explanations, I will say a bit more about this argument once we discuss generalized explanatory links below.

#### 6.5 Candidates for empty-base self-explanations

Now, what would empty-base self-explanations look like? Recall the suggestion that explanatory links of empty-base explanations have the form '◼P', where '◼P' stands for the result of the corresponding empty-base explanation. Since explanatory links of empty-base self-explanations are identical to the result of their explanation, it follows from this that their links have the form '◼P' and that the proposition [P] is identical to the proposition [P]. Call this the formal criterion.

Now the question is whether there can be propositions of this form. Using 'is Rrelated to' as a placeholder for relational predicates used to express explanatory links and 'is zero-R' as a placeholder for predicates used to express corresponding empty-base links. We can state the form of self-explanatory links as 'The proposition that P is zero-R', where the proposition expressed is identical with the proposition that P. Consider grounding as an example. Predicational zero-grounding statements have the form 'The proposition that P is zero-grounded'. Thus, if there are emptybase self-explanations of the grounding variety, the corresponding self-explanatory propositions have the form 'the proposition that P is zero-grounded', where the proposition that P is identical with the proposition that the proposition that P is zerogrounded. Indeed, here is a candidate that has this form:

(3) This proposition is zero-grounded.

Here, the expression 'This proposition' in (3) is intended to refer to the proposition expressed by (3). Note that while some propose that certain self-referential (e.g. paradoxical, liar-type) sentences do not express propositions, the self-referential nature of (3) alone is presumably not sufficient to assume that (3) expresses no proposition; after all, many (apparently) unproblematic self-referential sentences exist.222 But now note how (3) resembles the truth-teller 'This sentence is true': If we had to speculate about the truth-value of (3), it would not seem unreasonable to assign it the same truth-value as the truth-teller, which, many are inclined to believe, is defective and neither true nor false.223 And even if (3) were true, it presumably could not fulfill the high hopes some philosophers have put into self-explanatory propositions: Intuitively, (3) is somewhat thin in content, which is, perhaps, exactly what is to be expected of a zero-grounded proposition. Consequently, it is hard to see how it could serve the idea that a substantial class of truths are eventually explained by selfexplanatory propositions.

One might perhaps think that instances of the following schema could do better in this regard (let 'P' stand for an arbitrary proposition and '4' express the proposition labeled by '(4)'):

(4) The proposition that (P and 4) is zero-grounded.

But this is problematic because (4) seems to fail the formal criterion: If we eliminate the zero-grounding operator from (4), we obtain 'P and 4', which does not seem to be identical with (4), in part because (4) expresses a proposition with a zero-grounding operator having largest scope, whereas in 'P and 4', the conjunction operator has largest scope. We could perhaps allow that some conjunctions are identical (or at least suitably equivalent) to one of their conjuncts, this is for example possible according to certain worldly modes of identifying propositions or facts (e.g. Correia 2016). Then to vindicate the possibility of self-explanations of the above form, one would have to find a mode of individuation suited to deliver instances of (4) satisfying the formal criterion, but such an investigation goes beyond the scope of this chapter.

E.g. 'This proposition is a proposition', 'Every proposition is a proposition' and 'This proposition is such that 1+1=2'. Cf. Rosenkranz and Sarkohi (2006). As an anonymous commenter on the paper on which this chapter is based has stressed, it could be thought that the candidates considered here and in the next subsection would amount to objectionably ill-founded propositions. Development of a theory of propositions that would vindicate the existence of the candidates would go beyond the scope of this book, but let me note that the candidates are not obviously defective in this way and that at least with respect to (3), I am not alone in this assessment, cf. Lovett (2020). One reservation here might stem from an understanding of propositions as mereological wholes, but first this understanding is not mandatory, and second see Kearns (2011) for an argument that on such a view we should simply accept that at least certain (otherwise unproblematic) self-referential propositions are parts of themselves. For an investigation into the non-wellfounded mereology required for this, see Cotnoir and Bacon (2012).

Cf. Field (2008), but note also Field (2008, 277).

Instead, here are three further options to find (perhaps more substantial) candidates for empty-base self-explanations: First, one could attempt to find an explanatory relation R such that 'This fact is zero-R' is more substantial and less like the truthteller than (3). The second option invokes Dasgupta's (2014a) proposal that grounding is irreducibly plural, and the third considers laws as explanatory links.224 Setting aside the first option, we will now look at the second and third options in turn.

#### 6.5.1 Irreducibly plural grounding

According to Dasgupta (2014a), grounding is irreducibly plural in the following sense: (predicational) grounding statements have the form 'The Ys are grounded in the Xs', where 'Y' and 'X' are schema-letters for expressions denoting pluralities of facts, and it is possible that the Ys are grounded in the Xs, without any of the Ys on its own being grounded in the Xs. For example, Dasgupta argues that the individualistic facts (i.e. facts concerning particular individuals, like [Socrates is a philosopher] or [Obama is 75 kgs]) are together irreducibly plurally grounded in purely qualitative facts. That is, for example, individualistic facts about the mass of particular individuals are plurally grounded in purely qualitative facts capturing the mass relations between things, but no single fact about the mass of a particular individual is grounded in such facts on its own.

Correspondingly, plural zero-grounding statements can be expressed by having 'X' denote an empty plurality; alternatively, 'The Ys are zero-grounded' can be used. Dasgupta's proposal then allows for more contentful candidates for empty-base self-explanation by allowing for a plurality of propositions to occur as (joint) groundees in a grounding statement like this:

(5) This fact, [P] are zero-grounded.

Here, 'This fact' refers to the fact expressed by (5). Assuming with Dasgupta that there are irreducibly plural instances of grounding, an instance of (5) might in principle obtain without it being singularly zero-grounded, while at the same time being plurally zero-grounded together with [P].

A fourth option could perhaps be this: Returning to the assumption that links of empty-base explanations have the form '◼P', one might consider the possibility of prefixing a right-side infinite sequence of '◼P's to a sentence 'P' like this: '◼◼◼ … P'. Here, when the outermost '◼' is eliminated, arguably, a sentence of the same form '◼◼◼ … P' remains; but to my knowledge, a theory of non-well-founded propositions like this would yet have to be motivated and developed.

Now, is there any reason to assume there being self-explanatory facts of the form of (5)? What kind of facts would be suitable to be collectively zero-grounded, where one of the collectively zero-grounded facts is the corresponding collective zero-grounding fact itself? Dasgupta's examples for collectively grounded facts all involve facts that are similar in some respect (like the individualistic facts).

Therefore, a natural candidate for our collectively zero-grounded facts are other (non-factive) grounding facts. According to this idea, all non-factive grounding facts would be irreducibly collectively zero-grounded, including this collective nonfactive grounding fact itself. One tentative advantage this proposal has over Litland's (2017) original proposal (according to which non-factive grounding facts are zerogrounded) is that it avoids the following somewhat awkward regress: According to Litland's proposal, [P ! Q] is zero-grounded, [[P ! Q] is zero-grounded] is zerogrounded, [[[P ! Q] is zero-grounded] is zero-grounded], etc.; according to the present proposal there is just one self-referential collective zero-grounding fact here.225

#### 6.5.2 Generalized explanatory links

Let us finally consider how generalized links, such as laws of the following form might help (let '□L' stand for a law operator like the metaphysical law operator):

(LAW) □L∀x Fx ð Þ ! Gx

The idea is this: An ordinary generalized explanatory link can serve as an explanatory link of many explanations by linking different bases with different results. A generalized link of an empty-base explanation could in turn figure in explanations with several different results. Thus, in principle, there might be such a link which is the result of an empty-base explanation and which thus explains itself, but which in addition is the link of a further (possibly empty-base) explanation with a different result. Incidentally, the idea is reminiscent of Nozick's idea of "explanatory self-subsumption":

If one considers this regress to be more problematic than merely somewhat awkward, one might additionally reason as follows. What the regress shows is that some explanatory work remains to be done at each step and is hence deferred ad infinitum. Hence, a non-factive grounding fact cannot be fully zero-grounded on its own. But given the present idea, Litland's proposal can be amended: Non-factive grounding facts might not be individually zero-grounded, but they are all collectively zero-grounded.

The objectionable examples of explanatory self-deduction (total or partial) involve deductions that proceed via the propositional calculus. Would the explanation of a law be illegitimate automatically if instead the law was deduced from itself via quantification theory, as an instance of itself? If explanation is subsumption under a law, why may not a law be subsumed under itself? (Nozick 1981, 119ff.)

Here, Nozick appears to suggest that the permissibility of self-explanation somehow depends on whether the involved explanatory steps correspond to rules of the predicational calculus as opposed to the propositional calculus, but this does not seem very convincing: Just consider the question of whether universal generalizations are grounded in their instances or whether they ground their instances: While both options may have some initial plausibility, we should not accept both on pain of violating the asymmetry of grounding.

But we can ignore this part of Nozick's suggestion, and then the above considerations about empty-base self-explanation can help capture his idea of a selfsubsuming explanatory law. Nozick (1981, 119) does not properly distinguish between the roles of explanatory link and base; for example, he takes a self-subsuming principle to be an (explanatory) reason of itself. But if we make the distinction and understand explanatory self-subsumption as a kind of empty-base self-explanation, we can explain why explanatory self-subsumption may seem possible, namely because the simple arguments against self-explanation then do not apply to it. Moreover, the idea of explanatory self-subsumption gives us a further resource to address Kovacs' second argument: Someone who knows a self-subsuming explanatory principle might not have grasped it fully and thus might not have realized that it is self-subsuming. Thus, such a person may wonder why the selfsubsuming principle obtains. To understand why the principle obtains, this person then need not obtain further information, rather they need to grasp that the principle is self-subsuming and can thus take them from the empty base of reasons to the principle itself.

Let us think a little about the form self-explaining links à la Nozick would have to take. Let us consider unconditional links involving both quantification over entities and into sentence position. We can furthermore consider ordinary quantification or quantification into sentence position. Empty-base law-like links could then for example have one of the following forms (let 'O' schematically stand for a sentential operator):

(L1) □L∀x Gx ð Þ (L2) □L∀p Op ð Þ

It is unclear to me whether there could be an instance of (L1) that satisfies the formal criterion, i.e. an instance such that one of the instances of the involved quantification is identical to the proposition that is the whole link.226 But consider (L2): Could there be an instance for 'O' and a proposition [P] such that the proposition □L[∀p Op ð Þ] is identical to the proposition [OP]? Well, such instances are provided by the □L-operator and the proposition [∀p(□Lp)]:

#### (L3) □L∀p(□Lp)

If the quantifier is understood as ranging over all propositions, the result is absurd because for no false proposition [P] is it the case that □LP. This problem can be avoided if we instead understand the quantifier as ranging over all facts. The result is a candidate explanatory link according to which every fact is a law. While this will strike many as only marginally more plausible, the result is still interesting: Some philosophers have been moved to admit self-explanatory facts by their acceptance of the PSR. The PSR has also moved some to endorse necessitarianism, the idea that every fact is necessarily the case.227 (L3), properly understood, embodies these two rationalist ideas: It is self-explanatory and it states a variant of necessitarianism according to which every fact is a law.228

Let us take stock: While it is unclear whether there are more plausible candidates for empty-base self-explanation, we have made progress towards answering whether empty-base self-explanation is possible by clarifying what it would take for them to exist. If we are pessimistic about the prospects of empty-base selfexplanation, we have at least gained a better understanding of why this kind of self-explanation does not exist: Not because 'explainsinclusive' is irreflexive, as the arguments of section 6.2 would have it, but because it is hard to find substantial and plausible propositions of the required form.

226 If we assume, e.g., that [P] and [[P] is the case] to be identical, then 'L∀xðx is the caseÞ' is an instance of (L1) that satisfies the criterion, but this example faces similar issues to those discussed below. The issue here is to find an instance that satisfies the formal criterion without being too implausible.

Spinoza is an example for both moves, cf. Della Rocca (2010) and Lærke (2011), but see Schnieder and Steinberg (2015) on how proponents of the PSR can avoid either consequence.

One idea worth considering might be to restrict the quantifier in (L3) such that it still ranges over (L3) itself, but does not range over all facts, thereby avoiding the consequence that every fact is a law.

#### 6.6 Empty-base self-explanation meets philosophical theology

Let me end the chapter by showing how the notions of empty-base explanation and empty-base self-explanations might inform our understanding of certain ideas about the explanation of the existence of God. According to many scholastics like Aquinas, but also according to some later philosophers like Spinoza, God's essence involves God's existence.229 This alone suggests a way in which God's existence might be explained, namely by its status as being part of the essence of God. Using the conceptual apparatus developed above, the idea can be put like this: God's existence is empty-base explained, and the explanatory link of this explanation is the fact that it is part of God's essence that God exists.

Now, both Aquinas and Spinoza go further in that they also believe that God's existence is identical to God's essence.230 But this provides the material for a proposal for an empty-base self-explanation of God's existence: God's essence, i.e. the fact that it is part of God's essence that God exists, would be the empty-base link of this explanation and God's existence would be the explanatory result of this explanation. But according to both Aquinas and Spinoza, God's essence just is God's existence. If we understand this identity as the identity between the fact that God exists and the fact that it is part of God's essence that God exists, then the result is a proposal for an empty-base self-explanation.

Some remarks: First, by understanding their proposal as concerning emptybase self-explanations, both Aquinas and Spinoza might avoid the arguments against the intelligibility of self-explanation, as I have argued above. Second, the proposal is confronted with an issue we have encountered already: It is unclear that the required claim concerning the identity between the explanandum and the explanatory link can be made sense of. Third, while Aquinas' and Spinoza's shared assumptions allow for a proposal for a self-explanation of God's existence without the need to claim that God's existence is its own reason why (e.g. its own ground or cause), Spinoza appears to explicitly want to claim that God is her own cause, i.e. a causa sui and thus reason why.231

Cf. Lærke (2011).

Lærke (2011, 447f.).

Cf. McInerny and O'Callaghan (2018, sec. 11.3) for Aquinas and Lærke (2011, 456) for Spinoza.

### 6.7 Conclusion

Let us recapitulate: Using the tripartite account of the structure of explanations, I have distinguished two notions of self-explanation, defended one against several arguments against the possibility of self-explanation, and applied it in a solution of the circularity problem for Humeanism about laws of nature. In the remainder of the chapter, I have developed and defended the notion of an empty-base selfexplanation and suggested some applications for it.

# 7 The Epistemology of Empty-Base Explanation

The topic of this chapter is the epistemology of empty-base explanation. In the literature on zero-grounding (which provides one kind of empty-base explanation), several methods and kinds of arguments to establish instances of zerogrounding can be found (rather than being completely distinct, some of these overlap to an extent):

1. Extrapolation and application of general principles of grounding to the limiting cases with zero grounds.

As we have seen earlier, Fine (2012) takes a general principle of grounding and observes that the principle has a limiting instance which corresponds to an instance of zero-grounding, in Fine's case the zero-grounding of the empty conjunction.

2. Deriving instances of zero-ground within an explanatory calculus.

Litland (2017) argues for a calculus of explanatory inference rules and shows that given this calculus, the zero-grounding of non-factive grounding claims can be established.

3. Arguing both that a given proposition is grounded and possesses no grounds.

Muñoz (2020) argues for the (contingent) zero-grounding of negative existential facts by first arguing that negative existential facts cannot have grounds and then arguing that they must be grounded, leaving their being zero-grounded as the only option.

From the previous chapters, we can furthermore extract the following approaches:

4. Arguing first that a given (kind of) proposition has a certain kind of explanation and then arguing that this explanation is zero-ground (or more generally emptybase) explanation.

If we can argue that a given proposition has an explanation by status (for example, in the case of essentialist explanation, this idea seems to be backed by intuition), then given that I have argued in chapter 3 that such explanations should be understood as empty-base explanations, this amounts to an argument for an instance of the latter. Relatedly, in chapter 4 I have relied on a mix of intuition and theoretical considerations to argue that logical theorems possess a special kind of explanation that outstrips their ordinary grounding explanations, and then I have argued that either zero-grounding or other empty-base explanations provide such an explanation.

5. Using the notion of zero-ground together with the familiar (mixture of) philosophical methods of conceptual analysis, Carnapian explication, and abductive theorizing.

What I have done in chapters 3, 4, and 6 is an instance of this kind of general methodology. A further example is provided by De Rizzo (2020), who uses zeroground to give an account of the grounds of necessities. Moreover, presumably, the other methods will have to be accompanied by a mixture of familiar philosophical methods as well: As Rodriguez-Pereyra, Lo, and Skiles (manuscript) argue, the extension of familiar grounding principles to edge cases may yield a set of extended principles which together are inconsistent. In such a case, abduction or considerations concerning theoretical usefulness may help.232

More generally, we can observe that empty-base explanation is a type of explanation, and that the involved explanatory notions are the same notions that also figure in ordinary cases of explanation. For example, these are grounding in the case of zero-grounding explanations and laws of nature in the case of emptybase explanation involving a law of nature as link. Therefore, it is to be expected that the epistemology of empty-base explanation is an extension or part of the epistemology of ordinary explanations. Thus, however we can come to know ordinary grounding explanations presumably is also (or at least closely related to) how we can come to know zero-grounding explanations. Indeed, the methods mentioned above conform to this idea.

In this chapter I will focus on the epistemology of grounding explanations. Not much has been written explicitly about the epistemology of grounding and grounding explanations specifically, and I will take some steps towards remedying this situation. The goal is to explore the prospects of an account of how inference to the best explanation can establish grounding explanations. Some assumptions concerning inference to the best explanation will be developed that show how it might be used to establish empty-base explanations in general and zero-grounding explanations in particular. Of course, there may well be other (basic) methods of acquiring knowledge of grounding facts, but in the following I want to focus on inference to the best explanation.

This is the plan for the chapter: In section 7.1, to approach the use of inference to the best explanation (IBE) in a metaphysical context, I present and develop Stephen Biggs' account of how IBE can provide knowledge of metaphysical modalities on the

Thanks to Jonas Werner for discussion here.

basis of non-modal propositions, and formulate a dilemma for the account. In section 7.2 I argue that the problem raised by the dilemma for an abductive epistemology of metaphysical modality can be avoided by an abductive epistemology for explanatory notions such as grounding. I take some steps towards a theory of how inference to the best explanation can establish grounding claims on the basis of metaphysically innocuous, e.g. empirical claims. In section 7.3 I discuss how inference to the best explanation might establish identities and what a unified account of inference to the best explanation that can establish both grounding claims and identities might look like. In section 7.4 I make some suggestions as to how to flesh out and develop the account further, and I discuss the application of IBE vis-à-vis the PSR and a distinctive problem that arises for IBE given the existence of empty-base explanation.

#### 7.1 Abductive modal epistemology and a dilemma for Biggs' account

Stephen Biggs (2011) has argued that IBE can be used to establish metaphysical necessities on the basis of non-modal premises.233 At the core of his proposal lies the assumption that metaphysical necessities, such as [□∀x is scarlet ! x is red], sometimes provide the best explanation for their non-modal counterparts, in this case [∀x is scarlet ! x is red]. I argue that the account fails because the relevant necessities cannot explain their non-modal counterparts. Instead, I suggest that IBE might play a fruitful role in the epistemology of explanatory metaphysical notions such as grounding.

While philosophers have sometimes offered abductive considerations in modal metaphysics, few accounts of this practice exist in the literature on the epistemology of modality. Some applications and discussions of abductive considerations in modal metaphysics can be found in Lewis (1986b), Block and Stalnaker (1999), Hill and McLaughlin (1999), Shalkowski (2010), Hale (2013) and Williamson (2013). Some of the literature on the question of why there is anything at all can also be considered to be employing abduction to infer necessities from non-modal premises, e.g. see Rundle (2004). Relatedly, Rayo (2013) and Greco (2015) suggest that abduction might establish 'just is'-statements on the basis of non-modal premises, and Fine (2001) suggests that IBE might be used to establish grounding facts, but he does not develop the idea. One notable exception to this shortcoming is Biggs' (2011) "Abduction and Modality",

233 A comment on terminology: As I use the term, 'metaphysical necessity' refers to claims, propositions or facts of the form '□P', where 'P' stands for an arbitrary proposition and '□' expresses metaphysical necessity, as customary.

which contains an account of how IBE can establish metaphysical necessities, such as [□∀xðx is scarlet ! x is redÞ], using non-modal premises, such as [∀xðx is scarlet ! x is redÞ. Crucially, the account relies on the assumption that the relevant necessities can explain their non-modal counterparts.

I will now provide a sketch of Biggs' proposal and some questions it is confronted with. Under its alternative name 'abduction', Biggs characterizes IBE as follows:

Abduction can be thought of as consisting of three steps. First, identify a phenomenon that requires explanation. Second, generate theories that would explain the phenomenon. Third, choose the theory that best explains the phenomenon. Abduction, then, involves three main elements: the phenomenon to be explained (i.e. the explanandum), the competing theories qua explanations (i.e. the potential [explanantia]), and the principles for ranking theories. (Biggs 2011, 293)234

In accordance with this schema, Biggs proposes that metaphysical necessities can, in a nutshell, be established on the basis of non-modal premises along the following steps:

1. Identify non-modal explananda of the form '∀x Fx ð Þ ! Gx ', such as:

2. Generate salient candidate explanantia. These come in the following forms:

3. Considerations of parsimony rank the candidate explanantia: Provided that all three candidates are compatible with the non-modal evidence, the relevant instance of IDENTITY is best. Provided that this candidate explanans is excluded by the non-modal evidence (for example in the case of RED provided by things that are crimson in color), the relevant instance of META-PHYSICAL is best. Inference to the best explanation allows us to infer the best candidate explanandum, for example [Being a quantity of water = being a quantity of H2O] in case the explanans is WATER and [□M∀xðx is scarlet ! x is red)] in case it is RED.

A comment on terminology: Sometimes, it appears, 'abduction' is used in a sense in which it includes inferences on the basis of theoretical virtues such as simplicity and strength without any involvement of explanation (that is, at least not with any obvious or explicit such involvement). Biggs and I on the other hand are (at least initially) only concerned with IBE as characterized in the quote above. For a more thorough characterization of IBE see Lipton (2004).

Before I set up the dilemma for the assumption that the relevant instances of METAPHYSICAL are candidate explanantia for propositions such as RED and WATER, two comments are in order: First, I will postpone discussion of the case of IDENTITY until section 7.3. Second, Biggs' account raises a number of questions that his discussion leaves open, some of which are: (1) Can any arbitrary universally quantified conditional figure as a premise of an abductive argument as sketched above? (2) Is there a way to differentiate those that can from those that cannot? What kind of consideration allows to rank instances of NOMOLOGICAL higher than instances of METAPHYSICAL or IDENTITY? (3) What consideration guards against inferring necessities from contingent universal generalizations?

These questions need to be satisfactorily answered, not least to be confident that the sketched method does not objectionably overgenerate necessities; until then, the account remains incomplete. Biggs (2011, 312, fn. 26) seems confident that these questions can be answered using some ideas concerning theory choice from Lewis (1986b), but he does not provide details. We will come back to these questions in section 7.4, but for now, we turn our attention to a more fundamental flaw than incompleteness that Biggs' account suffers from: The alleged candidate explanantia are none, or so I will argue.

To set up the dilemma that confronts Biggs' account, I have to introduce a few assumptions about explanation and IBE. First, IBE deals with explanation why; the explanations that are considered and inferred in applications of IBE are explanations why a certain thing is the case and not, e.g., explanations how something is done or what something is.235 Accordingly, we will only be concerned with explanation why in what follows. Second, recall from chapter 1 the tripartite account of explanation according to which explanations involve claims, propositions, or facts in the roles of explanatory sources (comprising the explanation's base), link, and result. The tripartite account raises the question of how the traditional (and Biggs') explication of IBE as an inference from an explanandum to its best candidate explanans should be recast in terms of base, link, and result. While 'result' can simply replace 'explanandum' in the characterization, matters are less clear with 'base' and 'link' on the one hand and 'explanans' on the other. The salient questions are whether IBE is an inference from an explanatory result to

It is an interesting question whether this assumption is indeed correct (i.e. whether there could for example be something like inference to the best explanation what something is), but this is a question for a different occasion. In any case, the standard examples of IBE deal with explanation why.

– . . . sometimes the base, sometimes the link of the best candidate explanation, and sometimes both.

Examples of the first type come to mind most readily. For example, Lipton introduces IBE using the following examples:

Faced with tracks in the snow of a certain peculiar shape, I infer that a person on snowshoes has recently passed this way. There are other possibilities, but I make this inference because it provides the best explanation of what I see. Watching me pull my hand away from the stove, you infer that I am in pain, because this is the best explanation of my excited behavior. Having observed the motion of Uranus, the scientist infers that there is another hitherto unobserved planet with a particular mass and orbit, since that is the best explanation of Uranus's path. (Lipton 2004, 1)

But it seems plausible that IBE can also be used to infer the link of the best relevant explanation: Since explanations consist of both a base and a link, inferences to the best explanation should accordingly be able to license an inference to both a base and a link. And indeed it seems that, for instance, from the fact that the window is broken, Sherlock may not only abductively infer that Moriarty threw a ball at it, but also that the window is broken because Moriarty threw a ball at it and that Moriarty's throwing the ball caused the window to break.

Moreover, consider the toy-case of a wooden box to which a switch and a lightbulb are attached; the latter lights up when the former is pressed. Given suitable background assumptions, IBE seems to license to infer from this description of the situation that pressing the switch causes the lightbulb to light up. This already supports the assumption that IBE can establish the link of an explanation (here the causal link between the pressing of the switch and the lighting of the bulb) from a description of the result (the lighting of the bulb) and a base (consisting in the pressing of the button) of an explanation. Note that (again given suitable background assumptions) IBE appears to license inferring certain explanatory links over others in the described situation: Inference of a simple causal mechanism (perhaps the presence of a simple electric circuit that is closed by the push of a button) seems licensed over inference of more complex causal mechanisms (such as a miniature Rube Goldberg machine hidden in the wooden box).236

In addition to these cases, there are examples from the literature on IBE that support the point:

But if we try instead to explain why Jones rather than Smith contracted paresis, we will be led [. . .] to look for some possibly relevant difference in the medical histories of the two

The example of the box with the lightbulb is not mine originally, but I cannot recall where I first encountered it.

men. Thus we may infer that Jones's syphilis was a cause of his paresis, since this is an explanatory difference. (Lipton 2004, 73, my italics)

So I will assume that IBE can establish base alone, link alone or both together, depending on the case at hand. With respect to my case against Biggs' proposal this is fair since it provides a further option to understand the proposal. Additionally, we will see in the following sections that the assumption is theoretically fruitful when it comes to the epistemology of grounding.

With respect to Biggs' proposal now the question arises whether metaphysical necessities (more specifically: strict conditionals, possibly universally quantified) play the role of sources (constituting the base) or links in the abductive arguments suggested in his account. In what follows I argue that this question raises a dilemma for Biggs' account. According to the first horn, it is not sufficiently plausible that the relevant metaphysical necessities can figure in the bases of explanations like those suggested by Biggs: As I have argued in chapter 3, possibly ignoring some irrelevant exceptions, an instance of '□∀x Fx ð Þ ! Gx ' is not a reason for the corresponding instance of '∀x Fx ð Þ ! Gx '. 237 According to the second horn, strict conditionals cannot be the links of any explanation at all and hence cannot occur in that capacity in abductive arguments.

First horn of the dilemma: instances of '□∀x Fx ð Þ ! Gx ' do not normally occur as sources of explanations of the corresponding instances of '∀x Fx ð Þ ! Gx '.

The first horn of the dilemma consists in the fact argued for in chapter 3 that (with the possible exception of some extraordinary cases) for no [P], [□P] explains [P] in the sense of being a source in an explanation of [P]; or, in the idiom of reasons, (ignoring some extraordinary cases) for no [P], [□P] is a reason why [P]. Consequently, facts of the form '□∀x Fx ð Þ ! Gx ' are not (ignoring the extraordinary cases again) reasons why/sources in explanations of the corresponding instances of '∀x Fx ð Þ ! Gx '. The relation that obtains between the sources and the result of an explanation can be expressed by a because-sentence. Thus, Biggs' account requires that because-statements like the following are true:

(BECAUSE-RED) ∀xðx is scarlet ! x is red) because □∀xðx is scarlet ! x is redÞ.

To be clear, there are of course explanations with metaphysical necessities as sources (such as the partial explanation of my knowing that □P by it being that case that □P), the claim is merely that the explanations required by Biggs' account do not have metaphysical necessities as bases. On the other hand, as I have already argued in chapter 3, metaphysical necessities are never explanatory links due to their and explanation's nature.

In general, the account requires that there are true instances of the following schema:

(BECAUSE-□) P because □P.

But in chapter 3 I have argued against the relevant instances of BECAUSE-□, and suggested how to alternatively accommodate the intuitions in favor of those instances of BECAUSE-□. Moreover, if Biggs wanted to maintain the relevant instances of BE-CAUSE-□ he would have needed to provide an account of the explanatory link connecting [□P] and [P] (it cannot be grounding, as discussed in chapter 3) and, second, the resulting epistemological account (if combined with the thesis that we in fact sometimes come to know necessities as it describes), as well as the theories that are established along its lines carry commitment to the relevant explanatory links, avoiding which would be preferable.238

#### Second horn of the dilemma: strict conditionals cannot be explanatory links.

The second horn of the dilemma states that strict conditionals cannot be explanatory links. Before I argue for this, note that even if a strict conditional such as [□∀xðx is scarlet ! x is redÞ] could in principle figure as the explanatory link of an explanation, it is unclear how that explanation could have the form suggested by Biggs. This is because explanatory links are ordinarily thought of as in some way connecting the base and result of an explanation, and there does not appear to be a candidate explanatory base that [□∀xðx is scarlet ! x is redÞ] connects like this with [∀xðx is scarlet ! x is redÞ]. This problem can be solved by assuming that IBE can take the following form:

238 Note that even if the proposal from appendix B of chapter 3 could be made to work, it is unclear how this could help Biggs' cause: A way would have to be found that allows to weigh instances of '□P' against instances of 'P' in abductive arguments. Biggs offers nothing in this regard, he does not even consider instances of 'P' as explanatory candidates. Without such an account though, it remains unclear whether IBE could ever establish an instance of '□P' over an instance of 'P'.

239 There may be other candidate sources such as [a is crimson], but at least this option is excluded by our knowledge of [a is scarlet].

. . . and an explanatory link such as

# (METAPHYSICAL) □M∀xðx is scarlet ! x is redÞ.

3. The candidate explanations are ranked and if everything goes well, META-PHYSICAL might be inferred as the explanatory link of the best explanation for [a is red].

The wooden box example from above supports the point that IBE can indeed proceed in this fashion from a candidate explanandum to an explanation whose base is already known, thus only providing additional knowledge of an explanatory link. Biggs' proposal thus conceived would have two advantages over the version discussed before: First, it would carry no problematic commitment to additional explanatory relations. Second, one might have suspected from the beginning that the explanatory proposals that philosophers of mind such as Block and Stalnaker had in mind when discussing IBE to establish necessities concerned explaining why certain mental states obtain in terms of physical states. If the relevant explanatory proposals are indeed of this form, then it is somewhat natural to suspect that the strict conditionals that connect physical states with mental states are explanatory links for the relevant explanatory proposals.

Alas, the point that constitutes the second horn of the dilemma is that there are well-known reasons against the thesis that metaphysical necessity is an explanatory notion and thus against the thesis that strict conditionals can be explanatory links (this is often put in relational terms: The relation expressed by a strict conditional is not an explanatory one).240 Two such reasons are these: First, explanatory links are asymmetric, but modal dependence (as captured by strict conditionals) is not. Note that while some authors accept that there are instances of symmetric explanation, modal dependence has many symmetric instances where intuitively there is either no explanatory connection to be found, or it does not run in both directions.241 For example, modal dependence is reflexive, but explanation is not (even if we allow for some exceptional facts to be self-explanatory). Also consider the following two cases: It is plausible that [snow is white] is true because snow is white and not vice versa. But [snow is white] and [[snow is white] is true] are modally equivalent. Furthermore, any two necessary truths are modally equivalent, but neither need explain nor be a reason for the other.

I recap here my discussion from section 3.5.

For discussion of symmetric instances of dependence and grounding, see for example Jenkins (2011), Barnes (2012), and Thompson (2016).

This last case gets us to the second point against strict conditionals as explanatory links: Explanatory links only connect explanatorily relevant relata, but strict conditionals also connect explanatorily irrelevant relata: Each proposition necessitates any necessary proposition, even if the former does not explain the latter. Also, if [P] and [Q] are necessary and [R] is contingent, then [P ^ R] and [Q ^ R] are contingent, but necessarily equivalent and no explanatory connection seems to hold between them, at least if [P] and [Q] are explanatorily unconnected.

Biggs might propose a two-component view according to which what should be taken as the explanatory link is a strict conditional together with a further element that ensures that the two conditions above are satisfied. Here are some remarks on this view: First, I do not know of any such account in the literature, so the onus would be on the friend of strict conditionals as explanatory links to develop such a view. Second, taking a clue from Kim (1994), we may believe that explanatory links should somehow account for features of explanation such as irreflexivity, asymmetry, and relevance. Prima facie, the proposed two-component view does a worse because less unifying job at this than views that propose explanatory relations like causation or grounding that natively satisfy conditions such as asymmetry and relevance. Third, we might allow for such proposed sub-par explanatory links in the candidate explanations considered in an application for IBE, but it becomes unclear why they should ever be preferred to more unified candidate explanations using notions such as ground.

This completes the dilemma against Biggs' account: In chapter 3 and above I argued first that it is not sufficiently plausible that instances of '□∀x Fx ð Þ ! Gx ' are explanatory sources or reasons for the corresponding instances of '∀x Fx ð Þ ! Gx ' and then I argued that instances of '□∀x Fx ð Þ ! Gx ' are never explanatory links. But since, as argued above, Biggs' proposal is committed to them either being links or explanatory sources of the right sort, we should discard his proposal.

#### 7.2 Towards an abductive epistemology for grounding

Strict conditionals cannot be the links of any explanation and thus cannot be established by IBE in the way sketched in section 7.1, but truths involving explanatory metaphysical notions that can be explanatory links may well be established in this way. One such notion is grounding, as discussed for example in Fine (2012).242 Adopting the form of argument proposed at the end of the last section, an application of IBE that establishes a grounding claim could look like this:

Incidentally, some authors have suggested (but not developed that suggestion) that grounding might be abductively established, e.g. Fine (2001) and Schaffer (2017).

. . . and the following explanatory link:

(GROUNDING) [a is scarlet] grounds [a is red].

Note that there might be further candidate explanatory links, such as:

(NOMEX) [a is scarlet] nomologically explains [a is red].

3. The candidate explanations are ranked and if everything goes well, GROUNDING might be inferred as the explanatory link of the best explanation for [a is red].

Note that the sketched proposal does not rely on the use of grounding specifically – instances of any notion that can provide links of metaphysical explanations may in principle be established by an argument like this: Two examples that come to mind are metaphysical laws and perhaps some sort of essential dependence. Note as a possible further avenue of investigation that abduction might also take a somewhat different form: First, a form of abduction could be considered that starts with an explanatory base and a result, and just compares possible links.243 In contrast, the abduction above compares bases and links together. Second, a form of abduction could be considered that takes several instances of, i.e. 'x is red' and 'x is scarlet' as its starting points from which a metaphysical law or a plurality of grounding statements is then inferred. Third, one could consider understanding abduction as theory choice guided by explanatory principles (amongst other principles, possibly). I will say a bit more about this option in what follows.

At this point there are good news and bad news for an abductive epistemology of metaphysical necessity. The good news is that an abductive epistemology of explanatory notions such as grounding may form the basis for an epistemology of metaphysical necessity: While not everyone in the literature agrees, grounding truths are commonly taken to entail corresponding metaphysical necessities. In order to arrive at an at least partially abductive epistemology of metaphysical necessity, the abductive argument in favor of the relevant grounding or essence claims might then be supplemented by a method that can establish the link between grounding claims and the corresponding metaphysical necessities.

Recall the wooden box example from above.

The bad news is that even if this link is a case of grounding, i.e. if the strict conditional is grounded in the grounding claim, it is of a form that the method sketched above cannot establish, because the method establishes an explanatory link based on the knowledge of an explanatory result (and potentially an explanatory base that also is known already). In this case, though, the explanatory result (the strict conditional) is not yet known and thus cannot serve as a premise of an abductive argument (at least not if abductive arguments are understood as I have done above). Even if we grant that IBE is able to establish grounding claims as suggested above, the method remains limited in a notable way with respect to metaphysical necessities. In metaphysics, interest in notions such as grounding is at least partially due to the idea that merely modal notions cannot capture certain explanatory features of the metaphysical phenomena. In our context, this feature of notions such as grounding in principle allows for IBE to provide epistemic access to these notions. Somewhat ironically, the stronger, explanatory notions may in principle be accessed by IBE, while metaphysical necessity requires different or at least supplementary methods.

Let me give an idea what these could be: Aside from a sui generis capability of assessing modal propositions (or related propositions such as counterfactuals), some live options are conceptual analysis, Carnapian explication, and abductive methodology understood more broadly as a kind of theory choice: If these methods deliver accounts of modal propositions in terms of explanatory ones, IBE as understood above could establish the latter, from which modal truths could then be derived using the aforementioned accounts.

Finally, turning to the epistemology of empty-base explanation, note that the above outline of how IBE might establish grounding statements is flexible enough to show how IBE might establish zero-grounding explanations: One of the explanatory candidates might be a zero-grounding explanation. IBE may then establish that candidate, if it turns out to be the best explanatory candidate that is available.

#### 7.3 The case of identity

In the following sections I will try to refine and put some flesh on the bare bones of the proposal offered in the previous section. In this section let us consider whether the proposal can accommodate the idea that IBE is also able to establish identity statements: As in the case of metaphysical necessity, if an identity statement can be the conclusion of an instance of IBE, it has to either be the link of the corresponding explanation, or it has to figure in its base. More specifically, Biggs proposes that identity propositions such as [being a quantity of H2O is identical with being a quantity of water] can sometimes be abductively inferred from universally quantified biconditionals such as [∀xðx is a quantity of H2O \$ x is a quantity of waterÞ]. For this to be true, [being a quantity of H2O is identical with being a quantity of water] has to figure in the base or be the link of an explanation of [∀xðx is a quantity of H2O \$ x is a quantity of waterÞ].

So, might [being a quantity of H2O is identical with being a quantity of water] figure in the base of an explanation of [∀xðx is a quantity of H2O \$ x is a quantity of waterÞ]? Or, in the idiom of reasons, might [being a quantity of H2O is identical with being a quantity of water] be a reason [∀xðx is a quantity of H2O \$ x is a quantity of waterÞ]? Although I am not completely unsympathetic, one reason to be skeptical here is that to understand the relevant explanations as grounding explanations would commit us to non-standard instances of grounding, while understanding it differently commits us to novel explanatory relations (recall here my discussion of the explanation of logical theorems from chapter 4).

To the thesis that identity statements can be the links of explanations one might object that they relate entities (broadly construed, including individuals, properties, facts, and propositions), while explanatory links are properly expressed using sentential operators.244 But this objection would be insufficient for two reasons: First, while the operator view may well be correct for the case of grounding, this in itself does not suffice to show that all explanatory links are to be expressed using sentential operators – for example, perhaps causation is best understood as a relation between events. Second, as has been argued in the literature (e.g. by Rayo 2013), the notion of identity can be generalized to yield identity operators that apply to predicates and sentences, instead of merely to singular terms. Such a notion of generalized identity could then be used. But then it seems like the asymmetry-considerations from above apply here again: Explanatory links must be asymmetric, but identity statements are not, therefore identity statements cannot be explanatory links.245

Alternatively, we can try to invoke theoretical virtues such as ideological parsimony to argue in favor of generalized identity statements: A theory according to which being F just is being G is in this respect more ideologically parsimonious than a theory that does not identify them. While 'abduction' is sometimes used to refer to a general practice of theory choice on the basis of theoretical virtues such as parsimony, let us try to see whether we can understand the idea that identities can be established by abduction while understanding the latter as properly involving explanatory considerations as before.

As for example Fine (2012) proposes for the case of grounding.

Perhaps an asymmetric notion can be developed on the basis of generalized identity, but this is an idea that I will not pursue here.

So instead of going down either of the routes suggested above, I want to take a look at a method suggested by Rayo (2013) and developed by Greco (2015), that is related to IBE but somewhat more general, which may help to make sense of the idea of using something like IBE to establish identities. Rayo and Greco are concerned with the epistemology of 'just is'-statements, a kind of generalized identity statement. Instead of using IBE as understood above, they suggest an explanatory principle of theory choice which sometimes favors theories containing certain identity statements over other theories.246 I will now quickly present the principle and offer some comments on it:

#### (Methodological Maxim)

In choosing between rival sets of 'just is'-statements, the more why-questions you can pose and answer, the better. The more why-questions you can pose but can't answer, the worse.

Note first that it might be possible to use this as a basis to develop a unified abductive account of the epistemology of 'just is'-statements and properly explanatory notions such as grounding. Of course, as with Biggs' proposal, more would have to be said about how the candidate answers to the relevant why-questions are to be ranked. Note also Greco's explanation of how the Methodological Maxim allows to decide between rival sets of 'just is'-statements:

This double contrastivity of explanation – both in why questions, and in because answers – suggests a natural way of saying more about what sorts of costs and benefits are associated with accepting or rejecting a 'just is'-statement. In general, accepting a 'just is'-statement amounts to rejecting a distinction. On the doubly contrastive approach to explanation, distinctions (contrasts) are central to both questions, and answers. This will shed light on the sense in which rejecting or accepting a distinction has both costs and benefits. When we accept a new 'just is'-statement and thereby reject a distinction, we eliminate potential explanantia. We cannot explain anything by appeal to something's being F rather than G, once we accept that what it is to be F just is to be G. This is the cost side of the equation, and the sense in which accepting a 'just is'-statement involves a "decrease in the range of theoretical resources of one's disposal". But for essentially the same reason, rejecting a distinction also eliminates potentially pesky explananda. This is the benefit side of the equation – one cannot be under a burden to explain why something is F rather than G if what it is to be F just is to be G. (Greco 2015, 7)

But this account seems unsatisfactory because it cannot explain why the benefit of not being under a burden to explain why something is F rather than G counts in favor of the corresponding 'just is'-statement, rather than only in favor of the

In fact, Rayo and Greco are skeptical that their method ultimately succeeds, but this shall not concern us here.

ideologically cheaper corresponding universal biconditional [∀x Fx ð Þ \$ Gx ]. Let me explain: Once we accept that 'F' and 'G' are co-extensional, we have already removed the burden of explaining why something is F rather than G, because this question has as a presupposition that something is F and not G, but this presupposition is inconsistent with the assumption that everything is F iff it is G. Analogously, once we accept that everything is F iff it is G, we are already unable to explain anything by appeal to something's being F rather than G, because explaining thusly would require that we believe that there is something that is F and not G. Again, the consideration does not only apply to 'just is'-statements, but already to universally quantified biconditionals.

Therefore, let me propose a better account of the Methodological Maxim that does not rely on any considerations about contrastivity. We need to show how accepting 'just is'-statements affects how many why-questions we can pose and answer, even under the assumption that being F and being G are co-extensional. To do this, consider the following why-questions and because-answers:

If it is not the case that being F just is being G, then we deal with two distinct whyquestions and two distinct because-answers here. But if being F just is being G, then a case can be made that in the relevant sense, 1. and 2. are the same whyquestion, and 3. and 4. are the same because-answer. After all, being F just is being G. Therefore there are prospects of an abductive epistemology of identity statements via the move to theory choice and explanatory maxims such as Greco's Methodological Maxim: 'Just is'-statements reduce the number of why-questions as well as the number of possible answers to why-questions.

It may be worth pointing out that a move like this does not without further ado provide an abductive epistemology for metaphysical necessities: Since whyand because-contexts are hyperintensional, the acceptance of mere co-intensionality – that metaphysical necessities can deliver – does not allow for a reduction in the number of why-questions and because-answers that one can offer: The notion of identity that is required for such a reduction needs to be strong enough that whyquestions such as 1. and 2. are conflated if being F and being G are identical in the relevant sense.

Note that if this proposal works, the Methodological Maxim may provide the material to formulate an account of the theoretical virtue of ideological parsimony in explanatory terms: It may provide an answer to the question why a theory according to which being F and being G are identified and which is hence (in this respect) more ideologically parsimonious than a similar theory according to which being F and being G are not identified, is to be preferred over the latter theory: Namely because the Methodological Maxim advises us to prefer the theory that gives rise to fewer unanswered why-questions.247

#### 7.4 Fleshing out the account

Let us take a step back now and consider in what directions the above should be developed next. Note at the outset that there are at least two ways a proposal like the above can be understood: First, it can be understood as delivering an account of our judgments, intuitions, how they are justified, and what their underlying mechanisms are. A proposal with such an aim is for example given by Lipton (2004), who proposes IBE as an account of our scientific but also our ordinary, everyday life inferential practices. With respect to the present proposal, the idea would be that our intuitive judgments concerning grounding statements are due to and justified by an underlying application of IBE.248

Second, the present proposal can be understood as offering a method that can be reflectively applied to arrive at grounding statements. In order to offer such a method, the proposal needs to include specific enough instructions as to how to generate explanatory proposals, how to weigh competing explanatory proposals against each other and as to when we are in fact allowed to infer the best explanatory candidate. If we want to offer an account of the first kind, a similar account of how competing explanatory proposals are generated, weighted against each other, and finally inferred must be offered: Otherwise, it seems to me, we cannot be confident that our intuitions are indeed the result of a weighing of explanatory proposals. So, there are some open questions that an account of IBE that is able to establish grounding claims – whether the account proposes IBE as a method that can be reflectively applied or as an account for our intuitive judgments – needs to answer. Such questions include:

247 Let me mention one problem that further development of these ideas would have to address: There is some intuitive support for the idea that (assuming the identity of water and H2O) we should accept that x is water because x is H2O and still reject that x is water (or H2O for that matter) because x is water (for discussion see Schnieder 2010). The approach either has to either reject this, or develop a notion of generalized identity and a criterion of identity of why-question s and because-answers that avoids the problem.

In this case, one may wonder whether the account would be phenomenologically adequate.

These are also questions that a general account of IBE should address, but they nevertheless generate additional pressure for an abductive epistemology of grounding: We can be decently confident of applications of IBE in ordinary, everyday and scientific contexts, but here we want to argue that IBE can be used in a metaphysical context and, more specifically, to establish grounding claims. In order to do this, we need to argue that at least in principle, the above questions can be answered for applications of IBE that aim to establish grounding claims.

We should be able to show how IBE is constrained, not least because if it were unconstrained, it appears that an all too simple argument for a principle of sufficient reason could be attempted along the following lines: Suppose we wonder whether a fact [P] has a reason why it obtains. Then if the application of IBE is unconstrained and if explanatory candidates such as candidate grounds are easily stipulated (as they appear to be), then IBE can establish the best available explanatory candidate.249 Since [P] was arbitrarily chosen, it is unclear what should stop IBE from providing each fact with a reason why it obtains. It appears that if we let it come this far, the only option to stop this overgeneration is to claim that sometimes there must be several maximally good explanatory candidates and that then none of them – but also not their disjunction – can be inferred.

Luckily, there are several options for how the application of IBE to infer instances of grounding is constrained are available:

Let us ignore empty-base explanation for the sake of convenience for now.

– IBE might only be applicable once it has been established that an explanation exists.250

A complete account of IBE as a method to establish grounding claims would have to choose and defend at least one of these options. In the remainder of this section, I will first look at how Biggs' account fares with respect to the questions raised above, and then I will offer some thoughts on how IBE might establish whether a given proposition is grounded in something, zero-grounded, or ungrounded.

#### 7.4.1 Can Biggs' account deal with these questions?

Returning to Biggs' proposal briefly, we can observe that he does not sufficiently account for how explanatory candidates are determined, how the best amongst the candidates is determined, and how IBE is constrained such that sometimes we cannot infer a corresponding modal claim from a universally generalized conditional. While he (2011, 312, fn. 26) gestures at Lewis (1986b) for an idea to answer the two questions above, he does not properly formulate or defend an account here.

The existence of this lacuna in Biggs' account also means that he does not succeed in his secondary aim, namely to provide an abductive argument against dualist theories in the philosophy of mind and in favor of physicalism concerning the mental. At least, his case remains incomplete at this critical juncture, because he provides no criterion that tells us (nor method to determine) when the abductive inference is warranted and when it is not. Since it cannot be always warranted, lest IBE objectionably overgenerate necessities (or, in our case, grounding claims), the question remains whether it is warranted in the case of the mindbody problem. A related problem arises from the existence of dualist intuitions that conflict with the (physicalist) explanatory candidates that Biggs would like to establish: Biggs does not tell us why IBE should be able to trump such intuitions. Yet, he probably cannot completely abandon all use of intuition, for presumably some sort of intuitive judgment about what counts as an explanatory candidate and about relative explanatory strength is required to identify the explanatory candidates, rank them, and constrain the application of IBE. It may turn out that the dualist intuition that the physicalist candidate is not an explanatory candidate

But note that this may seem to conflict with our everyday practice, where we regularly use IBE arguably without knowing that an explanation does exist.

or not a sufficiently good explanatory candidate is the very kind of intuition that has to be considered in metaphysical applications of IBE.

Staying with the topic of intuition for a while: Inference to the best explanation as a method for metaphysics has partially been motivated by various broadly 'empiricist' skeptical concerns about intuition- and conceivability-based rationalist views.251 In fact, it is sometimes explicitly suggested that abductive methods can address the empiricist worries, see for example Biggs (2011) and Roca-Royes (2017, 242). But IBE as a method for metaphysics can only respect the motivation to avoid these empiricist worries if the required methods for determining what the explanatory candidates are and how they are weighted to determine the best explanatory candidate are not similarly suspect. For if they are, it becomes unclear how the empiricist could accept the use of IBE in metaphysics (e.g. to establish grounding claims) although it involves the very kind of methods the empiricist finds objectionable.

Note that the point is not intended to be about labels, e.g. that IBE should count as a rationalist method rather than an empiricist method (for example Bon-Jour (1998) seems to see it that way). Rather, if IBE is offered as a method for metaphysics that is able to resist broadly empiricist concerns leveled against applications of intuition and conceivability in modal metaphysics and elsewhere, it should be established that how we determine and weigh explanatory candidates does not rely on methods (such as certain forms of intuition) that are equally suspect by empiricist lights as proposals that directly invoke such methods (such as an intuitive faculty that allows us to determine the essential truths or what grounds what) for metaphysics. As far as I can see, this has not yet been established; for example, no such account is to be found in Biggs (2011). In fact, it seems quite plausible that intuition plays some important role in determining and weighing the explanatory candidates of an application of IBE – yet it is an open question why these intuitions should be more trustworthy than direct intuitions about what is necessary or what grounds what.

Note that even certain dialectical moves do not free the empiricist from this burden: For example, if we start with the assumption that IBE in general is respectable and then find out that its use in metaphysics involves aspects that are problematic for the empiricist, one reaction would be to conclude that the empiricist is mistaken, not only about IBE, but about intuition-based, rationalist methods as well. On the other hand, the empiricist might discard even IBE, at least as a method for metaphysics. Either way, the empiricist does not get from IBE what they wanted. A similar consideration seems to apply to some other broadly

For a number of works in this vein concerning modal epistemology see, e.g., the papers in Fischer and Leon (2017) and the papers mentioned in Roca-Royes (2017, 221).

empiricist accounts such as Roca-Royes' (2017) inductive account of knowledge of de re possibility for concrete entities. In this particular case, the open question is whether the similarity-judgments required by applications of induction according to Roca-Royes are admissible by empiricist lights.

#### 7.4.2 Grounded, zero-grounded or ungrounded?

To close the chapter on more of a constructive note, let us think about whether and how explanatory considerations can help to decide whether a fact is ungrounded, grounded in something, or zero-grounded. More precisely, suppose there are three theories M1, M2, and M3 which differ with respect to the grounding status of a fact [P]: According to M1, [P] is ungrounded, according to M2, [P] is grounded in something, and according to M3, [P] is zero-grounded. Without going into the messy details, the theories are intended to be as similar as possible, but they need not only differ with respect to the proposition that [P] is grounded, for M2 should also contain the grounds of [P], if there are any.

We can then ask whether explanatory considerations at least sometimes tell us to favor M2 (according to which [P] is grounded in something) over M1 (according to which [P] is ungrounded). The previous sections suggest that the answer is yes: M2 contains the material for a grounding explanation of [P], while M1 does not. But immediately, two further questions arise: First, what constrains explanatory considerations such that M2 is not always valued over M1? Second, suppose that explanatory considerations sometimes, but not always, value M2 over M1. Then it is a further question whether explanatory considerations ever value M1 over M2: Can explanatory considerations establish that certain facts are ungrounded, and if so, how?252

I want to focus on the latter question here. One idea as to how explanatory considerations might establish ungroundedness is that the fact that a certain fact [P] is ungrounded itself figures as a ground in a grounding explanation of some fact [Q] and can be established by an inference to the best explanation for [Q]. For example, suppose the abductive method sketched above does not establish a ground for a certain fact [P] – perhaps there are no candidate explanations, or perhaps none of the candidate explanations is good enough to be inferred. One might think that the very fact that no ground for [P] has been established so far is itself best explained by the fact that [P] is ungrounded. Formulated like this, it

252 Note that in asking this I do not want to suggest that there are no other possibilities of how ungroundedness can be established. For example, the fact that [Q] does not ground [P] might be derivable from other grounding facts that are incompatible with [Q]'s grounding [P].

seems plausible that if there are ungrounded facts, there are explanations like this, but further investigation would be required to clarify exactly how this kind of explanation operates.

A further idea is to apply a variant of the Methodological Maxim from the previous section:

#### (Methodological Maxim✶ )

In choosing between M1 and M2, the more why-questions you can pose and answer, the better. The more why-questions you can pose but can't answer, the worse.

But this maxim alone can often not favor a theory M1 according to which [P] is ungrounded over a theory M2 according to which [P] is grounded in [Q]. Let me explain:

It appears that the maxim is silent on whether to accept M1 or M2: While M1 does not answer why P, M2 appears to merely have traded the answer for the cost of a new why-question it does not answer, namely why Q. Note first that it does not make a difference if we consider the relevant (negated) grounding statements as well: For M1 the question arises why [P] is not grounded and for M2 the question arises why [Q] grounds [P].253 It seems the only case in which the maxim favors a theory M1 according to which [P] is ungrounded over a theory M2 according to which [P] is grounded (in something) is if, according to the latter theory, [P] is fully grounded only in two or more facts taken together for which why-questions arise that do not arise for M1.

So perhaps we have not counted correctly and we have to suppose that M1 either contains [Q] or its negation. But this would only increase the amount of why-questions that arise for M1 and are not answered (as we may stipulate in analogy to our stipulation about Q for M2). Therefore, this change does not show that Methodological Maxim✶ alone can favor M1 over M2. It appears that a principle like Methodological Maxim✶ that involves merely counting why-questions and their answers can often not help decide in favor of a theory according to which a given fact [P] is ungrounded, rather than in favor of a similar theory according to which [P] has a ground.

253 If anything, this spells trouble for using the maxim to establish ungroundedness, because [P]'s not being grounded is plausibly grounded in [P]'s not being grounded in x, for every x, and this gives rise to a plenitude of why-questions.

Of course, alternatives are available: Perhaps M2 will involve concepts that M1 does not involve. If F is such a concept, then globally, M2 will give rise to many why-questions that M1 does not. For instance, suppose M2 stipulates as a ground for [P] that some entity is F. Once the concept F is introduced, questions arise concerning what other things stipulated by M2 are F and why they are F. Also, suppose that according to M2 no other things are F. Then why-questions arise with respect to those things not being F. Depending on whether or not M2 offers satisfactory answers, M1 might be valued higher with respect to the Methodological Maxim✶ .

If we look beyond Methodological Maxim✶ , further options seem available: Perhaps M2's being more (for example ideologically) can somehow be weighted against its answering more why-questions, such that it comes out as less attractive than M1. A problem with this is that it is not quite clear how this weighting works. Here, an attraction of the Methodological Maxim✶ was that it suggested a unification of the theoretical value or disvalue of complexity and power to answer why-questions. Another (if perhaps not completely satisfactory) option would be to accept a certain bruteness of explanatory theoretical value – one theory is just explanatorily better than another with nothing more to be said, and we have some epistemic faculty (e.g. intuition) that can tell us which theory that is. But before we go that far, we should remember that we only enter the business of weighing two theories if they are epistemic possibilities: It is presumably at least possible to rule out a grounding proposal like M2 on intuitive or conceptual grounds before any application of IBE or the Methodological Maxim✶ .

Now, once zero-grounding enters the field, things change: Let us see whether we can identify explanatory virtues that favor theory M3 according to which [P] is zero-grounded over the similar theory M2 according to which [P] is grounded in something, and over a similar theory M1 according to which [P] is ungrounded. M3 answers the question why P and apparently only gives rise to the why-question concerning the grounding statement. In this respect it does better than both M1 which gives rise to both the question why P and why P is not grounded, and it does better than M2 which answers why P but gives rise to the new questions why the ground(s) of [P] obtain and why the relevant grounding fact obtains.

So, focusing on the first case, it appears that for a given theory according to which certain facts are fundamental by way of being ungrounded, zero-grounding has the potential to generate alternative, explanatorily stronger, proposals. For each theory according to which a fact [P] is ungrounded, there is an alternative theory according to which [P] is zero-grounded. In contrast to ordinary grounding explanations, zero-grounding explanations do not involve any grounds, and hence they do not incur new explanatory burdens with respect to such grounds. Prima facie, the latter theory has therefore more explanatory power than the former theory, since the latter allows for a zero-grounding explanation of [P], whereas according to the former theory, [P] does not have a grounding explanation; other things being equal, this seems to favor the latter over the former theory.

Are there now any other broadly explanatory virtues that rank the zerogrounding involving theory M3 lower than either of its two rivals? It seems complexity will not do: M3 does not involve any concepts or entities that are not involved in M1 and M2, and with respect to what facts it involves it seems similarly complex to M1. It appears we are again left with three possibilities: First, there are other explanatory virtues; second, explanatory virtue is an at least partially fundamental matter; or third, it is only possible to decide against proposals for zerogrounding before the stage of weighing relative explanatory value is reached – for example by deciding on intuitive or conceptual grounds against the possibility of the relevant zero-grounding theory. Unsurprisingly perhaps, an upshot of this discussion is that a crucial part of an account of an abductive method that is able to establish grounding statements will be an account of how exactly the candidate grounding explanations are determined. One idea to be pursued here may be (as towards the end of chapter 5 to take more seriously the connection between explanation, understanding, and general explanatory principles, or unification more generally): The hope is that if candidates for instances of zero-grounding (and empty-base explanation in general) must be limiting instances of more general explanatory principles, then brute 'zero-ification' of the fundamental postulates of some theory will not be possible in general.

# Conclusion

Let us recap: The goal of this book was to argue for the legitimacy of empty-base explanation and offer philosophically interesting applications of the idea. Aside from the overall argument, some things that I developed along the way include:

Novel philosophical ideas (especially those of the more 'out there' persuasion) have to earn their keep via their applications and theoretical fruitfulness. Otherwise they may turn out to be intelligible given some open-mindedness and conceptual flexibility, but remain more a curiosity than of serious philosophical interest. When I started working on this project, the only material discussing zero-ground were Fine's original paper and Litland's account of the grounds of ground. Since then, as reflected above, the literature has steadily grown. I hope that this book can make a further contribution in arguing that the idea of emptybase explanation really has intriguing applications.

Aside from investigating further applications of empty-base explanation, some areas in which I believe further research could be valuable are the metaphysics of multigrade and variably-adic relations, the epistemology of empty-base explanation (e.g. to better understand what lets us decide between a fact's being unexplained and the fact's being empty-base explained), the form of explanatory laws or generalizations that allow for instances of empty-base explanation (for a relevant recent discussion of the laws of ground see Litland and Haderlie (manuscript)).

Furthermore, the characterization of metaphysically explanatory relations besides grounding that allow for empty-base explanations could be improved (and it might be interesting to consider whether alternative notions of grounding such as Fine's (2012) natural and normative varieties allow for their own varieties of zero-grounding). Speaking of varieties: First, Muñoz's (2020) intriguing use of the combination of grounding contingentism and zero-ground deserves more attention; second, I have argued that Litland's (2017) argument concerning the grounds of ground likely generalizes to other explanatory notions; it would be interesting to further investigate whether we should embrace the generalized conclusion.

Finally, more realistic scientific examples of causation and laws of nature could be examined for potential applications of empty-base explanation. Most prominently, the suggestion by Hicks and Wilson (2021) based on Albert's and Loewer's mentaculus (cf. Loewer forthcoming), according to which the first event is empty-base explained by chancy law of nature, requires more detailed attention. Another starting point here could be the discussion in McKenzie (2017). Relatedly, cosmological arguments and considerations concerning the PSR could be reevaluated with the notion of empty-base explanation in hand, and for concrete proposals for explanatorily brute truths, the merits of alternative theories according to which these truths are empty-base explained could be investigated.

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# Index

'just is'-statement 163, 174 abduction. See inference to the best explanation alethic essence 115, 117, 120 anything at all. See Why is there anything at all? asymmetry of grounding 76 autonomy. See explanatory autonomy axiarchism 71 because-sentence 4, 11, 18, 20–21, 26–27, 52, 133, 145, 167, 176 beginning of the universe 47, 51, 62–63, 137 Boltzmannian statistics 51, 126, 135–136, 142 Carnapian explication 162, 172 causa sui 159 causal circles 131 causal explanation. See causation causal powers 135, 139 causation 2, 6, 12, 14, 17, 21, 64, 125, 185 causation ex nihilo 6, 64, 123, 125, 137, 184 causation, interventionist theory of 137 causation, irreflexivity of 146 causation, mark transmission account of 136 chains of reasons why 33 circular argument 150 collective essence 118 colour 118 communicative act of explaining why 36 conceivability 179 conceptual analysis 162, 172 conceptual engineering 9, 54, 65, 142 conditional with empty antecedent 65 contrastivism 79 cosmological arguments 2, 60, 185 determinate-determinable 80, 95 determination constraint 73, 92 disjunction 30, 38, 41, 75, 98, 102, 123 dispositions 128, 135, 139

empiricism 179 empty conjunction 42, 48, 55, 138 empty set 1–2, 41 empty-base causal explanation 125 empty-base explanation by law of nature 184 empty-base explanation of logical truths 184 empty-base self-explanation 150 epistemic reasons 151 epistemic value 4, 37, 55, 59 epistemology of empty-base explanation 7, 161, 184 epistemology of essence 116 epistemology of grounding 167 epistemology of identity 172 essence 1, 4–5, 22, 40, 45, 50, 68, 70, 72, 76, 81, 97, 101, 113, 132, 135, 139, 171 essence, direction of 120 essence, individual vs. generic 116 essence, localization of 116 essence of logical operators 120 essential conditional 114 essential dependence 171 essentialist explanation 113 existence of anything at all. See Why is there anything at all? explanation as a communicative act 10, 27, 39 explanation by constraint 71, 82 explanation by essence 97 explanation by law of nature 136 explanation by reductive truthmaking 109 explanation by status 4–5, 8, 30, 35, 49, 59, 63, 70, 97, 150, 184 explanation of causal facts 128, 133 explanation of laws of nature 62, 133 explanation of logical theorems 64, 75, 97 explanation why without reasons why 9, 61, 68, 141, 150 explanation, 'wh-' 10 explanation, complete vs. incomplete 73 explanation, contrastive 16, 19, 31 explanation, inclusive vs. restrictive sense 53, 146 explanation, partial vs. full 13 explanation, relata of 15 explanation, relational sense of 14, 35

downwards explanatory relation 141 dualism concerning the mental 178

explanation, senses of 10 explanation, structural features 33 explanation, theories of 17 explanation, vehicle and content senses 10 explanations by status 100 explanatory argument 4, 39, 48 explanatory autonomy 54, 57, 60, 101, 144 explanatory base 17 explanatory bruteness 54, 57, 60, 185 explanatory calculus 49, 130, 161 explanatory circle 77 explanatory dependence 145 explanatory link 2, 12, 17–19, 22, 24, 30, 37, 45–46, 55 explanatory principles 1–2, 18, 48, 136–137, 143, 156, 161, 184 explanatory regress 79, 88 explanatory self-subsumption 7, 144, 156 explanatory sources 17 explanatory value 75, 88, 101 externality of causation 128 extraordinary ground 103

final explanation. See ultimate explanation fundamentality 143

generalized identity 173 God 1, 30, 41, 60, 78, 143, 159 ground of normative facts 89 grounding 2, 4–5, 7, 12, 14, 21, 64, 135 grounding as a relation 68 grounding circles 131 grounding contingentism 184 grounding explanation. See grounding grounding fact 18 grounding necessitarianism 62 grounding of grounding facts 6, 33, 49, 59, 61, 127, 156, 184 grounding, factive vs. non-factive 49 grounding, irreflexivity of 146 grounding-essence link 118 grounds of identities 63

harmony of explanatory notions 83 Humean supervenience 149 Humeanism about laws of nature 6, 133, 148 identity 163 ill-founded propositions 154 inference to the best explanation 7, 9, 60, 64, 70, 162 inference-ticket conception of natural laws 32 infinite regress 1, 61, 75, 95 intelligibility of zero-grounding 65 intuition 176 intuitionistic conditional 63 irreducibly plural grounding 155 just because 36, 53, 85, 100, 109, 137 law of metaphysics 2, 4, 7, 40, 45, 50, 68, 70, 72, 81, 113, 171 law of nature 2, 6, 24, 40, 45, 50, 62, 70, 72, 81, 128, 135, 139, 185 laws of metaphysics 1, 138 laws of nature 1, 4, 17, 127, 162 levels of reasons why 24 logic of ground 75, 98, 102–103, 134, 149 logic of 'because' 98 logical theorems. See logical truth logical truth 1, 5, 97

mathematical fact 63 mentaculus 51, 185 metaphor of explanation as a machine 33, 42, 45, 127, 129 metaphysical causation 123, 126, 140 metaphysical explanation 12 metaphysical law 5, 142 metaphysical specification 123 modal dependence 86 modal epistemology 163 modality 1, 4, 8, 50, 63, 70, 72, 86, 95, 162

necessitarian explanation 83 necessitarianism 144, 158 necessitism 7 necessity. See modality 4 negative existential fact 63 negative existential facts 161 non-classical logic 99 non-factive grounding 144 non-well-founded mereology 154

Index 195

normative covering law explanation 89 normative fact 30 null-explanation 91

operator view 16, 41, 64, 173

phenomenal consciousness 119 philosophical theology 159 physicalism concerning the mental 119, 178 powers 128 principle of sufficient reason 2, 7, 61, 144, 158, 163, 185 priority relation 21, 25, 46–47, 55, 109 probabilistic law of nature 51 probability 5, 50, 58, 63, 71, 90, 135–136 profound explanation 7 proposition 103 puzzles of ground 79, 146

rationalism 158, 179 reason why 2, 4, 18–20, 23, 26–27, 145 reductive theories of modality 76 reference magnetism 29 relations, multigrade and variably polyadic 69, 184 revelation of essence 119 rule-circular justification 7, 151 Russellian monism 119

second-order reasons why 24, 26, 29 self-causation 147 self-evidence 144 self-explanation 6, 9, 35, 64, 143, 184 self-grounding 147 self-reference 35, 153 sentential quantification 138 speech act of explanation 4 square brackets 15

status problem 59, 133 strict conditionals 168 structural properties of explanation 83, 86, 109, 115, 120, 131, 147, 170, 173 surprise 58

theoretical virtues 172–173 tripartite structure of explanation 2, 4, 10, 17, 27, 37, 147, 157 truthmaker semantics 42 truthmaking 5, 42, 97, 103, 105 truth-teller 154

ultimate explanation 1, 4, 8, 33, 37, 59, 61, 76, 78, 84, 150, 184 understanding why 1, 4, 12, 25, 36, 55, 60, 88, 153 ungrounded 180 unificationist theory of metaphysical explanation 71 universal generalization 5, 63, 77, 91 upwards explanatory relation 140

value of explanation 60 varieties of grounding 123, 184

well-foundedness 33, 61 What causes instances of causation? 128 Why is there anything at all? 2, 5, 7–8, 70, 90, 127, 135, 142, 163 why-question 1, 4, 7, 18–19, 26–27, 51, 176

zero-causation 125, 134, 184 zero-grounding 3–4, 5, 6, 7, 36, 38, 45, 50, 55, 59, 61, 63, 65, 68, 85, 97, 100, 115, 120, 141, 153, 161, 184