The thesis that every grounding fact is grounded gives rise to an infinite series of grounding facts. According to Frugé (“Janus-Faced Grounding”), this series of grounds of ground amounts to a vicious regress. This paper (1) responds to Frugé’s argument, (2) argues for a more plausible motivation for the vicious regress, and then (3) deploys a Bolzanian regress argument against this to defend the innocence of the series of grounds of ground.
Theories according to which every grounding fact is grounded give
rise to the following kind of infinite series of grounding facts (let
‘
Against these accounts, Frugé (2023) attempts to show that the resulting infinite series of grounds of grounds constitute vicious regresses by arguing that they involve a kind of metaphysical dependence that allows to apply an analogue of Schaffer’s (2010, 2016) consideration for the well-foundedness of grounding to the infinite series of grounds of ground. In what follows, I will develop Frugé’s argument, argue against it, and draw a general lesson about the well-foundedness of metaphysical explanation from this discussion.
This is the plan: Section 1 presents Frugé’s argument and argues that it fails. Section 2 discusses a related argument by (1) considerations akin to Frugé’s counterfactual considerations, (2) introducing the inclusive sense of ‘explains’ and showing how each element in the series of grounds of ground is explained by its successor in this sense, and (3) arguing that the metaphor of explanation as a machine has a natural reading given which Schaffer’s consideration for the well-foundedness of grounding applies mutatis mutandis to the infinite series of grounds of ground.2 Against these considerations, section 3 fields a regress argument by Bolzano (2014b, para. 199) to argue for the innocence of the series of grounds of ground.3
1 Frugé’s Argument
In a nutshell, Frugé
(2023)
argues for the viciousness of the series of grounds of ground by arguing
(1) that what he calls a kind of “genuine dependence” holds between
Proponents of the well-foundedness of grounding reject the existence of infinite regresses (i.e., downwardly non-terminating grounding chains) and circles of grounding, at least as long as the involved facts (or propositions, if you prefer) are not appropriately tethered to the fundamental. An important metaphorical consideration that motivates well-foundedness stems from Schaffer, according to whom grounding regresses are objectionable because in them, being would be “infinitely deferred [and] never achieved” (Schaffer 2010, 62):
Grounding must be well-founded because a grounded entity inherits its reality from its grounds, and where there is inheritance there must be a source. […] [S]omething cannot be real merely by having a limitless sequence of ancestors, each claiming reality from its parents. There must actually be a source of reality somewhere. (Schaffer 2016, 95)
I will call this ‘Schaffer’s consideration’.4 Before we continue, note that infinite series of grounds of ground are not downwardly non-terminating grounding chains (and thus well-foundedness of grounding is not sufficient to argue against them): no element of these series is grounded in the next element. Accordingly, proponents of accounts according to which every grounding fact is grounded have insisted that the resulting infinite series are unproblematic and that we can accept their accounts while remaining neutral on whether grounding is well-founded.
Against this, Frugé
(2023, sec.
2) argues that Schaffer’s consideration can be
extended to the infinite series of grounds of ground once we realize
that
Why is connection dependence a genuine form of dependence? Suppose the following is the case:
grounds , where grounds that grounds . Then, doesn’t only depend on . Instead, it also depends on , because only generates given . If there were no to put grounds in place, then even if there were there would be no , since wouldn’t generate because it wouldn’t be the case that grounds . For example, if a collection of particles ground the composite whole of those particles only via a composition operation grounding this grounding fact, then if, perhaps counterpossibly, there were no composition operation then those particles would not ground that whole, because there would be no composition. (Frugé 2023, 976–977)
Having thus argued that
Similar reasoning applies at each step in the stepwise path. If
grounds that grounds grounds , then if there were no , then even if there were and , then there would be no because would not generate that grounds , and so would not generate . And so on for each ground in the stepwise grounding path [i.e., in our terminology, the ]. Thus, connection dependence is a genuine form of dependence. needs in order to come about, and it also needs in order to come about, and so on down the stepwise path. So metaphysically requires each ground in its stepwise grounding path. (Frugé 2023, 977)5
Frugé then argues that (an analogue of) Schaffer’s consideration applies:
As Jonathan Schaffer says in the context of defending well-foundedness, if grounding did not terminate in an ungrounded ground, then “being would be infinitely deferred, never achieved” (2010: 62). But given connection dependence, then the same can be said for an infinite stepwise path of ever more grounding of grounding facts [i.e., series of ground of ground]. Even if ground were well-founded, if the grounding of grounding facts had no end, then ‘being would be infinitely deferred, never achieved’, since there would be no point at which it’s ultimately settled that the grounded is generated. It would always need a further ground of a grounding fact. Therefore, if one thinks that violating the well-foundedness of ground is vicious, then one should also think that the fact regress is vicious—given that the grounded depends not just on its grounds but also on the grounds of its grounding facts, and, more generally, on the grounds in its entire stepwise path of grounding facts. (Frugé 2023, 978)
Now, I believe Frugé’s argument misses its mark: Even if we set all
worries about his counterfactual argument aside and simply grant that
Since the structure of metaphysical priority that would seem to be
required for Schaffer’s consideration to apply is indeed that of an
infinitely descending (and non-tethered) chain of dependence, one would
have expected Frugé to argue that the elements of the series of ground
of ground (i.e., those at the beginning of this paper) stand in a
relation of metaphysical priority, but he does not do so. To an extent,
this problem is perhaps obfuscated by Frugé’s talk of “
Additionally, Frugé seems to take issue with there seemingly being an
infinity of
While I thus conclude that Frugé’s argument fails, I will now show how one might attempt to substantiate in a different way the idea that series of ground of ground involve a relation of productive metaphysical priority that allows an analogue of Schaffer’s consideration to apply and show the regress to be vicious.
2 Three Better Considerations?
Let us discuss three considerations in favor of the thesis that the elements of the series of ground of ground stand in a relation of metaphysical priority (grounding or other) to which Schaffer’s consideration applies. While I ultimately reject these considerations, I believe that they (or something close enough) are plausibly what motivates uneasiness about the series of grounds of ground.
2.1 Counterfactual Considerations
It may be tempting to think that a case can be made for the claim
that each element of the series of grounds of ground counterfactually
depends on its successor. For example, one might think that had
Now, rather than getting bogged down in thinking about counterfactuals further, let us consider two further attempts to argue that there is a relation of productive metaphysical priority (to which Schaffer’s consideration applies) that holds between the elements of the series of grounds of ground—staying neutral for now on the question whether this alleged priority relation would be grounding or not; let us call it ‘gog-priority’. The first attempt stems from an inclusive sense of ‘explains’ and the second from a particular reading of the metaphor of the machine.
2.2 The Inclusive Sense of ‘Explains’
I assume that explanation why has the following tripartite structure (see, for example, Schaffer 2017):
Base. A set of reasons why the explanandum obtains, e.g., causes or grounds.
Link. An explanatory connection between the reasons in the base and the explanandum; these could either be instances of explanatory relations such as causation or grounding (call these ‘type 1’) or explanatory generalizations such as laws of nature or metaphysics, (explanatory) schemata, or (explanatory) inference rules (call these ‘type 2’).
Explanandum. That which is being explained.
For an example, consider an explanation why a certain rose is red (Explanandum) in terms of its being scarlet (a ground that constitutes the explanation’s Base) and the grounding fact of the rose’s being scarlet grounding it’s being red (or a metaphysical principle that states that instantiations of determinates ground instantiations of corresponding determinables).
In a restrictive sense, only the elements of the Base explain the Explanandum—the rose’s being red is explained by its being scarlet, while the grounding claim or metaphysical principle plays a different (for example, explanation-backing) role. It is this restrictive sense that corresponds to ‘because’, which connects a sentence that expresses a reason why with a sentence that expresses an Explanandum (cf. Schnieder 2010; and Skow 2016).
But there also exists another sense of ‘explains’, in which links also (partially) explain their explananda. In this sense, the rose’s being red is explained by it’s being scarlet and the corresponding grounding fact of metaphysical principle together: Base and Link(s) together explaininclusive the Explanandum. This sense is, for example, operative in how the DN-model of explanation is often framed: Boundary conditions and laws (or lawlike generalizations) together form the explanans, and the explanans (or what is contained therein) explains the Explanandum (in the inclusive sense).
Equipped with this inclusive sense of ‘explains’ and assuming that
instances of grounding correspond to instances of explanation, we can
observe that the elements of the series of grounds of ground are
(partially) explainedinclusive by their successor, which is a
type 1 Link of a grounding explanation of its
precursor. For example,
Hence, if there are series of grounds of ground, then the inclusive ‘explains’ allows for infinitely descending chains. Moreover, ‘explainsinclusive’ would presumably not be well-founded in the sense that any explainedinclusive fact is ultimately explainedinclusive by unexplainedinclusive facts.8 For consider a series of grounds of ground: Perhaps all elements could have a fundamental ground outside the series, but then the involved grounding relations give rise to further series (assuming that all instances of grounding are grounded, of course), and so on.9
Now, given the inclusive sense of ‘explains’ and the fact that the explanation in question is a metaphysical one, some metaphysical explanatory relation (i.e., a relation that can be called such in some good sense) holds between the elements of the series of grounds of ground. Together with the previous counterfactual observation, this could lead one to think that the relation in question is a relation of productive metaphysical priority, i.e., gog-priority. Assuming further that all such priority relations are subject to a variant of Schaffer’s consideration, the viciousness of the series of grounds of ground would then follow.
2.3 Metaphor of the Machine
Indeed, I believe that there is something of a sense that Schaffer’s consideration or a variant of it applies to gog-priority (if it applies at all) and that this can be brought out by a particular, yet arguably natural, way of construing the metaphor of grounding (or explanation) as a machine (cf. Litland 2017):
Think of grounding (or explanation) as a machine: Instances of grounding are machines that take inputs (grounds) and use them to generate outputs (groundees). But for a machine to be able to generate something, it either has to exist without having been generated, or it has to be generated first. But this means that the series of grounds of ground corresponds to a series of machines, each generated by a previous machine, and so on ad infinitum. It seems like each machine inherits its reality from a further machine that generates it, and thus its reality is infinitely deferred and never achieved.10
Now, this understanding of the metaphor of the machine is not mandatory. First, it is just not clear why the causal-temporal relation between the machine and its output within the metaphor should have an analogue in a relation of productive metaphysical priority within reality: after all, metaphors break down somewhere, and this might well be where this one does.11
Second, as one of the anonymous referees for this paper has
thankfully pointed out, if we assume explanatory Links only to be of type
2, that is, explanatory inference rules (as, e.g., Litland does) or
certain laws (as, e.g., Schaffer does), rather than instances of
grounding, we should presumably understand the metaphor as involving
these rules or laws as their machines. But since, e.g., in Litland’s
case, roughly speaking, a general rule for grounding introduction is
sufficient to generate all statements of higher order ground, no
hierarchy of ever-descending explanatory machines is required. As the
referee has pointed out, moreover, these accounts can avoid the regress
of inclusive explanation: Roughly, in a case of
Now, while I find this very compelling, my aim in this section was to attempt to come up with possible reasons that could be what motivates uneasiness concerning the series of grounds of ground, and this I believe the above version of the metaphor of the machine achieves even in the light of the previous paragraph.
2.4 Taking Stock
The series of grounds of ground has been considered unproblematic by those committed to it (cf. Bennett 2011; deRosset 2013a; Dasgupta 2014; Litland 2017). For one, it is not an infinitely descending series of grounds and not obviously problematic in any other way. But more importantly, its proponents assume there to be a strong theoretical reason to allow for it: otherwise, it seems there must be at least some ungrounded grounding facts. But together with a principle of purity of the fundamental, this leads to the result that every entity (and other constituent of facts) is fundamental.12
Above, I have developed potential reasons in favor of the claim that the elements of the series of grounds of ground stand in a relation of gog-priority such that an application of Schaffer’s consideration reveals the series to be objectionable. Though I ultimately reject these reasons, I take them to (1) provide a plausible diagnosis for the uneasiness concerning the series of grounds of ground that one occasionally encounters outside of print and which is likely shared by Frugé, and (2) substantiate this uneasiness to a point that is worth further engaging with.
While I have already mentioned some possible objections above, I will now argue that gog-priority runs into a version of Bolzano’s regress.
3 With Bolzano against Gog-Priority
Let us suppose for the sake of argument that grounding is well-founded and that at least part of what reveals this is Schaffer’s consideration. Given these assumptions, let us turn to gog-priority and see whether it is a notion to which an analogue of Schaffer’s consideration applies (given the above considerations that motivated considering gog-priority as a genuine kind of metaphysical priority in the first place).
Now, either gog-priority just is grounding too or it is not: In the former case, insofar as Schaffer’s consideration applies to grounding, so does it to gog-priority, because the latter just is (a subcase of) grounding. In that case, the series of grounds of ground would constitute a vicious regress. Alternatively, gog-priority is a sui generis metaphysical priority relation besides grounding, for which we want to investigate whether Schaffer’s consideration applies or not. Therefore, let us first consider whether gog-dependence could be grounding and then generalize our argument.
3.1 Could Gog-Priority Be Grounding?
I argue that in this case, grounding facts are not fully grounded in
their conventional grounds (the non-gog-priority grounds), and at least
the corresponding higher-order grounding facts (which are gog-priority
grounds) must be added. For example,
On this understanding, both the input of the machine and the machine
(i.e., the grounding fact) ground the output. In a way, the causal
relations that hold within the fiction of the metaphor between the input
and the output as well as the machine and the output stand, on this
view, simply for grounding. But again, within the metaphor, input and
machine can cause the output only together. Therefore, on this
understanding, it would seem that the metaphor suggests that
Now, to simplify, we write ‘
This result (i.e., that
Now, if what I have just said is correct, then we run into a version
of Bolzano’s (2014a, para. 199;
cf. Rusnock
and George 2014) regress:
Now, for our opponent, who set out to avoid an allegedly problematic
infinite series, the situation is already somewhat awkward, but could
they perhaps bite the bullet and declare the plurality of all the
grounds constructed above to be a full ground
One might now consider whether a full ground of
In this subsection, I have argued that given (1) the motivation (from the previous section) for gog-priority being a genuine kind of metaphysical priority to which Schaffer’s consideration applies and (2) the assumption that gog-priority just is grounding, Bolzano’s regress arises. Since Bolzano’s regress must be avoided, (1) is undermined given (2). Next, we will drop assumption (2) and argue that a Bolzanian regress arises (and hence (1) is undermined even if gog-priority is not grounding).15
3.2 Suppose Gog-Priority Is Not Grounding
To round off the argument, assume now that gog-priority is
not grounding but some sui generis kind of
metaphysical priority relation. Suppose
Therefore, it seems our opponent should rather hold that neither only
something’s being grounded in something fundamental nor only
its being gog-posterior to something fundamental can be sufficient for
the thing’s having achieved reality, or (we might say) “having been made
real.” In our example case, this means that while
Now, whatever this relation of real-making would be (perhaps it could just be the disjunction of grounding and gog-priority?), it looks like the opponent of the series of grounds of ground must hold that an analogue of Schaffer’s consideration applies to it, and hence that for something to be real, it must either not be made real by anything or be fully made real by some things that are not made real by anything.
But then it seems like we can run the crux argument with real-making
instead of grounding and thus construct the Bolzanian regress for this
relation of real-making: If
It seems that the reasons our opponent has to believe that
But then our opponent would have to produce a good reason to stop the
metaphor thus understood from applying to real-making as well: Like
grounding, real-making can metaphorically appear as a machine that takes
inputs (the “real-makers”) and puts out real things. But why would the
metaphorical parallel between causation and grounding in the original
(stretched) metaphor then not have its equivalent here? If our opponent
cannot produce a good answer, it would seem that
Again, one might criticize this argument for its dependence on an intuitive assessment of a particular understanding of the metaphor of the machine. But consider the dialectical situation once more: This is the very kind of consideration that, according to my diagnosis in section 2, underlies the idea that the series of grounds of ground is problematic. Thus, at least unless the opponent of the series of grounds of ground comes up with a different argument, they are confronted with the Bolzanian regress whether they understand gog-priority as grounding or as a sui generis relation of metaphysical priority.
4 Conclusion
Let us recapitulate: Section 1 argued against Frugé’s argument for the viciousness of the series of ground of ground: Even if his notion of connection dependence corresponds to a genuine form of metaphysical priority, the resulting structure does not allow for an application of Schaffer’s consideration to the series of ground of ground.
Section 2 discussed three related considerations in favor of the thesis that a well-founded relation of metaphysical priority holds between the elements of the series of ground of ground. These concerned certain counterfactuals, the inclusive sense of ‘explains’, and a natural reading of the metaphor of explanation as a machine. While I ultimately argued that we should not endorse these three considerations, I suggested that we should take them seriously as likely underlying the claim that the series of ground of ground is vicious.
Finally, in section 3 I argued that endorsing the viciousness of the series of ground of ground based on the considerations identified in the previous section runs into a variant of Bolzano’s regress. Hence, we should reject the problematic considerations and can maintain the innocence of the series of ground of ground.
Yannic Kappes
0000-0002-8065-2935
University of Vienna
yannic.kappes@univie.ac.at
Acknowledgements
Many thanks to Philipp Blum, Fabrice Correia, Stephan Leuenberger, Steph Rennick, and two anonymous referees for this journal, as well as the organizers and audience of the 2022 Varieties of Non-Wellfoundedness Workshop in Ligerz for very valuable comments and suggestions for improvements! I also thank the Deutsche Forschungsgemeinschaft for funding work on this paper through the project The Structure of Fundamentality (Grant No. SCHN 1137/5-1).
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