Abstract
Those who wish to claim that all facts about grounding are themselves grounded (“the meta-grounding thesis”) must defend against the charge that such a claim leads to infinite regress and violates the well-foundedness of ground. In this paper, we defend. First, we explore three distinct but related notions of “well-founded”, which are often conflated, and three corresponding notions of infinite regress. We explore the entailment relations between these notions. We conclude that the meta-grounding thesis need not lead to tension with any of the three notions of “well-founded”. Finally, we explore the details of and motivations for further conditions on ground that one might add to generate a conflict between the meta-grounding thesis and a well-founded constraint. We explore these topics by developing and utilizing a formal framework based on the notion of a grounding structure.
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Notes
A third possibility is that grounding facts aren’t the type of facts that can enter into grounding relations - they are somehow “outside” the hierarchy of grounding. On this picture, these facts are neither fundamental nor non-fundamental. Attempts to apply the notion of fundamentality to them, and the search for their metaphysical ground, involve a category mistake. Dasgupta [8] for such a view.
We leave open the possibility that the three examples involve different relations. There may be a genus - grounding - of which the relations in the three examples are species. Or there may be no such genus. Either way, we take the examples to involve conceptually, metaphysically, and formally similar relations that share enough important features that it makes sense to study them together. They all involve a metaphysical “in virtue of” claim. We shall, for the most part, speak as if there is one grounding relation (which we call ‘grounding’). But no substantive claims turn on this manner of speaking. In fact, we are sympathetic to the idea that there are many grounding relations, each governed by different principles. (Cf. Section 7 for more on this idea).
Schaffer [18].
Of course, some grounding facts might be grounded, others ungrounded. We do not consider this option, but that choice has negligible import here.
There is a meta-grounding thesis that says that all grounding facts are merely partly grounded. We don’t consider such a thesis here, mainly because we find it far less theoretically interesting.
The “grounding facts” we target are those facts in which the grounding relation is the “main connective”. Perhaps all such facts will be expressible using a sentence of the form ‘p grounds q’, where ‘p’ and ‘q’ refer to facts. These facts aren’t the same as the facts that contain the grounding relation. The fact that reading about grounding is fun contains the grounding relation, but isn’t a “grounding fact” in our sense. The meta-grounding thesis does not entail that the fact that reading about grounding is fun is grounded.
This result does require some minimal claims about the individuation of facts, and specifically about the relationship between sentences that express facts and the facts themselves. Without any guidelines whatsoever, it’s unclear that the facts \(p, g_{1}, g_{2}, g_{3}, \dots \) are distinct. Two assumptions enable the proof that these facts are distinct. First assumption: if fact P concerns grounding, and fact Q does not, then P and Q are not the same fact. Second assumption: if P and Q are both facts about grounding, then they are the same fact if and only if they relate the same two relata via the grounding relation. Both assumptions are tremendously plausible. These assumptions imply that, for each i, g i and g i+1 are not the same fact. The proof proceeds by induction.
We admit that even the claim that there are infinitely many mathematical facts requires some minimal assumptions about the individuation of facts. We’re willing to commit ourselves to the claim that any reasonable individuation of mathematical facts will yield the result that there are infinitely many mathematical facts (but not necessarily infinitely many fundamental, or ungrounded, facts).
Cameron [6], Schaffer [18–20], and Sider [21] all agree that non-fundamental, derivative, grounded facts are an “ontological free lunch”. The ontological commitments of a theory should be determined by measuring what the theory takes as fundamental, or ungrounded. The non-fundamental, derivative, or grounded posits of the theory incur no ontological cost. For a dissenting view, cf. Audi [2].
Cameron [5], Nolan [15], and Schaffer [17] for more on this topic. To add some fuel to the fire, we offer an example of a world whose grounds appear not to be well-founded.
Let the world Escher be a near-duplicate of the actual world, in so far as this is possible given the rest of the construction. Escher has microscopic fundamental particles, called fundamentons, facts about which ground the macroscopic facts about Escher. However, every fundamenton at Escher is itself a miniature replica of Escher. This miniature replica also contains fundamentons, which are themselves miniature miniature replicas. The 2 nd level of replicas is grounded in its fundamentons, which are themselves replicas of Escher. And so on ad inifinitum. You get the idea.
We see nothing wrong with the grounding structure of Escher. We’re liberal about modality, and see no reason to rule out a world like Escher as metaphysically impossible.
Importantly, we assume that the < part of a grounding structure 〈A, < 〉 includes all the facts about grounding, not just the facts about grounding relations between members of A. This is slightly odd, but makes the definitions simpler and exposition clearer.
This definition requires us to divide up the grounding facts from the non-grounding facts. Cf. footnote 7.
This would be structurally similar to what Hans Herzberger called “groundlessness”, which he describes as being like “the bureaucratic regress in which each clerk endlessly refers you to the next to settle your accounts” [14].
A binary relation R is well-founded on a domain D iff every non-empty subset of D has a minimal element with respect to R. Assuming the axiom of dependent choice (thanks to an anonymous referee for pointing out this need), this is equivalent to the descending chain condition on partially ordered sets, which demands that there be no infinite descending chains. A set S is a well-founded set iff the set membership relation is a well-founded relation on the transitive closure of S.
Bennett [3] claims that the ground of g 0, i.e. p < q, is p itself. She claims that, in the general case, when A grounds B, A also grounds the fact that A grounds B, and the fact that A grounds the fact that A grounds B, etc. Such a structure provides another example of a grounding structure that satisfies the meta-grounding thesis without generating infinite descending chains.
The grounding relation on a non-finitely grounded structure will fail to be well-founded in the relation- or set- theoretic sense. The grounding relation on that structure will not be a well-founded relation precisely because it contains infinite chains. On such a structure, grounding will not provide a well-founded partial ordering. Schaffer [18]: 376 demands that grounding provide such an ordering. Thus, non finitely-grounded structures may run afoul of the Schafferian well-founded constraint.
We don’t think this point should be that controversial. Cameron [5]: 4 certainly appears to agree. He writes, “The intuition under discussion [well-foundedness] does not demand that we should be able to reach the ultimate ontological ground in a finite number of steps... It demands only that there is a fundamental ground. This is compatible with a situation where no entity depends directly on that fundamental ground, but every entity has a chain of dependence with an infinite number of steps taking that entity to the ultimate ground.” Not every philosopher is as careful as Cameron in discussing what the well-founded constraint amounts to. It’s common to hear or read the constraint glossed as a ban on infinite descending chains of ground.
One need not believe in infinitely disjunctive facts in order to generate structures that have a foundation but are not bounded from below. We choose the case of infinite disjunction to provide a concrete illustration of a formal claim.
In fairness to Schaffer, he says only that well-foundedness is imposed by requiring that all priority chains terminate. One can reasonably interpret Schaffer as endorsing only the mild conditional that if all priority chains terminate, then grounding is well-founded, rather than claiming that well-foundedness amounts to all priority chains terminating. We agree with the conditional. Theorem 4 and Corollary 8 demonstrate that any structure that is finitely grounded, i.e. in which all priority chains terminate, is also bounded from below and has a foundation. It is thus well-founded in all three senses.
Thanks are to due here to an anonymous referee.
The type of foundation defined here is a foundation that fully grounds a set of facts. On could define the notion of a partial foundation - a set of perfectly fundamental facts that at least partly grounds every member of the facts founded thereby. Such a notion might serve some theoretical purposes. However, we think it clear that a partial foundation is not what anyone who favors a well-founded constraint on ground is after. We do not investigate partial foundations here.
Bennett [3]: 36-7 discusses something quite close to the LC-condition. She motivates the principle in much the same way we do. However, her discussion is limited to deflecting an LC-motivated objection to her super-internality principle, which says that if p < q then p<[p < q]. Our discussion of LC is independent of super-internality.
Bennett [4] for a good overview.
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Acknowledgments
Thanks for helpful comments and discussion to Karen Bennett, Jonathan Cusbert, Louis deRosset, Kit Fine, Holger Thiel, Daniel Nolan and participants in Karen Bennett’s metaphysics seminar at NYU in autumn 2010. Thanks also to audiences at the Conference on the Philosophy of Kit Fine, Sinaia, Romania, May 2012, the 2012 meeting of the Australasian Association of Philosophy in Dunedin, New Zealand, and the “Fundamentality and Metaphysical Infinitism” Workshop at the University of Helsinki, June 2014.
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Rabin, G.O., Rabern, B. Well Founding Grounding Grounding. J Philos Logic 45, 349–379 (2016). https://doi.org/10.1007/s10992-015-9376-4
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DOI: https://doi.org/10.1007/s10992-015-9376-4