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Grounding and Supplementation

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Abstract

Partial grounding is often thought to be formally analogous to proper parthood in certain ways. Both relations are typically understood to be asymmetric (and hence irreflexive) and transitive, and as such, are thought to be strict partial orders. But how far does this analogy extend? Proper parthood is often said to obey the weak supplementation principle. There is reason to wonder whether partial grounding, or, more precisely, proper partial grounding, obeys a ground-theoretic version of this principle. In what follows, I argue that it does not. The cases that cause problems for the supplementation principle for grounding also serve as counterexamples to another principle, minimality, defended by Paul Audi.

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Notes

  1. While I take the fundamental notion of grounding to be expressed by a predicate, and not an operator, what is said below will be of interest to those who work in an operator framework as well. See Correia (2010: 253–54) for a discussion of the difference between operationalist and predicationalist conceptions of grounding. It will also be of interest to those who work in a predicationalist framework but think that grounding may hold between things other than facts. See, for example, Schaffer (2009: 375–76).

  2. See, for example, Witmer et al. (2005: 332), Trogdon (2009: 128, 2013a), deRosset (2010: 91), Rosen (2010: 118), Bennett (2011: 36), Correia (2011: 3), Audi (2012b: 697), Fine (2012b: 1), Raven (2012: 690–91), and Bliss (2014: 147). It should be noted here that this claim has been challenged. See, for example, Schaffer (2010a) and Leuenberger (2014).

  3. I allow a single object to be assigned to any plural term, and assume that, for any \(\Gamma \), there is an x such that x is among \(\Gamma \). That is, I assume that there are no empty pluralities.

  4. See Rosen (2010: 115), Audi (2012b: 698), Fine (2012a: 50), and Raven (2013: 194) for endorsements of this definition.

  5. See Audi (2012b: 698) and Fine (2012a: 50, b: 3–5) for more discussion on the difference between full and partial grounding. I have Fine’s notion of a strict ground in mind, rather than weak ground. See Fine (2012a: 51–53, b: 3–7).

  6. In what follows, I allow singular as well as plural terms to fill the second argument places of each of the grounding predicates. Similarly, I allow singular as well as plural terms to flank the ‘is/are among’ predicate. I also allow the construction of complex plural terms from simpler singular or plural terms via lists. So, for example, ‘x, y’, ‘\(x, \Gamma \)’, and ‘\(\Gamma , \Delta \)’ are all plural terms.

  7. For explicit reference to the similarity of proper parthood and grounding in this respect, see Trogdon (2013b: 106). For characterizations of partial grounding as asymmetric, irreflexive, and transitive, or as a strict partial order, see Cameron (2008: 3), Rosen (2010: 115–16), Schaffer (2010b: 37), and Raven (2012: 689, 2013). Correia (2010: 262, 2011: 3–4), Fine (2010: 100), and Schnieder (2011: 451) provide inference schemas that characterize the behavior of grounding in an operator framework in an analogous way.

  8. For endorsements of this principle, see Simons (1987: 28 and 116), Olson (2006: 743), Uzquiano (2006: 431), Sider (2007: 69–70), Effingham and Robson (2007: 635), Varzi (2008: 110–11, 2009: 599), Bohn (2009: fn. 3), McDaniel (2009: fn. 48), and Bynoe (2010: fn. 8). van Inwagen (1990: 39) and Lewis (1991: 74) actually endorse Uniqueness of Composition, but (WSP) follows from this principle along with the standard definition of composition.

  9. I follow Fine (2012a: 67–68) in my use of the term-forming operator ‘\(\Lambda \)’.

  10. See Rosen (2010: 126), Audi (2012b: 686 and 689), and Schaffer (2012: 126–27) for endorsements of this principle. Schnieder (2006: 32–33) may endorse a version of this principle, formulated in terms of metaphysical explanation. It should be noted that Wilson (2012) discusses reasons to think this principle is false. I will not discuss the details of her argument here. Instead, I direct the reader who is sympathetic to Wilson’s criticism to the next section, wherein I develop a counterexample to (WSG) that does not rely on (DG).

  11. Adapted from Correia (2010: 268, 2011: 5), Rosen (2010: 117), Schnieder (2011: 449), and Fine (2012a: 58).

  12. The parenthetical restriction is necessary, since, if, for example, \( [p] = [p \; \& \; q]\), then (&I) would yield the result that \( [p \; \& \; q]\) is partially grounded by \( [p \; \& \; q]\), or, equivalently, [p] is partially grounded by [p]. This would violate the irreflexivity of grounding. See Correia (2010: 268) for an endorsement of an operationalist version of this restriction.

  13. Given a certain assumption, this principle may best be understood to apply only when \(\varphi \) and \(\psi \) are atomic sentences. Suppose that \(p,\,q,\,r\), and s, and that \([p],\,[q],\,[r]\), and [s] are pairwise distinct. Given this, it follows that \( [p \; \& \; q],\,[r \; \& \; s]\), and \( [(p \; \& \; q) \; \& \; (r \; \& \; s)]\) are also pairwise distinct. By (&I), \( [(p \; \& \; q) \; \& \; (r \; \& \; s)]\) is fully grounded by \( [p \; \& \; q], [r \; \& \; s]\), and so is partially grounded by each of them. And, while this is not guaranteed by (&I), if one thinks that \( [(p \; \& \; q) \; \& \; (r \; \& \; s)]\) is also partially grounded by \( [q \; \& \; r]\), a potential problem arises for (&N). This is because \( [q \; \& \; r]\) cannot be identical to, nor partially ground, either \( [p \; \& \; q]\) or \( [r \; \& \; s]\). Given that \([p],\,[q],\,[r]\), and [s] are pairwise distinct, \( [q \; \& \; r]\) is not identical to either \( [p \; \& \; q]\) or \( [r \; \& \; s]\). And it is usually thought that, for x to be partially grounded by \(y,\,y\) must be explanatorily relevant to x (See, for example, Correia 2010: 263, Schnieder 2011: 450, Audi 2012b: 693 and 699, Fine 2012a: 56, b: 2, Raven 2013: 198, Dasgupta 2014: 4). So, as long as [r] is not explanatorily relevant to \( [p \; \& \; q]\), and [q] is not explanatorily relevant to \( [r \; \& \; s],\,[q \; \& \; r]\) does not partially ground either \( [p \; \& \; q]\) or \( [r \; \& \; s]\). Thus, given the assumption that \( [p \; \& \; q], [r \; \& \; s]\) is partially grounded by \( [q \; \& \; r]\), there is a violation of (&N). Thanks to Daniel Nolan for this example.

  14. This principle, or an operationalist version of it, is endorsed or follows from principles endorsed in Correia (2010: 262, 2011: 3), Fine (2010: 100, 2012a: 56, 2012b: 5), Rosen (2010: 116), Schnieder (2011: 451), and Raven (2013: 198).

  15. Tahko (2013) provides reason to think that truth-grounding is not transitive, but provides a possible way out of saying that grounding is not transitive, by denying that truth-grounding is grounding.

  16. See, for example, Litland (2013) and Raven (2013: 198–200). While Javier-Castellanos (2014) does not argue directly against the effectiveness of Schaffer’s counterexamples, he argues that Schaffer’s alternative—his contrastive analog of transitivity—is susceptible to counterexamples as well.

  17. Gilmore (2009) argues that parthood is a four-place relation, and formulates a four-place analog of (WSP). Kleinschmidt (2011: 268–271) independently formulates such a principle. Either of these would be a good starting point for formulating a four-place analog of (WSG).

  18. It’s worth noting that one could get this result in another way. In particular, one could conclude that [grass is green & grass is colored] is not fully grounded by [grass is colored] on the basis of the fact that the proposition that grass is colored does not metaphysically necessitate the proposition that grass is green & grass is colored. As noted in the introduction, many are of the mind that full grounding implies metaphysical necessitation.

  19. Regarding possibility (d): while I have supposed that [grass is green] is fundamental, I have done so only for the simplicity it afforded in the presentation of the case. I recognize that being green is probably not fundamental. Furthermore, I want the case to be conceivably regarded as a counterexample to (WSG) even by those who deny that there are any fundamental facts. Thus I take this possibility just as seriously as the other three.

  20. Thanks to Cody Gilmore and an anonymous referee for bringing this counterexample to my attention.

  21. See Correia (2010: 268), Schnieder (2011: 449), and Fine (2012a: 63) for endorsements of this principle or of an operationalist version of it.

  22. Note that the claim that grounding implies metaphysical necessitation is of no help here either, since \(\sim \sim \!p\) metaphysically necessitates p (See fn. 18).

  23. It is hard to hide the fact that this argument relies on the claim that partial grounding is irreflexive and asymmetric. And there are certainly some who would deny this. Jenkins (2011), for example, argues that grounding is not irreflexive. I will sidestep this issue nonetheless. After all, anyone who denies that (proper) partial grounding is irreflexive and asymmetric will presumably have no interest in the argument against (WSG) anyway, since she would not see an interesting analogy between proper parthood and (proper) partial grounding in the first place. After all, on her view, only the former is irreflexive and asymmetric, and so only the former would be a strict partial order.

  24. As before, taking possibility (d) seriously ensures that the assumption made earlier, that [p] is fundamental, is innocuous.

  25. For a good discussion of fine- versus coarse-grained conceptions of facts, see Correia and Schnieder (2012).

  26. I take this principle from Rosen (2010: 116). See Correia (2010: 262, 2011: 3) for operationalist versions of this principle. The operationalist version is derivable in Fine’s system PLG as well. See Fine (2012b: 6).

  27. As evidence that this last inference is sound, consider the following. It is standard in plural logic to define identity in terms of the ‘is/are among’ predicate. In particular,

    Identity \(\Gamma = \Delta =_{df}\) (i) \(\Gamma \) are among \(\Delta \) and (ii) \(\Delta \) are among \(\Gamma \) (See McKay 2006: 122, Oliver and Smiley 2013: 109).

    In addition, it is plausible that

    (AX 1) For any \(\Gamma \) and \(\Delta ,\,\Gamma \) are among \(\Delta \) iff, for any x, if x is among \(\Gamma \) then x is among \(\Delta \) (See McKay 2006: 121).

    I take it as obvious that [grass is green] is among [grass is green], [grass is green]. But it is also the case that [grass is green], [grass is green] are among [grass is green]. Because [grass is green] is the only thing among [grass is green], [grass is green], for any x, if x is among [grass is green], [grass is green], then x is among [grass is green]. So it is (AX 1) that guarantees that [grass is green], [grass is green] are among [grass is green]. By the above definition of identity, then, [grass is green] = [grass is green], [grass is green]. Thus, the footnoted inference is just an instance of good old-fashioned identity elimination.

  28. Similar remarks apply to a non-monotonicity principle governing partial grounding.

    Partial Non-Monotonicity If something is partially grounded by something (else), then it is not the case that, for any \(x,\,y\), and \(\Gamma \), if x is partially grounded by \(\Gamma \), then x is partially grounded by \(y, \Gamma \).

    Such a principle is just as plausible as full non-monotonicity. After all, while [the shirt is red and expensive] is partially grounded by [the shirt is maroon], it is not partially grounded by [the shirt is maroon], [the shirt is cotton]. Further, this principle, like full non-monotonicity, is vacuously true when there exist only fundamental facts. And it also results from other formal features of grounding, given the existence of at least one non-fundamental fact.

  29. I’d like to thank Cody Gilmore for helpful suggestions and for reading numerous drafts of this paper. I’d also like to thank an audience at the 2014 Australasian Association of Philosophy Conference, an audience at a workshop in the Philosophy Department at the University of California, Davis, and an audience at the 2015 Central Division Meeting of the American Philosophical Association for helpful comments and suggestions. I would like to give special thanks to Aldo Antonelli, David Copp, Li Kang, Daniel Nolan, Gabriel Rabin, and Jonathan Schaffer.

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Dixon, T.S. Grounding and Supplementation. Erkenn 81, 375–389 (2016). https://doi.org/10.1007/s10670-015-9744-z

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