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.
Similar content being viewed by others
Notes
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).
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).
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.
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.
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.
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.
I follow Fine (2012a: 67–68) in my use of the term-forming operator ‘\(\Lambda \)’.
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).
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.
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.
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.
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.
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.
Thanks to Cody Gilmore and an anonymous referee for bringing this counterexample to my attention.
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).
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.
As before, taking possibility (d) seriously ensures that the assumption made earlier, that [p] is fundamental, is innocuous.
For a good discussion of fine- versus coarse-grained conceptions of facts, see Correia and Schnieder (2012).
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,
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.
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.
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.
References
Audi, P. (2012a). A clarification and defense of the notion of grounding. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding: Understanding the structure of reality (pp. 101–121). Cambridge: Cambridge University Press.
Audi, P. (2012b). Grounding: Toward a theory of the in-virtue-of relation. The Journal of Philosophy, 109, 685–711.
Bennett, K. (2011). By our bootstraps. Philosophical Perspectives, 25, 27–41.
Bliss, R. L. (2014). Viciousness and circles of ground. Metaphilosophy, 45, 245–256.
Bohn, E. D. (2009). An argument against the necessity of unrestricted composition. Analysis, 69, 27–31.
Bynoe, W. (2010). Against the compositional view of facts. Australasian Journal of Philosophy, 89, 91–100.
Cameron, R. P. (2008). Turtles all the way down: Regress, priority, and fundamentality in metaphysics. The Philosophical Quarterly, 58, 1–14.
Correia, F. (2011). From grounding to truth-making: Some thoughts. In Philosophical Papers Dedicated to Kevin Mulligan. http://www.philosophie.ch/kevin/festschrift/Correia-paper.
Correia, F. (2010). Grounding and truth-functions. Logique et Analyse, 53, 251–279.
Correia, F., & Schnieder, B. (2012). Grounding: An opinionated introduction. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding: Understanding the structure of reality (pp. 1–36). Cambridge: Cambridge University Press.
Dasgupta, S. (2014). On the plurality of grounds. Philosopher’s Imprint, 14, 1–28.
deRosset, L. (2010). Getting priority straight. Philosophical Studies, 149, 73–97.
Effingham, N., & Robson, J. (2007). A merological challenge to endurantism. Australasian Journal of Philosophy, 85, 633–640.
Fine, K. (2010). Some puzzles of ground. Notre Dame Journal of Formal Logic, 51, 97–118.
Fine, K. (2012a). Guide to ground. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding: Understanding the structure of reality (pp. 37–80). Cambridge: Cambridge University Press.
Fine, K. (2012b). The pure logic of ground. The Review of Symbolic Logic, 5, 1–25.
Gilmore, C. S. (2009). Why parthood might be a four-place relation, and how it behaves if it is. In L. Honnefelder, B. Schick, & E. Runggaldier (Eds.), Unity and time in metaphysics (pp. 83–133). Berlin: de Gruyter.
Javier-Castellanos, A. A. (2014). Some challenges to a contrastive treatment of grounding. Thought, 3, 184–192.
Jenkins, C. S. (2011). Is metaphysical dependence irreflexive? The Monist, 94, 267–276.
Kleinschmidt, S. (2011). Multilocation and mereology. Philosophical Perspectives, 25, 253–276.
Leuenberger, S. (2014). Grounding and necessity. Inquiry, 57, 151–174.
Lewis, D. (1991). Parts of classes. Oxford: Basil Blackwell.
Litland, J. E. (2013). On some counterexamples to the transitivity of grounding. Essays in Philosophy, 14, 19–32.
McDaniel, K. (2009). Structure-making. Australasian Journal of Philosophy, 87, 251–274.
McKay, T. J. (2006). Plural predication. Oxford: Oxford University Press.
Oliver, A., & Smiley, T. (2013). Plural logic. Oxford: Oxford University Press.
Olson, E. T. (2006). Temporal parts and timeless parthood. Nôus, 40, 738–752.
Raven, M. J. (2012). In defense of ground. Australasian Journal of Philosophy, 90, 687–701.
Raven, M. J. (2013). Is ground a strict partial order? American Philosophical Quarterly, 50, 193–201.
Rosen, G. (2010). Metaphysical dependence: Grounding and reduction. In B. Hale & A. Hoffman (Eds.), Modality: Metaphysics, logic, and epistemology (pp. 109–135). Oxford: Oxford University Press.
Schaffer, J. (2009). On what grounds what. In D. Manley, D. Chalmers, & R. Wasserman (Eds.), Metametaphysics: New essays on the foundations of ontology (pp. 347–383). Oxford: Oxford University Press.
Schaffer, J. (2010a). The least discerning and most promiscuous truthmaker. The Philosophical Quarterly, 60, 307–324.
Schaffer, J. (2010b). Monism: The priority of the whole. The Philosophical Review, 119, 31–76.
Schaffer, J. (2012). Grounding, transitivity and contrastivity. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding: Understanding the structure of reality (pp. 122–138). Cambridge: Cambridge University Press.
Schnieder, B. (2006). Truth-making without truth-makers. Synthese, 152, 21–46.
Schnieder, B. (2011). A logic for ‘because’. The Review of Symbolic Logic, 4, 445–465.
Sider, T. (2007). Parthood. The Philosophical Review, 116, 51–91.
Simons, P. (1987). Parts: A Study in Ontology. Oxford: Clarendon Press.
Tahko, T. (2013). Truth-grounding and transitivity. Thought, 2, 332–340.
Trogdon, K. (2009). Monism and intrinsicality. Australasian Journal of Philosophy, 87, 127–148.
Trogdon, K. (2013a). Grounding: Necessary or contingent? Pacific Philosophical Quarterly, 94, 465–485.
Trogdon, K. (2013b). An introduction to grounding. In M. Hoeltje, B. Schnieder, & A. Steinberg (Eds.), Varieties of dependence: Ontological dependence, grounding, supervenience, response-dependence. Munich: Philosophia Verlag.
Uzquiano, G. (2006). Receptacles. Philosophical Perspectives, 20, 427–451.
van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press.
Varzi, A. (2008). The extensionality of parthood and composition. The Philosophical Quarterly, 58, 108–133.
Varzi, A. (2009). Universalism entails extensionalism. Analysis, 69, 599–604.
Wilson, J. (2012). Fundamental determinables. Philosopher’s Imprint, 12(4), 1–17.
Witmer, D. G., Butchard, B., & Trogdon, K. (2005). Intrinsicality without naturalness. Philosophy and Phenomenological Research, 70, 326–350.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dixon, T.S. Grounding and Supplementation. Erkenn 81, 375–389 (2016). https://doi.org/10.1007/s10670-015-9744-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-015-9744-z