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Abstract
It is plausible that the universe exists: a thing such that absolutely everything is a part of it. It is also plausible that singular, structured propositions exist: propositions that literally have individuals as parts. Furthermore, it is plausible that for each thing, there is a singular, structured proposition that has it as a part. Finally, it is plausible that parthood is a partial ordering: reflexive, transitive, and anti-symmetric. These plausible claims cannot all be correct. We canvass some costs of denying each claim and conclude that parthood is not a partial ordering. Provided that the relevant entities exist, parthood is not anti-symmetric and proper parthood is neither asymmetric nor transitive.
Notes
1We will use italics for quotes and bold italics for corner quotes. We will also use italics to italicize.
2More generally, instances of the schema U is F semantically encode a singular, structured proposition that has U itself as a constituent as well as the relevant property. If you doubt existence is a property, feel free to substitute an uncontroversial example.
3This assumes that for any x, if x is a constituent of a proposition p, then x is a part of p. (More on this below.) If x is a part of p and x is distinct from p, then x is a proper part of p.
4A relation R is a partial ordering iff R is reflexive, transitive, and antisymmetric.
5Antisymmetry of parthood says that that for all x,y if x is a part of y and y is a part of x, then x = y. Asymmetry of proper parthood says that that for all x,y if x is a proper part of y, then it's not the case that y is a proper part of x.
6A relation R is a strict partial ordering iff R is irreflexive, transitive, and asymmetric.
7We assumed that there is a unique thing such that absolutely everything is a part of it. Suppose you reject that. Then you hold that more than one thing has everything as a part. Anyone who accepts that there is a universe but denies that there is only one is committed to the denial of asymmetry in virtue of that fact. Suppose that there are two universes, U and U′. Since U and U′ are universes, absolutely everything is a part of each of them. So, U is a part of U′ and U′ is a part of U. But since they are two, U is distinct from U′ and U′ is distinct from U. Therefore, U is a proper part of U′ and U′ is a proper part of U.
8One might take the truthmaker for U exists to be some big conjunctive state of affairs concerning a number of U's parts. One who does so might then go on to deny U is a constituent of the truthmaker for U exists. (Thanks here to an anonymous referee.) We don't think it is obvious that the truthmaker for U exists, provided there is such a thing, has U as a constituent. But it's pretty plausible. For instance, it is plausible that truthmakers are facts and that (e.g.) the fact that Obama is human has Obama and the property of being human as constituents.
9That there is such a pair violates constraints on classical set theory. One might offer an argument for non-well-founded set theory that parallels the argument for non-well-founded mereology above. We don't pursue this line since we think the case for the existence of problematic complexes is stronger than the case for the existence of the problematic set.
10These are not, strictly speaking, the only options. We'll discuss some others below. Some we'll completely ignore.
11We are not concerned here with whether ‘ordinary’ material objects are complex. We are concerned with the sorts of entities listed above. We focus on the special case of propositions, but some of what we say generalizes to other entities on the list. (Thanks here to an anonymous referee.)
12See Soames Citation1987 for a classic defence of structured propositions. See also Richard Citation1990 and King Citation2001.
13This is a bit of stipulation of how we intend to understand the technical terms singular proposition and structured proposition for the purposes of our argument.
14See especially Kaplan Citation1977, Kripke Citation1980, Salmon Citation1986 and Soames Citation2002. For an overview see Fitch and Nelson Citation2007.
15This sort of argument appears in Salmon Citation1986 and Cartwright Citation1997.
16We do not take these arguments to be irresistible. But our main goal is not to present a case for structured propositions, but rather a prima facie case for the plausibility of our premise that propositions are complex.
17Lewis Citation1986b objects to structures on mereological grounds, but the grounds are different from those offered here.
18One might also deny that U functions as a Millian proper name for the world. Anti-Millians may take this line because they hold that no proper name is a Millian proper name. Millians may take this line if they hold that one must be acquainted with an individual in order to introduce a name for it, and we're not acquainted with the relevant individual. We hope it is clear why this response is hopeless. At best it would show that our argument didn't express what we took it to express. It would not make the problem go away as long as the relevant proposition (or suitable complex entity) exists.
19Reflexivity and antisymmetry follow from these theses, so classical extensional mereology is committed to the claim that parthood is a partial ordering and that mereology is well-founded.
20Here we are sympathetic with Smith Citation2009.
21More carefully, Lewis Citation1986a identifies propositions with sets. And Lewis Citation1991 says that sets have parts. We should be clear that we only agree superficially with Lewis. Lewis's account gets the facts about propositional parthood wrong. Suppose propositions P and Q do not have the same intension. Then on Lewis's account, the proposition P&Q is a proper part of the proposition P and a proper part of the proposition Q . This is intuitively backwards, and is a result of Lewis's account of propositions as unstructured sets of worlds. Also, on Lewis's account, the sets that are propositions do not have their elements as parts. (Thanks to an anonymous referee here.) The account of propositions we favour differs drastically from Lewis's account. But we share at least one view: propositions are parts of other propositions. We think it is a (perhaps weak) point in favour of this view that proponents of such diverse accounts of propositions endorse it.
22One might also follow McDaniel Citation2009 in citing the Humean stricture forbidding necessary connections between distinct entities as motivation for the view that complexity is mereological complexity. If Obama must exist provided the proposition that Obama exists exists, then there is a necessary connection between distinct entities: Obama and the proposition that Obama exists. This is a violation of Humeanism. The Humean avoids the problem if Obama is a part of the proposition. Then Obama and the proposition are distinct but they overlap, which is enough to satisfy Hume's dictum.
23This claim implies that propositions are fusions. (Something is a fusion of some things just in case it has each of those things as parts and each of its parts overlaps at least one of those things.) According to King Citation2007: 9], if propositions are fusions, then there is no principled way of saying which fusions are propositions and why certain fusions have truth conditions, can be objects of attitudes, etc., while others cannot. We do not find King's argument convincing. To be a fusion is to be a thing. As van Inwagen Citation2006 points out, if something exists, it is a part of itself. And anything with parts is a fusion. But now consider the following argument:
If propositions are things, then there is no principled way of saying which things are propositions and why certain things have truth conditions, can be objects of attitudes, etc., while others cannot.
But by our lights, a good (partial) response to this argument is that what it is to be a proposition is to be the sort of thing that has truth conditions, can be an object of attitudes, etc. A candidate for a more complete answer is King's theory of propositions.
24Provided there is more than one thing. (Thanks to Joshua Spencer here.)
25For suppose there are only two simples: a and b. Then there are at least three objects: [a] (a thing with a as its sole part), [b], and [ab] (a thing with a and b as its only proper parts). Suppose there are three simples: a, b, and c. Then there are at least seven objects: [a], [b], [c], [ab], [bc], [ac], and [abc]. And so on. (As is standard, we assume here that n ≥ 1.) Note that if composition is identity, we may get a different result: [ab] = the plurality [a], [b]. It is ‘them’, non-distributively, so to speak. We are agnostic about whether on this view a ‘plural’ proposition of the form <[ab], F > ‘really is’ the proposition <[a], F > and <[b], F > and instead assume without argument that composition is not identity. (Thanks here to an anonymous referee.)
26This argument is inspired by the argument Rosen Citation1995 uses to expose an inconsistency in Armstrong's theory of classes as states of affairs.
27See Soames Citation1987, Citation2008 for criticisms of propositions as sets of truth-supporting circumstances.
28Our case for there being such a 1–1 mapping relies on intuitions according to which propositions are abundant and fine-grained. Some may be suspicious of our case on the grounds that similar intuitions lead to apparent paradox. Consider the intuition that there is a distinct proposition corresponding to each plurality of things. If this intuition is correct, there are at least as many propositions—and hence at least as many things—as there are pluralities of things. But if there are n things, there will always be 2 n pluralities of things. And 2 n is strictly greater than n, for any n. (Thanks here to anonymous referee. See McGee and Rayo Citation2000 for a similar line of reasoning.) It would take us too far afield to adequately address this sort of worry here. For further discussion, see Spencer forthcoming.
29For example, Markosian Citation1998.
30For example, Koslicki Citation2008 and Korman Citation2008, Citation2010.
31See especially Sider Citation2001 and Hawthorne Citation2006. For a reply, see Korman Citation2010.
32Maybe it is common sense that some maximal concrete object exists but it is not common sense to suppose something exists that includes all the concreta, all the numbers, etc., as parts. (Thanks here to an anonymous referee.) But if it is common sense that there is a maximal concrete object, then provided there are no hard restrictions on ‘inter-category’ composition, it is plausible that U exists, too. And there must not be any hard restrictions on inter-category composition if there are singular, structured propositions with ordinary concreta, properties, and even numbers among their parts. So, more carefully: provided it is common sense that some maximal concrete object exists, and there is no law against inter-category composition, it is plausible that U exists.
33This is a moral drawn in Uzquiano Citation2006, which shows that if the field of classical mereology is unrestricted, the cardinality of the universe is 2n−1 while if the field of set theory is unrestricted, the cardinality of the universe is strongly inaccessible. (2n−1 is not strongly inaccessible for any n.)
34See also Rosen Citation2001.
35For arguments to this effect, see Rosen [ibid.].
36See (e.g.) Hume Citation1739: Book I, Part I, §vi].
37See (e.g.) Fine Citation1999 and Koslicki Citation2008.
38See Lewis Citation1991 and Caplan, Tillman, and Reeder Citation2010.
39Or perhaps we should say that the thoroughgoing bundle theorist is a compositional nihilist!
40See Armstrong Citation1989.
41See Fine Citation1994, Citation1999 and Koslicki Citation2008 for representatives.
42One might have a different sense of the dialectic. Perhaps rejecting such entities flows from the motivations behind a restriction of fusions to concreta, and these views should be rejected because of the liberal view of composition that they require. (Thanks to an anonymous referee here.) While we think there may be a sound argument of the form composition is restricted to Fs, therefore view V is false, we do not think such arguments will help move the debate forward since we know too little about the true principles of composition. If there are powerful reasons in favour of the premise, this sort of argument may be just fine. But in the absence of such reasons in favour of a view of composition that would rule out the views mentioned, it seems we should evaluate each of these views on their own merits, and not rule them out by fiat.
43We have in mind Thomson Citation1983.
44Sider Citation2001: 155]. Sider calls strong supplementation ‘PO’. He considers a temporally relativized version of strong supplementation. We omit the temporal relativization.
45We assume here that parthood is transitive. If parthood is intransitive, weak supplementation may be retained.
46See Appendix for proof sketch of Claim 1.
47See Simons Citation1987 for one of the many examples of this sort of view. Koslicki Citation2008 also makes heavy use of this assumption.
48See Smith Citation2009 for a similar attitude towards Effingham and Robson's Citation2007 objection to endurantism involving a time-travelling brick composing a wall at a time, thus violating weak supplementation. See also Caplan, Tillman, and Reeder Citation2010.
49See Appendix for proof sketch of Claim 2.
50As McDaniel Citation2009 points out, a parthood relation has a closely associated proper parthood relation that is irreflexive and transitive iff that parthood relation is anti-symmetric. So if parthood is non-well-founded and proper parthood is, by definition, irreflexive, then it must be intransitive. See Appendix for proof sketch of Claim 3.
51The intransitivity of proper parthood plus the transitivity of parthood provides another route to the failure of weak supplementation. See Appendix for proof sketch of Claim 4.
52See McDaniel Citation2009: 254].
53As suggested above, the allegedly conceptually necessary conditions on fundamental parthood relations simply rule out certain coincidentalist and endurantist views. We do not think coincidentalists and endurantists are conceptually confused about parthood. And we find them good company to keep. Mutatis mutandis for the view of singletons defended in Caplan, Tillman, Reeder Citation2010.
54Thanks here to an anonymous referee.
55We think the most promising options for resisting non-well-founded mereology are either to make the case that constituents are not parts or to hold that there are multiple fundamental parthood relations, one of which is constituent-hood. But we can't think of any good reason to endorse either option, and we accept the arguments set forth in this paper. We suspect taking constituents not to be parts in order to preserve well-foundedness would be rather like insisting that all mammals bear live young by denying that monotremes (like the platypus) are mammals, instead of the correct response (that there are strange mammals which don't conform to our earlier expectations vis-à-vis mammals). If it turns out that, contrary to our expectations, good reasons are found for holding that constituents are not parts or that constituent-hood is a sort of parthood distinct from well-founded parthood, we hope that the arguments in this paper can be seen as groundwork for a theory of the otherwise ill-understood notion of constituent-hood. Alternatively, one may be able to preserve analogues of classical principles of minimal mereology by following Gilmore Citation2009, forthcoming in holding that the fundamental parthood relation is a four-place relation between an object, o1, and its location, l1, and an object, o2, and its location, l2.
56Thanks to audiences at the University of Manitoba, the 2010 Central APA, the Society for Exact Philosophy, David Braun, Dan Korman, Philip Kremer, Bernard Linsky, Michael McGlone, Michael Rea, Gillian Russell, and Gabriel Uzquiano for very helpful discussion of ancestors of this paper. Special thanks to Ben Caplan, Cody Gilmore, Daniel Rabinoff, Joshua Spencer, and two anonymous referees for this journal.