An elegant universe


David Lewis famously endorsed Unrestricted Composition. His defense of such a controversial principle builds on the alleged innocence of mereology. This innocence defense has come under different attacks in the last decades. In this paper I pursue another line of defense, that stems from some early remarks by van Inwagen. I argue that Unrestricted Composition leads to a better metaphysics. In particular I provide new arguments for the following claims: Unrestricted Composition entails extensionality of composition, functionality of location and four-dimensionalism in the metaphysics of persistence. Its endorsement yields an impressively coherent and powerful metaphysical picture. This picture shows a universe that might not be innocent but it is certainly elegant.

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  1. 1.

    At a first approximation the principle says that any non-empty collection of entities has a mereological fusion. The official formulation is the one in (9). Arguments in favor of Unrestricted Composition can be found in Rea (1998), Sider (2001) and Van Cleve (2008). Different critiques of the arguments are in Koslicki (2003), Simons (2006), Elder (2008).

  2. 2.

    There is in the literature a stronger version of this thesis, known as Composition is Identity. This thesis maintains that a composite object is, strict and literally, just its parts. For a recent overview see Baxter and Cotnoir (2014).

  3. 3.

    Interested readers can start from Simons (1987), Hovda (2009) or Varzi (2014). Lewis’ most explicit formulation is in Lewis (1991).

  4. 4.

    Lewis himself writes: “proper parts—parts not identical to the whole” (Lewis 1991, p. 2, italics in the original).

  5. 5.

    I will return, albeit briefly, to this definition of fusion in Sect. 3 as well. Let me just note here that this is a notion of fusion that Lewis himself sometimes adopt: “The fusion of all cats is that large, scattered chunk of cat-stuff which is composed of all cats there are and nothing else. It has all cats as parts. There are other things that have all cats as parts. But the cat-fusion is the least such thing” (1991, p. 1). Note that, strictly speaking, (3) is a schema.

  6. 6.

    As in the case of (3), this is a schema as well.

  7. 7.

    A referee for this journal rightly notes that the left-to-right direction of the biconditional requires Leibniz’s indiscernibility of identicals (LII) to be proven. Suppose now I endorse multilocation—which will be discussed thoroughly in Sect. 4, and I claim that one and the same object can have different properties at different locations. This would seem to undermine LII. I do not think friends of multilocation have to abandon LII. They could retain LII and then relativize property-instantiation to particular regions, for example the object’s exact locations. I will not take side here, for it is not crucial for the purpose of the paper. I will rest content in laying down two different options. The first option consists of the following: (i) endorse LII, (ii) relativize property-instantiation to regions and (iii) phrase the extensionality theorem with (10)—as is done in Varzi (2014) for example. The second option consists of the following instead: (i) abandon LII and (ii) phrase the extensionality theorem using only the right-to-left direction of the biconditional in (10)—as is done in Varzi (2008) for example. That direction is the most interesting one in any case and it is this direction we will be interested on in what follows. Thanks to anonymous referee for this journal here.

  8. 8.

    I will return to this later on in this section.

  9. 9.

    As we will see, “entails” might be too strong a word. I will rest content to suggest that it (strongly) favors Extensionality.

  10. 10.

    Varzi considers yet another argument, which is due to Simons (1987, pp. 30–31). The Unrestricted Composition Principle entails that that two overlapping entities have a merelogical product, i.e. something which is composed by all and only those things that are part of both: \(O(x,y)\rightarrow \exists z\forall w(w\prec z\leftrightarrow w\prec x\wedge w\prec y)\). This in turn entails the Strong Supplementation, and therefore, Extensionality.

  11. 11.

    This argument assumes the definition of proper parthood given in (1). As we shall see in a moment this is strictly related to Anti-Symmetry.

  12. 12.

    This follows by the definition of fusion. The fusion of x and y overlaps all and only those things that overlap either x or y. Thus if \(z_1 \) is disjoint from x it has to overlap y.

  13. 13.

    Adapted to the present context.

  14. 14.

    A referee for this journal correctly notes that this establishes only the right-to-left direction of (10). The left-to-right direction is a simple application of LII. See footnote 7 for a somewhat long discussion of different options regarding the extensionality theorem and LII.

  15. 15.

    Which is in turn related to Anti-Symmetry as we will see in a moment.

  16. 16.

    And, as we saw, endorsed by Lewis himself.

  17. 17.

    See Cotnoir (2014, p. 8). He claims that the difference is “merely terminological”.

  18. 18.

    I will be somewhat sloppy and use them interchangeably, thus ignoring the more technical sense of “composition”, as defined, for example in Inwagen (1990).

  19. 19.

    See for example Fine (2003).

  20. 20.

    I consider two “atomistic” cases for the sake of simplicity. I believe the arguments can be given even if there are only gunky objects. Even if electrons turn out to be gunky the same considerations would still apply. Furthermore I doubt that any mutual parthood theorist would want her approach to rule out atomism.

  21. 21.

    A referee of this journal suggested that in such cases we simply do not have any adequate intuitions about the persistence conditions of fusions.

  22. 22.

    The question—posed in Inwagen (1990)—about the necessary and sufficient conditions that have to be met for any-non empty plurality to have a mereological fusion.

  23. 23.

    I take atom to be an entity without proper parts.

  24. 24.

    She could maybe argue that mutual parthood enters the picture only when we deal with mereologically complex entities. But we need an argument for this last claim, and there are none in the literature, as far as I know. Furthermore my previous argument would still apply.

  25. 25.

    Cotnoir (2014), the forerunner of the mutual parthood approach, uses examples from physics himself.

  26. 26.

    See for example Casati and Varzi (1999) and Parsons (2007).

  27. 27.

    In what follows I will downplay this modal element.

  28. 28.

    For another argument see Calosi (2014).

  29. 29.

    I’m in good company. Here’s a list of some other philosophers who understand the relation in the same way: Casati and Varzi (1999), Hudson (2001), Gilmore (2006), Sattig (2006), Hawthorne (2008), and Donnelly (2010).

  30. 30.

    For an authoritative and extensive overview of the relation between mereology and location see Gilmore (2013).

  31. 31.

    Gilmore (2013) rightly notes that Exactness follows by definition (16). Hence, it should not be considered an assumption after all.

  32. 32.

    For a different, yet similar, argument see Calosi (2014).

  33. 33.

    I shall give another argument for this very conclusion at the end of Sect. 5.

  34. 34.

    I will also assume throughout the argument that nothing else besides x and y is exactly located at the relevant regions.

  35. 35.

    I’m using the set-theoretic notion of union for spacetime regions for the sake of familiarity. This does not mean that a further primitive set-theoretical notion is needed, for the argument could be cast using the notion of mereological fusion instead. See Uzquiano (2011, p. 201).

  36. 36.

    A referee for this journal suggested the following: the fusion of x and y is exactly located at \(R_1 \cup R_2 \cup R_3 \) and there is nothing exactly located at \(R_1 \cup R_3 \). This is a nice suggestion, yet it does not provide a (complete) solution to the problems discussed in the argument in the main text. For it entails that there is no material object, nor any part of any material object that has the same size, area, volume of \(R_1 \cup R_3 \). But in fact we want to say that there is. Consider the case in which x is singly located at \(R_1 \) and y singly located at \(R_3 \). In this case we would readily admit that their fusion is located at \(R_1 \cup R_3 \). Hence this fusion has the same size, are, volume of that region. In the multilocation case just focus on x at one of its exact locations, namely \(R_1 \) and the case is entirely analogous.

  37. 37.

    Thanks to an anonymous referee here.

  38. 38.

    Not to mention that independent arguments in favor of the Region Dissection Principle can be given. See for example Calosi (2014).

  39. 39.

    Given Anti-Symmetry.

  40. 40.

    We just need to change the roles of \(z_1 \) and \(z_2 \) in the arguments.

  41. 41.

    The arguments in this section presuppose some sort of eternalism, i.e. the view that all the tenses are ontologically on a par. It is beyond the scope of this paper to argue in favor of this claim. Lewis gives an (admittedly weak) argument in favor of this claim in Lewis (1986, p. 204).

  42. 42.

    It follows from Exactness that it has at least one exact location. Note that the argument assumes that material objects are at least weakly located somewhere.

  43. 43.

    A referee for this journal has noted that this argument relies on a (not so implicit) assumption, namely that the relation between a material object and (space)time is that of location. This is correct. Now, if this is relatively uncontroversial in the case of space–modulo, for example, supersubstantivalism—it is controversial in the case of time. Giordani and Costa (2013) discuss a very interesting alternative option. They call it “temporal trascendentism”. According to temporal trascendentism “events are properly located [...] at times whereas objects are derivatively present at times by being participants of events” (Giordani and Costa, 2014: 213, italics mine). In other words, objects are not located in time, events are. Objects strictly speaking transcend the temporal dimension. Addressing payoffs and limits of trascendentist persistence goes beyond the scope of this paper. I will simply note that, on the one hand, it seems to offer a way to resist the argument in the main text. On the other hand it should be considered what is the relation of material objects to spacetime, rather than time simpliciter. If this relation is indeed location that argument still applies. Thanks to an anonymous referee for this journal for having drawn my attention to this point.

  44. 44.

    It could be argued that what the definition forbids is that a 4D object is exactly located at its path and one of its proper subregions, but it allows a 4D object to have more than one path. Suppose \(Path_1\) and \(Path_2 \) are two such distinct regions. Can a 4D object be exactly located at both \(Path_1 \) and \(Path_2 \)? Suppose it is the case. Since the path of an object is defined as the union of its exact locations Path(x) would be in this case \(Path(x)=Path_1 \cup Path_2 \). Now, as I mentioned already, the definition of a 4D object does not allow for it to be exactly located at its path and one of its proper subregions. But both \(Path_1 \) and \(Path_1 \) are proper subregions of \(Path(x)=Path_1 \cup Path_2 \). Hence the 4D object is exactly located neither at \(Path_1 \) nor \(Path_1 \), against our assumption. This reductio argument shows that uniqueness of location is indeed inbuilt into the definition of a 4D object. Thanks to an anonymous referee here.

  45. 45.

    A referee for this journal rightly notes that Functionality here is redundant. The argument would go through even without R being the only one of z’s exact locations.

  46. 46.

    Which is actually what we expected all along. This is because with Functionality at hands we could (should?) require exact location to obey the following Additivity axiom:

    \(ExL(x,R_1 )\wedge ExL(y,R_2 )\rightarrow ExL(S(z,\varphi (w)),R_1 \cup R_2 )\), where

    \(\varphi (w)=w=x\vee w=y\). This formulation assumes Unrestricted Composition in that it assumes that there always exists a fusion of x and y. The Additivity axiom is problematic in the context of multilocation. See Sattig (2006) and Calosi and Costa (2014).

  47. 47.

    A referee for this journal suggested the following. Many three-dimensionalists see themselves –at least when they do not talk about multi-location- to advocate a more commonsensical view of the world. And on this view there is plenty of four-dimensional entities that extend through time. I agree. But it seems that these temporally extended entities are not material objects but rather events, on the commonsensical picture the three-dimensionalist is advocating. So the argument in the main text—if correct—still stands, for it establishes that Unrestricted Composition entails the existence of a four-dimensional material object. Thanks to an anonymous referee here.

  48. 48.

    Which is good news, for there are other influential arguments from Unrestricted Composition to four-dimensionalism, the most celebrated of which is probably the argument from vagueness in Sider (2001). For a critique of this last argument see Koslicki (2003) and Simons (2006). It should be noted however that the arguments presented here and Sider’s argument are entirely different. Sider’s argument—if found compelling—is actually much stronger for it entails that a 4D object divides into temporal parts. Nothing of this sort is entailed by the arguments I presented. It is actually a distinct possibility to have 4D-objects according to the definition I gave that do not divide into temporal parts. This is something advocated already in Parsons (2000). Thanks to an anonymous referee for having pushed this point.

  49. 49.

    Keats, Ode to a Grecian Urn, verses 49–50.


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I am extremely grateful to two anonymous referees for this journal for detailed and insightful comments on different drafts of the paper which led to substantive revisions and improvements. Part of this work was funded by the Swiss National Science Foundation, Project Number BSCGI0_157792.

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Correspondence to Claudio Calosi.

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Calosi, C. An elegant universe. Synthese (2015).

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  • Unrestricted composition
  • Extensionality
  • Functionality
  • Four-dimensionalism