Abstract
Paul (Noûs 36:578–596, 2002; Noûs 40:623–659, 2006, The Handbook of Mereology, forthcoming) has argued for a bundle theory of objects that analyzes the bundling relation between properties and objects in terms of parthood relations. In this paper I argue that any mereological bundle theory with the explanatory power of Paul’s theory will entail the principle of the identity of indiscernibles (PII). This is problematic, since similar bundle theories seem to fall to Max Black’s two sphere counterexample to (PII). I argue, however, that a fully developed mereological bundle theory provides a new way of interpreting Black’s two sphere universe that dispels the counterexample. I argue that this solution to Black’s puzzle is superior to other solutions on offer, and consequently that mereological bundle theory is an attractive ontological strategy for friends of (PII).
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Notes
By properties Paul means something like immanent universals, though she thinks that her theory can accommodate tropes as well. Throughout this essay when I refer to properties I will mean something like immanent universals as well, though it will become clear later in the paper that I do not believe these universals are multiply located in the usual sense.
These definitions are given in (Paul 2006, p. 5)
Not just any properties can be fused, since being a circle and being a square are not compossible. What determines whether two properties are compossible is an important, but unanswered, question for the mereological bundle theory. Since I don’t know the answer, I will only note the question here. A related question is, When does composition occur on the mereological bundle theory? While no principled answer suggests itself, we know that the mereological bundle theorist cannot accept unrestricted composition, since this would entail fusions of properties that aren’t compossible. Here I will assume that composition is restricted by compossibility, but is otherwise unrestricted.
Note that by (D5) properties are objects on this theory as well, since they are qualitative parts of themselves by (D1) and not qualitatively disjoint from themselves by (D3).
Lemma 1 Weak supplementation principle (WSP): for all x and y, if x is a proper part of y, then there is a part of y that is disjoint from x. Proof Suppose that x is a proper part of y. Suppose, for reductio, that there is no part of y disjoint from x. By (A2), y is not a proper part of x. Hence, by (SSP), there is a part p of x that doesn’t overlap y. Either p is identical to x or p is a proper part of x.
Case 1: Suppose p is x. Then by hypothesis p is a proper part of y. Since p is a proper part of y, by (D2) p overlaps y. But p doesn’t overlap y. (contradiction)
Case 2: Suppose p is a proper part of x. By hypothesis, x is a proper part of y, so, by (A3), p is a proper part of y. Hence, by (D2), p overlaps y. But p doesn’t overlap y. (contradiction)
Either case results in absurdity. So, by reductio, there is a part of y that is disjoint from x. \(\square \)
There is some controversy over how to draw the distinction between intrinsic and extrinsic properties (see Weatherson and Marshall (2012), §2.1), but I don’t think we will go too far wrong with this definition.
The existence of a universe containing exactly three objects, two qualitatively identical spheres that differ only in their pure extrinsic relations to a cube, seems possible and unproblematic, for example.
Rodriguez-Pereyra (2006) has a much more fine-grained analysis of the proper interpretation of (PII). He defines a non-trivial interpretation of (PII) to be one that excludes trivial properties from its domain, where a property F is trivial if and only if “differing with respect to F is or may be differing numerically.” (219) While sorting out the details of this definition he argues that not all impure properties are trivializing. Thus, since I exclude all impure properties, one might think that my analysis is incorrect. It is not clear to me, however, that this analysis is extensionally different from the one I intend to be giving, for it is the properties that involve unanalyzable numerical identity or numerical distinctness that are meant to be excluded by excluding the impure properties. Consequently, I do not disagree with Rodriguez-Pereyra, but instead interpret him as offering a more precise, albeit idiosyncratic, definition of impure property than the one relied on in this paper.
See Wilson (2010).
Obviously, distinctness has to be understood as mereological distinctness rather than numerical distinctness here, since an object does depend on its numerically distinct proper parts for its own identity, it has necessary connections to numerically distinct entities. If Hume’s Dictum is interpreted in the sense of numerical distinctness, then it seems that either composition is impossible or composition is identity. Since we’re here assuming that composition is both possible and not identity, we conclude that distinctness should not be interpreted as numerical distinctness. For more on this very interesting issue see Bøhn (xxx).
Not only are these weakly discerning location properties distinct, they are also incompossible. That is, like being a circle and being a square, no object can have both of them. Otherwise, the object in question would be both some distance from itself and not some distance from itself. This also shows why there is not a co-located sphere in Black’s universe in any straightforward sense. Such an object would be the sum of the located spheres, and would thus be both some distance from itself and not some distance from itself.
An anonymous reviewer notes that you might use the overlap defense to make sense of the distinctness of particles in a fermion system, which are apparently weakly discernible by their spin properties in the same way that Black’s spheres are weakly discernible by being one mile from a sphere. I am interested in whether the overlap strategy could be applied in the quantum realm, but I lack the expertise to confidently draw any conclusions on this issue.
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Acknowledgments
Thanks to Einar Duenger Bøhn, the audience at the 2012 Arché/CSMN Graduate Conference at the University of Oslo, and three anonymous referees for helpful comments on earlier drafts of this paper.
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Shiver, A. Mereological bundle theory and the identity of indiscernibles. Synthese 191, 901–913 (2014). https://doi.org/10.1007/s11229-013-0298-9
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DOI: https://doi.org/10.1007/s11229-013-0298-9