Vague Parthood
With this in mind, let us examine another criterion that is often cited as sufficient for the existence of vague objects: the existence of objects standing in borderline parthood relations.[185] [186] As a criterion for being a vague object, much needs to be clarified here, but let us begin by getting a handle on the phenomenon.
Following McGee [103], let us suppose that Mt Kilimanjaro is a vague object.
It's not a precise matter where the mountain begins and ends, it's not a precise matter exactly how many feet above sea level it is, it's not precise which rocks and stones are parts of it, and so on. It's possible that Mt Kilimanjaro also figures in vague identities: you might think that there is something that's determinately 19,341 feet tall, but that Mt Kilimanjaro, being indeterminately 19,341 feet tall, is indeterminately colocated with that thing. If you also thought that, determinately, colocation of distinct things was impossible, then you might think that Mt Kilimanjaro is indeterminately distinct from that thing. Indeed, one might go further and attempt to explain the indeterminacies about height, boundary, and parthood in terms of the indeterminate identities. Perhaps there are a bunch of entities with precise boundaries and heights and it is indeterminate which of them Mt Kilimanjaro is identical to.11 But it is unclear how far this strategy will carry us towards the view that Mt Kilimanj aro is uncontroversially a vague object. The picture described seems to be one in which, for all we know, there are no vague objects. For, according to the picture, it's borderline whether our putative examples ofvague objects just are precise objects. For example, it is borderline whether Mt Kilimanjaro is identical to a precise object (and thus not a vague object after all).Another strategy would be to attempt to explain the vagueness in Mt Kilimanjaro in mereological terms.
Brian Weatherson, however, has argued that, when combined with the assumptions of classical mereology, it's unclear whether this idea is coherent (Weatherson [150]). Let me illustrate some of these difficulties with a simplifying assumption: that there are finitely many objects. Now, given classical mereology we know that the structure of the parthood relation is severely constrained. Indeed, any two scenarios with the same finite number of objects must be mereologically isomorphic: for instance, any two worlds that contain exactly three atoms—a, b, and c—must be the same in the sense that they contain the fusions a + b, b + c, a + c, and a + b + c, and that these entities stand in the same configuration of parthood relations. One version of the problem can be stated as follows: assuming that it is not vague what exists and not vague what is identical to what, then it is a determinate matter how many objects there are (since finite numerical claims can be stated using only the quantifiers, identity, and connectives, which we may assume all to be precise). So every admissible precisification must agree on the number of individuals (in keeping with this literature I will temporarily adopt the supervaluationist terminology; all of this can be rephrased in our terminology of accessible indices too). Assume also that classical mereology is determinate, and so true at every admissible precisification: then, by the above argument they must each precisify the parthood relation in an isomorphic way. It follows that the parthood relation imposes a rigid structure on the individuals: a structure that can't vary between precisifications. It seems, then, that vagueness in the parthood relation is severely limited.However, as Barnes and Williams [8] note, this does not necessarily mean that there can't be any vague parthood at all. Although every precisification has to be mereologically isomorphic in a finite world, there could still be vague parthood because it could be vague which object occupies which mereological role even when the mereological roles form a fixed structure: it could be determinate that there are three objects, a, b and c, that obey classical mereology, but borderline whether a is a part of c because, relative to one precisification, a and b are atoms and c is their fusion, and, relative to another, b and c are atoms and a is their fusion.
Matters change also when one moves to the infinite case. One can construct, consistently with standard set theory (i.e. ZFC), two precisifications that agree about how many objects there are, agree that classical mereology is satisfied, and even agree that everything is composed of atoms, but disagree about how many atoms there are, making the two precisifications non-isomorphic.12 In these cases, even the structure of the parthood relation isn't rigid.It seems unlikely, however, that these limited kinds of vagueness about parthood could support the sorts of things that those who posit vagueness in parthood typically want to say. It's unclear, for example, how to model the following idea, which is surely consistent: that Mt Kilimanjaro is mostly precise except for a single atom which is such that it is borderline whether it is part of Mt Kilimanjaro. On the Barnes and Williams
12 Indeed, Cohen's original forcing argument was a model in which 220 = 2 21 (= X2) (even though lang=EN-US style='font-size:5.5pt;line-height:110%'>1!). In this universe, the powerset of 20 and of 21 are both classical models of mereology with the same number of objects, but are non-isomorphic because they have a different number of atoms. (See also Easton's theorem [37].) model, such vagueness can only arise at the expense of vagueness elsewhere—if the atom isn't a part of Mt Kilimanjaro relative to some precisification, some other object has to be substituted in its stead. And it is even less clear how the models that play on the strange properties of infinities in consistent extensions of standard set theory help us here.
An extremely natural response to these sorts of worries is to relax classical mere- ology.
Indeed, the puzzle we have been considering above is formally closely analogous to famous temporal and modal puzzles about mereology (see the problem of the statue Goliath and the lump of clay that constitutes it, Gibbard [63], or Geach's puzzle of temporary coincidence outlined in Wiggins [151]). In that context it is not unusual to relax classical mereology to something weaker; however, this is not the place to explore those options in more detail.[187]It seems, then, that borderline identity and borderline parthood are both at least coherent. However, even granting their coherence it is unclear how either of these things delivers an unambiguous criterion for vague objecthood. The issue is that both notions are relational—vagueness associated with them usually involves at least two objects—and so it is unclear how to identify the source of the vagueness. If it is borderline whether x is identical to y, does it follow that x is a vague object, or that y is a vague object, or both? On a view in which Mt Kilimanjaro is a vague object that is borderline identical to a precise object, X, there can simply be no identity- theoretic answer to this question. Identity is a symmetric relation after all: whatever makes Kilimanjaro the vague object and X the precise one can't be spelled out in terms of identity. A similar question arises for parthood, even though it is not symmetric: if x is a borderline part of y, does that make y the vague object or x, or both?
The general moral follows from a principle we discussed at the end of section 16.1: when it's borderline whether a complex proposition is true, that's due to vagueness in one of the things that compose it. Thus, in particular, for a binary relation R, if it's borderline whether Rxy, then either R is vague, or x is vague, or y is vague. Knowing that a proposition involving parthood or an identity claim is borderline gives us no clue as to what the source of the vagueness is.
Even assuming we know it's not in the identity or parthood relations, we do not know whether the vagueness is due to the first or second relatum.Notice, moreover, that there are other plausible sufficient conditions for an object to be vague which involve only monadic properties. For example, if it is borderline whether an object is exactly 19,341 feet high, then it is also natural to think that that's because the object in question is vague. Or, alternatively, there are relational properties where at least one of the relata is the sort of thing that is (intuitively) not normally vague. It could, for instance, be vague where Mt Kilimanjaro is located.[188] It is not plausible to blame the vagueness here on the location relation, or on the space-time regions that stand in this relation to Mt Kilimanjaro. What we are lacking, however, is some kind of theory that ties all these different kinds of vagueness together: we may have identified some plausible sufficient conditions, but we have not identified a single overarching feature of this sort of vagueness that unifies them as being distinctively to do with vague objects.
16.4