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Informally, Necessitism is the view that it is necessary what there is. Necessitists should endorse the following principle: Modal Plenitude: If it is possible that there be some cardinal number κ, it is possible that there are exactly κ-many angels.6 Unfortunately, Modal Plenitude is inconsistent with CIE and SIE. Given Necessitism, Modal Plenitude implies that if it is possible for there to be some cardinality κ, then in the actual world there are at least κ-many things. Now consider all of the things in the actual world, the tt. It is possible that they form a set by SIE. So, it is possible that there is a cardinal that is the cardinality of that set, call it λ. By CIE, it is possible that there is some larger cardinal λ+. By Modal Plenitude, it is possible for there to be exactly λ+ distinct angels. By Necessitism, it follows that there are at least λ+ objects in the actual world. This contradicts the fact that cardinality of all the objects in the world is λ.7 By now, the pattern should be familiar. Similar arguments can plausibly be run, for example, on maximalist views about the existence of fictional characters. Perhaps for every cardinal κ, there could be a fictional character who strongly desires that the world includes exactly κ angels. Another argument can be run on the existence of abstract sentence types. Infinitary logic, for example, is the study of sentences of arbitrary cardinal length.8 Maximalist views which seek to posit the existence of sentence types corresponding to the sentences of these kinds of languages are also vulnerable to this kind of argument. And the list goes on. 6Note that while the Necessitist thinks that there are the same things at every possibility, they still typically think that the number of concrete things (and the number of angels) varies between possibilities. 7Hawthorne and Uzquiano (2011) and Sider (2009) have developed somewhat related problems for Necessitism and Modal Realism, although they do not appeal to SIE or CIE. The argument schema developed here can be seen as a generalization of these other arguments, which applies to many other ontological debates besides Necessitism and Modal Realism. 8See Bell (2016) for an introduction to infinitary logic. 5. Concrete Objects 8 One might naturally think that this style of argument can only be run against the existence of abstract objects, but that would be a mistake. Consider the famous Statue and Lump, which are made of the same matter and occupy the very same spatiotemporal location. There is prima facie pressure to say that they are distinct: after all, the Lump could survive being squashed but the Statue cannot. So, it seems that their essential properties differ. We must then ask: exactly how many objects are located exactly where the Statue is located? According to a minimalist view, there is only one object (or perhaps zero, if mereological nihilism is true). According to intermediate views, perhaps there are only two such objects (the Statue and the Lump), or perhaps there are 17 or 2,521 such objects. According to a maximalist view, defended by Yablo (1987), Fine (1999), Johnston (2006), Koslicki (2008) and Fairchild (2019), there are a plenitude of material objects located exactly where the Statue is located. The basic idea is that the Statue and Lump have different essential profiles (different sets of essential properties). So, instead of thinking that there are only two essential profiles had by two distinct objects, maximalists think that any collection of properties had by the Statue should count as the essential profile of some distinct object located exactly where the Statue is.9 Again, the primary motivation for maximalist views is anti-arbitrariness. Intermediate views need an account of exactly which properties count as the essential profile of some co-located object and which don’t. Maximalists need not draw any arbitrary lines. In order to not draw any lines, the maximalist should endorse the following: Essential Plenitude: If it is possible that there be some cardinal number κ, then for any concrete object C, there is an object Cκ that is co-located with C and has the essential property of being in a world with at most κ angels. Just as before, Essential Plenitude can easily shown to be inconsistent with SIE and CIE. One last case. Are there any concrete possible worlds? Minimalists say there is only the actual world. If one does believe in other concrete possible worlds, however, one shouldn’t believe that there are just 17 of them Instead, one should endorse a maximalist view according to which there is a plenitude of possible worlds. In particular, one should at least endorse the following: World Plenitude: If it is possible that there be some cardinal number κ, then there is a possible world in which there are exactly κ-many angels. 9There are a number of subtleties about how to exactly formulate this maximalist view. For example, there cannot be an object which is essentially red but not essentially colored. See Fairchild (2019) for a recent treatment of these issues. Again, as before, CIE and SIE imply that World Plenitude is false.10 9 6. A Lewisian Response The argument schema that I have so far been developing has important precursors. For example, in the Plurality of Worlds David Lewis considers a challenge from Forrest and Armstrong (1986) to the effect that natural recombination principles on the space of worlds ‘entrap modal realism in paradoxes akin to those that refute naïve set theory’ (1986: 101). The kind of recombination principle that Forrest and Armstrong proposed is substantially different from any principle that I have discussed so far, and it is a matter of controversy whether it successfully refutes a plenitudinous version of Modal Realism.11 Nevertheless, Lewis’ response to the threat of inconsistency is instructive. Here is how he responds: My proviso, if spelled out, would have to put some restriction on the possible size of spacetime. Among the mathematical structures that might be offered as isomorphs of possible spacetimes, some would be admitted, and others would be rejected as oversized. Forrest and Armstrong say that such a restriction ‘seems to be ad hoc’. Maybe so; the least arbitrary restriction we could possibly imagine is none at all, and compared to that any restriction whatever will seem at least somewhat ad hoc. But some will seem worse than others. A restriction to four-dimensional, or to seventeen-dimensional, manifolds looks badly arbitrary; a restriction to finite-dimensional manifolds looks much more tolerable. Maybe that is too much of a restriction, and disqualifies some shapes and sizes of spacetime that we would firmly believe to be possible. If so, then I hope there is some equally natural break a bit higher up: high enough to make room for all the possibilities we really need to believe in, but enough of a natural break to make it not intolerably ad hoc as a boundary… If study of the mathematical generalisations of ordinary spacetime manifolds revealed one salient break, and one only, I would dare to say that it was the right break – that there were worlds with all the shapes and sizes of spacetime below it, and no worlds with any other shapes and sizes. If study revealed no suitable breaks, I would regard that as serious trouble. If study revealed more than one suitable break, I would be content to profess ignorance – incurable ignorance, most likely. (1986: 103) We might call this kind of view Intermediate Realism about possible worlds, in contrast to maximalism and minimalism about possible worlds. Intermediate Realist views concede that a 10 It is important to note that this argument works regardless of what possible worlds are taken to be. They may be concrete objects (as in Lewis (1986)), properties (as in Stalnaker (2011)) or other kinds of abstract objects (as in Plantinga (1974) and Adams (1974)). 11 Forrest and Armstrong’s recombination principle states that for any concrete worlds ww, there should be a ‘big’ world W that contains a duplicate of each of the worlds in ww as distinct parts. Nolan (1996) has argued that such a principle does not land Modal Realism in contradictions. 10 maximalist ontology is impossible, and instead they attempt to draw an ontological line between those logically possible objects that exist and those that do not. For any philosopher who wishes to avoid arbitrariness, the tenability of Intermediate Realism hinges on there being a unique, non-arbitrary ‘suitable break’ in logical space, corresponding to the kind of object in question. In the absence of any suitable break, any version of Intermediate Realism would be wholly arbitrary. In the presence of multiple equally salient suitable breaks, the Intermediate Realist would still be saddled with an arbitrary choice between them. Even if the Intermediate Realist wishes to remain agnostic between these multiple breaks (as Lewis does), they must still think that the world’s true ontology, whichever break it happens to correspond to, is ultimately arbitrary. As Lewis also notes, there are also epistemological worries about Intermediate Realism. If there is no theoretical reason for preferring one break over some other, then it’s hard to see how any belief about where such a break lies could ever be justified. So, is there a unique, non-arbitrary break for the kinds of objects that we have been considering? One shouldn’t expect a simple, uniform answer to this kind of question. Depending on one’s first-order views, different proposals will be more or less attractive. For example, perhaps some version of Finitism is a defensible philosophy of mathematics, in which case it would be reasonable to believe in only finitary mathematical structures.12 In the case of possible worlds, some philosophers have defended the view that the space of metaphysically possible worlds coincides with the space of nomologically possible worlds.13 Combining Modal Realism with this kind of view would easily avoid any set-theoretic paradoxes. Assessing these and other Intermediate Realist views is far beyond the scope of this paper. The important point to make here is that it is Intermediate Realism, rather than maximalism, which deserves to be the main rival to a minimalist ontology. 7. Conclusion Unsurprisingly, there are other ways to resist the arguments that I have been developing other than Lewis’ Intermediate Realism. One could, for example, insist that the only intelligible reading of the modal operators in SIE corresponds to metaphysical modality, while at the same time adopting the view that the ontological truths of metaphysics are metaphysically necessary. Because of the myriad other candidate interpretations of such operators that have been defended by a wide variety of different philosophers (logical possibility, conceptual possibility, idealized conceivability, and other broader kinds of possibility than metaphysical possibility), 12 See Incurvati (2015) for a discussion of different forms of Finitism. See Builes (forthcoming-b) for a general argument against the existence of any abstract mathematical objects. 13 See Bird (2007) and Wilson (2013, 2020) for defenses of this view. Builes (forthcoming-a) defends a related view. 11 such a response strikes me as implausibly skeptical.14 Alternatively, one might question the possibility of quantifying over absolutely everything that there is, which was essential to the arguments that I presented. However, the possibility of unrestricted quantification has been explicitly defended by many metaphysicians engaged in the ontological debates I have been considering (see note 1). A common attitude to the denial of unrestricted quantification was expressed by Lewis himself, when he wondered whether those who denied the possibility of unrestricted quantification needed to resort to the idea that ‘some mystical censor stops us from quantifying over absolutely everything without restriction’ (1991: 68). My primary goal here has been to develop a way forward for philosophers who are committed to having a non-arbitrary ontology, but who have yet to decide between maximalism and minimalism. There is a path towards minimalism that only rests on one methodological assumption: metaphysics shouldn’t be arbitrary.15