Arguing about the world ’ s cardinality : Priority , existence , and metaphysical necessity 1

Metaphysicians still discuss about the number of things. According to monists, there is one thing, either fundamental (Priority Monism) or exclusionary (Existence Monism). According to pluralists, there are many things, either fundamental (Priority Pluralism) or exclusionary (Existence Pluralism). The claims of cardinality of these views are, presumably, metaphysical claims, which means, presumably, that they are necessarily true, if true at all. In this paper, I unravel a common pattern of some of the main arguments used by the parties involved and challenge their assumptions. By doing this, I intend to show that those arguments are all equally impotent to settle the question about the number of things because their conclusions are not necessary truths as they are meant to be. These views, at the very most, can be presented and defended as consistent ways of saying how many things, apparently, might be.


The world's cardinality
Let W be our concrete material world.Let the ps be a l the concrete material individuals/ objects/ things (I wi l use these expressions interchangea ly) that compose it.W is the cosmos itself; the ps are its many planets, pe bles, particles, etc.Now quantify over W. Are there many things or only one thing?
Priority Monism (PM), the view that has been championed by Schaffer (2009Schaffer ( , 2010)), says that both the ps and W exist, but that W is metaphysically p io to the ps.The parts are grounded in the whole.Fundamentality has a top-down direction.In contrast, Priority Pluralism (PP) answers, just like PM, that both the ps and W exist, but with the op osite qualification: the ps are metaphysically p io to W. The whole is grounded in the parts.Fundamentality has a bottom-up direction.We can attribute this view to Kim (1993Kim ( , 1998) ) and to Op enheim and Putnam (1991). 3PP honours the Newtonian tradition that ap eals to the sma lest bodies as the building locks of a l larger bodies: the fundamental ontological level of W consists in many atomic ps, many externa ly related point-like individuals, presuma ly space-time points or point-like particles, upon which a l composite individuals, including W itself, are grounded.
PM and PP are not the only possi le answers.There is also Existence Monism (EM) and Existence Pluralism (EP).EM rejects the assumed ps.EM understands the ps as fictions of decomposition of W, and answers that there is, really, only one individual, W, which is a mereological simple, a metaphysical atom.Of course, without rejecting such assumption, EM would be a non-starter.This was, according to some (e.g., Bennett, 2001, ch. 7), the view held by Spinoza (1994); it is now the view defended by Horgan and Potrč (2008); and it is recognised by Rea (2001) as a perfectly coherent and tena le alternative.Since W is not a composite object, its local variation is not in virtue of its distinct proper parts instantiating distinct properties."Rather, it is a matter of [W] itself instantiating in spatiotempora ly local ways various properties and relations" (Horgan and Potrč, 2008, p. 169).Strictly eaking, there are no planets, but only W being planet-lish there-ishly.EM can paraphrase the ta k of mi dle sized goods in terms of "complex adverbial qualifications of the world" (Schaffer, 2007, p. 179n).Objects shorter than W are not individuals, but only Spinozistic finite mo es or affections of W. They only count as individuals in virtue of distinctions of reason or imagination, not in virtue of real distinctions. 4 In contrast, EP rejects a l composite objects, including W itself, as fictions of composition, and answers that there are, really, only many ps, which are mereological simples, metaphysical atoms.Typica ly, it is the view embraced by compositional nihilists like Rosen and Dor (2002), Sider (2013), and van Inwagen (1990Inwagen ( , 1994Inwagen ( , 2002) ) 5 -though we should keep in mind that EM is also a variant of compositional nihilism, the most parsimonious availa le (cf.Schaffer, 2007).Many ps never compose a distinct object.What hap ens is that the ps instantiate lots of different properties and relations, they are ar anged in many different ways, but, strictly speaking, there are no planets, but 3 Oppenheim and Putnam claim, without providing any arguments, that "there must be several levels", and that "there must be a unique lowest level", which they identify with the level of "elementary particles" (Oppenheim and Putnam, 1991, p. 409).Kim, on the other hand, defends a worldview that understands the world as an "array of levels"; the structure of it is given by "the mereological relation being part of: entities belonging to a given layer are mereologically composed of entities belonging to the lower levels, and this relation generates a hierarchical ordering of the levels"; and it "carries the assumption that there is a bottom tier, a layer of entities that have no physically significant parts" (Kim, 1993, p. 337;cf. Kim, 1998, p. 15).For more textual evidence and examples, see Schaffer (2003).With qualifications, I think we can also see an instance of PP in Lewis's "Humean Supervenience" thesis (1986a, 1986b).I say "with qualifications" because the attribution demands reconciling some aspects of Lewis's thesis with the understanding that PP and PM, qua variants of metaphysical foundationism, have of grounding (cf.Bliss and Trogdon, 2016).First, Lewis's alleged "ontologically innocent" understanding of both supervenience and composition would need to be rejected, since it is incompatible with the metaphysical foundationist understanding of grounding, which takes the relata of the latter relation as distinct existents, in such a way that, when mapped in terms of parthood, either the whole grounds the parts (PM) or the parts ground the whole (PP), yet both the whole and the parts exist (more on this later).Second, we should exclude the possibility according to which supervenience could be understood as a symmetric relation (cf.Hall, 2016), since the metaphysical foundationist understanding of grounding takes it as an asymmetric relation (more on this also later).Another way to attribute PP to Lewis could be by appealing to his notion of "naturalness", and understand that the relation being more natural than can play the same role of grounding as understood by metaphysical foundationism.But it is doubtful that Lewis understood it like that.For start, although he did rank properties in terms of naturalness, it does not seem that he intended to rank individuals in those very same terms (cf.Lewis, 1999;Hall, 2016). 4That fragmentary understandings of W are not the results of real distinctions, but the result of distinctions of reason, seems to have been Spinoza's position.Thus, he writes to Meyer: "[F]rom the fact that we separate the affections of Substance from Substance itself, and arrange them in classes so that we can easily imagine them as far as possible, there arises Number, whereby we delimit them.Hence, it can clearly be seen that Measure, Time and Number are nothing other than modes of thinking, or rather, modes of imagining" (Spinoza, 2002, p. 789;cf. Spinoza, 1994, Part I, Appendix). 5I include van Inwagen as a compositional nihilist, although his position is not strictly this one.He thinks that two things never compose a third thing unless this third thing is a living organism.But, leaving aside living organisms, it is quite clear that van Inwagen does support compositional nihilism.His concrete material world is a world of atoms and living organisms.No tables, planets or pebbles around.
only some ps arranged planet-wise; there is no W, but only the ps arranged W-wise. 6 The expression "fiction of decomposition" is borrowed from Schaffer (2007).The expression "fiction of composition" is bor owed from Rosen and Dor (2002).Before taking any further step, it is important to highlight the important role of these two expressions.EP and EM embrace compositional nihilism.Strictly speaking, according to them, there are none of the mi dle-sized goods that populate Moorean ap earances (e.g., hands, chairs, cars, stones, planets).But both acknowledge that we do need, somehow, to find a place for them in our ontology.Although they obviously reject the route of mereological complexity and metaphysical foundationism embraced by PP and PM, they also, at least implicitly, reject the route of straight eliminativism.This latter route would be manes (cf.Schaffer, 2009).Thus, EP and EM recognise the existence of many composite objects, such as hands and chairs, at least as fictions, i lusions, or the like.They do so by a ding a fictional operator (or something along these lines) to their existential claims about them.Thus, as it can be ap reciated, in the debate about the number of things, the parties make, at least implicitly, the fo lowing restrictions to the existential quantifier.First, they restrict quantification to ou concrete mate ial world, whose existence is presup osed. 7Second, they restrict quantification to what really exists, 8 and this reality is meant to include our concrete material world and exclude fictions, i lusions and the like.So, when we make these presup ositions explicit, it hap ens that PM, PP, EM, and EP do not ar ue about how many things exist simplicite ; they ar ue about how many things exist really or funda entally at W. PM and PP both agree that the ps and W exist simplicite and really; their disagreement is about what grounds what.EP and EM both agree that the ps and W exist simplicite , but they disagree about which of these really exist; yet they agree that, once this latter question is settled, there is no question about what grounds what, because real existents have equal ontological rights.
There is wide agreement among the participants of this debate on the idea that metaphysical claims are necessarily true, if true at a l, and on the idea that the claims about cardinality that they hap en to defend or attack are indeed metaphysical claims (cf.Rosen, 2006;Schaffer, 2010;Sider, 1993;Ta lant, 2013;van Inwagen, 1990;Wi liams, 2006).A second basic shared commitment is the idea that, p ima facie, if something is conceiva le and logica ly possi le, then it is metaphysica ly possi le-yet there is no need to think that conceivability entails metaphysical possibility (cf.Schaffer, 2010, p. 61).We can state these assumptions thus: (A1) Metaphysical claims are necessarily true, if true at a l.Claims about the number of things are metaphysical claims.(A2) Conceivability and logical possibility are, considered jointly, our best uides to metaphysical possibility.
For now and for the sake of ar ument, let's assume that PM, PP, EM, and EP co lectively exhaust the possi le answers to the question of which is the number of things.How can we decide between them?Not at a l clear.The four alternatives, p ima facie, make sense; at least none of them seems to be logica ly inconsistent.But since a l of them claim to be necessarily true, if true at a l, then only one of them can be true.Which one?PM and PP need, as a condition of sense, the existence of a fundamental layer of being.
6 One could say that a planet is "nothing over and above" some pps arranged planet-wise, or that a planet just is some pps taken collectively and, conversely, that some pps taken collectively just are a planet.That is, one could embrace the idea that composition is identity, on the understanding that mereology, unlike set theory, is "ontologically innocent" (cf.Armstrong, 1997, p. 11-13;Baxter, 1988;Lewis, 1991, § 3.6).But this would not do.If we take the thesis that composition is identity in its weaker form, that is, if we understand that composition is like identity in many respects, but not strictly identity, the thesis is not "ontologically innocent", because being committed to many things (some pps) and being committed to one thing (a planet) are two distinct commitments, even if they happen to be invariant across possible worlds.In contrast, if we take it literally, in its stronger form, the thesis is contradictory, since some parts are many things, that is, not one; yet a whole is one thing, that is, not many.There is no such thing as a plural object: many things are not one thing, composition is not identity, and composition, even when modally invariant, is not ontologically innocent, on pain of contradiction (cf.Priest, 2014, p. 51;Yi, 1999aYi, , 2014)).Appealing to some mechanism of plural reference represents no way out from this contradiction, because "[t]he machinery does not allow us to refer to objects that are plural, but to a plurality of objects.Thus, when we say that Russell and Whitehead wrote Principia, we are not referring to some strange object, Russell and Whitehead; we are referring to Russell and to Whitehead" (Priest, 2014, p. 51).It seems that the only way left is that the relation being arranged planet-wise were, as Baxter puts it, a "one-making relation", that is, a relation capable of making many things one thing (cf.Baxter, 1996).But this route faces Bradley's dilemma.On one hand, if we take the relation as actually relating, then we simply beg the question about how the many are one, since we have posited a mysterious creature to account for oneness without explaining in virtue of what it is capable of making many things (including itself) one thing.On the other hand, if we take the relation in itself, as distinct from the many relata, then it is quite obvious that it doesn't make the many things one thing, unless we either again beg the question by saying that it does so because it does so, or embark ourselves in a vicious regress: the one-making relation makes many things one thing in virtue of a second one-making relation, and so on, ad infinitum (cf.Bradley, 1930, ch. II).Under the influence of Yi (1999aYi ( , 1999bYi ( , 2014)), I have partially changed my previous stance on these matters (cf.Briceño, 2016). 7Schaffer claims not to be "concerned" with things that are concrete but not material, or with things that exist but are non-concrete, such as abstracta or possibilia (cf.Schaffer, 2010, p. 33;in similar vein, Bohn, 2012;van Inwagen, 1990).In what follows, I assume, following Williamson (2013) and pace Lewis (1986a), that we cannot make full sense of possibilia only in terms of concreta. 8We can take them as adhering to something like the primitive sense of reality used by Fine (2001Fine ( , 2009)).
This need is more or less clear: without a fundamental layer of being, no objects would exist, because there would be no ground for them.But there is at least one of them, namely: I, since I am thinking now.So either I am fundamental or I am grounded in one or many other fundamental objects.On the other hand, EM and EP need, as a condition of sense, the existence of either one or many simples.This need is also more or less clear: without it, no objects would exist.But there is at least one of them, namely: I, since I am thinking now.So, either one or many simples exist: either I am a simple, or there is one simple being I-lish here-ishly, or there are many simples arranged I-wise (cf.van Inwagen, 1990, p. 73;Rosen and Dor , 2002, p. 159).
But how can these views uarantee that their particular conditions of sense actua ly obtain?In what fo lows I wi l try to show that some of the main ar uments of the parties involved are a l equa ly impotent to answer the question.The alternatives, at most, can be presented and defended as equa ly consistent ways of saying how many things, ap arently, might be-given certain crucial but quite arbitrary assumptions that presup ose already a minimal answer to the question.This should not be surprising.From the armchair we can reflect on the internal coherence of these alternative views and on the conditions that each of them needs to meet in order to make sense at a l.We can also clarify what possibilities represent a threat for the obtainment of those conditions.But we cannot te l what the number of things is, at least not if this is sup osed to be necessarily true, if true at a l.This seems crystal clear to me.But the sup orters of these theses seem to think differently.Let's check their credentials. 9

The ways of priority and mereological complexity
Let's start by examining PM and PP, who share, at least implicitly, the fo lowing assumptions: 10 (A3) W and the ps really exist and stand in some relation of composition.(A4) Composition is not identity.A whole is never identical to its parts. 11(A5) W and the ps instantiate at least one grounding relation.Grounding is a relation that "imposes a strict partial ordering (SPO) on the entities in its domain: grounding is ir eflexive, asymmetric, and transitive" (Bliss and Trogdon, 2016).This hierarchical metaphysical structure can be map ed in terms of parthood.(A6) Grounding is a we l-founded relation: there is a fundamental level of being.There are no circular or infinite chains of grounding. 12(A7) Fundamental objects leave no gaps and do not overlap.They cover the whole world in a sufficient and non-redundant way.
Given (A1)-(A7), PM and PP are the two jointly exhaustive and mutua ly exclusive answers to the question about the number of (fundamental) things.But there is an ar ument that favours PM over PP, since it puts into question the a leged necessary chara er of PP.This is the ar ument based on the possibility of unk (cf.Schaffer, 2010;Sider, 1993;Zimmerman, 1996).
A unky world is a world in which everything has proper parts, an atomless world that enjoys mereological complexity all the ay down.Schaffer's version of the ar ument from unk includes the fo lowing sup lementary assumption: (A8) W is a unique maximal object, the one fusion of a l the ps. 13he existence of W qua unique maximal object fo lows from the standard mereological principle of unrestricted composition, according to which any co lection of objects always composes an object, a whole, namely: their fusion or mereological sum.This principle is also meant to be necessarily true, if true at a l.Thus, neces a ily, the co lection of every object composes one object.W, regar less of how many ps it has, is not a proper part of other object.It is a maximal whole.
Given (A1)-(A8), the ar ument goes like this: If PP is true, then mereological atomism is true.If mereological atomism is true, then it is necessarily true.But unk is possi le.And if unk is possi le, then PP is not necessarily true as it is meant to be.In contrast, if W is the one fundamental object, 9 Each party involved has more arguments than the ones I will discuss here.Those arguments typically appeal to empirical considerations and theoretical virtues such as parsimony or explanatory power.I cannot address all these considerations here.I have chosen those arguments that seem to me most typically metaphysical and more central to the debate.These arguments aspire to convince with independence of those other secondary considerations. 10These assumptions are made explicit by Schaffer (2010), and I follow him closely; but they can also be traced in Lewis's "Humean Supervenience" thesis, once we qualify it (cf.supra, fn 3; Lewis, 1986aLewis, , 1986b). 11Cf.supra, fn 6.If whole and parts are distinct things, we are pressed to give an account of how they are so intimately related.This is precisely what PM and PP attempt to do in the next assumptions. 12(A5)-(A6) express a dominant way of understanding grounding.Cf.Bliss andTrogdon (2016), andCorreia andSchnieder (2012).
as PM claims, the possibility of unk can be accommodated.Since, given assumptions (A1)-(A8), PM is the only other alternative to PP, then PM is true, and necessarily so. 14ereological atomism receives sup ort from (A1) and (A6) and from the fact that a view that admits many non-overlap ing mi dle-sized composite objects as fundamental seems to be intolera ly arbitrary, because at what molecular level should we draw the line to distribute the privileges of fundamentality?It is no coincidence that our most consistent pluralists have been atomists.Ar uments for the metaphysical possibility of unk are also availa le.Gunk has been conceived in logica ly consistent ways (e.g., Descartes, 1985, II.20).Gunk seems also to be an empirica ly open scenario and science sometimes seems to point in that direction (cf.Schaffer, 2003, with references).Fina ly, mereology has coherent unky models (cf.Simons, 1987, § 1.6).Thus, it seems that PM, unlike PP, can offer an object that is not a proper part of other object, an object that is capa le of accommodating the possibility of unk and, alternatively, the possibility of mereological atomism.Of course, the ar ument only works under the sup lementary assumption that W is a maximal whole-(A8).Otherwise, such a whole would not be basic: it would be a proper part of other object, so it would overlap with it and would need sup lementation to cover everything-contra (A7).
The ar ument is valid, but doubtfu ly sound.Its weakness surely lies at the heart of its assumptions.Let's start with the last one introduced by PM: (A8).
By hypothesis, the sup orter of PP accepts (A1)-(A7).However, why should he also accept (A8)?Isn't PM just be ging the question, or at least taking for granted a very important part of the answer by the mere stipulation of (A8)?After a l, there seems to be no privileged reason to assume the existence of one maximal object as an axiom rather than assuming the existence of many minimal objects as a distinct axiom.So, as a first objection, the sup orter of PP might attempt to even the score.While PM may introduce as an axiom the existence of a maximal object, PP can introduce as an axiom the existence of many minimal objects, i.e., many mereological simples or atoms, through an alternative sup lementary assumption: (A9) There are many minimal ps that compose both W and every other composite object.
Obviously, (A9) is not a proof of PP simplicite but only the (key!) assumption of the existence of many minimal ob-jects, just like (A8) is not a proof of PM simplicite , but only the (key!) assumption of the existence of one maximal object.PM and PP are both intere ed in what grounds what.But if PM can introduce (A8) to grant the existence of a maximal object and then ar ue for a top-down direction of priority, there is no reason why PP shouldn't be a lowed to introduce (A9) to grant the existence of many minimal objects and then ar ue for a bottom-up direction of priority.PM might complain and say that PP cannot exclude the possibility of unk by decree, so there must be something wrong in assuming (A9).Of course it seems wrong to exclude the possibility of unk by decree!But then, for the very same reason, PP might complain against PM.After a l, PM is also using a decree (namely, the product of (A2) and the possibility of unk) to exclude the possibility of mereological simples, and another decree (namely, (A8)) to include a maximal object that excludes an analogous possibility to that of unk, namely: the possibility of junk.
A junky world is a world in which everything is a proper part of something else, a world that is not a maximal whole, a world that enjoys mereological complexity all the ay up.In a junky world there is no such thing as the fusion of everything.And since a junky world sti l has mereological structure, only some form of restricted composition can take place (cf.Bohn, 2009aBohn, , 2009b)).Thus, a junky world is ir educi ly plural. 15While a unky world lacks minimal parts, a junky world lacks a maximal whole.And the metaphysical possibility of junk seems straightforward.In fact, junk has been conceived in logica ly consistent ways (e.g., Descartes, 1985, II.21); its existence also seems to be an empirica ly open scenario, and science, now and then, has pointed in that direction (as the history of science shows, the pendulum seems to go back and forth, from a closed universe to an open universe; cf.Koyré, 1957); and mereology has coherent non-standard models for it (cf.Bohn, 2009aBohn, , 2009bBohn, , 2012;;Simons, 1987, ch. 2).
Thus, PP can raise the ar ument of the possibility of junk against PM in more or less the same fashion as PM raises the ar ument of the possibility of unk against PP.In fact, admitting (A1)-(A7) as a common ground, PP can introduce (A9) instead of (A8) and then ar ue in the op osite direction: If PM is true, then W is a unique maximal object.If W is a unique maximal object, then it is necessarily so.But junk is possi le.And if junk is possi le, then PM is not necessarily true as it is meant to be.In contrast, if the fundamental ps are mereological simples, as PP claims, then the possibility of junk can be accommodated.Since, given assumptions (A1)-(A7) and (A9), PP is the only alternative to PM, then PP is true, and necessarily so. 16he possibility of junk points to the heart of PM.Because if the world is junky, then there is no maximal whole, since every whole is always a proper part of some other whole.And if one is wi ling to accept the possibility of unk, then there are no principled reasons for rejecting the possibility of junk.In fact, on what grounds should PP accept W as being maximal instead of junky?If PM can assume (A8), why shouldn't PP be a lowed to assume (A9)?This is not to defend (A9) in itself.As far as I can see, (A9) is as dogmatic as (A8), and, therefore, equa ly unjustified (at least neither maximal nor minimal things belong to Moorean appearances; their existence is not obvious or evident).My point is that if we want to treat both PM and PP with fairness, then we must either a low or reject both assumptions.Both a unky world and a junky world are logica ly consistent and conceiva le, so they both seem metaphysica ly possi le; and they both seem equa ly compati le with the shared assumptions.Yet they cannot be both metaphysica ly possi le if PM or PP are meant to be necessarily true, if true at a l.It is true that PP is threatened by the possibility of unk.This is a scenario that mereology tolerates but that PP, qua ontology, is forced to reject in order to make sense.In contrast, it is true that PM is threatened by the possibility of junk, a scenario that mereology also tolerates but that PM, qua ontology, must reject in order to make sense.P ima facie, PM and PP are on a par.They both accept a layered ontology and the idea that either the top or the bottom layer must be fundamental.So, the most we can do is to honour (A6) and insist that there must be a fundamental level, either a maximal whole (W) or many minimal parts (a l the minimal ps).But the crucial pro lem is which one.Since there is no principled way to choose between (A8) and (A9), there is no non-arbitrary reason to prefer PM over PP, or vice-versa (cf.Ta lant, 2013).
If we include both (A8) and (A9), then the ar uments from unk and junk don't cut any ice, because both the possibility of unk and the possibility of junk are excluded from the start by two distinct decrees, and the real dispute is only about the direction of priority, on which, without a ditional ar uments, we are sti l in the dark.In contrast, the exclusion of both would mean admitting the joint possibility of unk and junk and, thereby, the admission that, possi ly, there is neither a maximal whole nor many minimal parts.This would mean giving up (A6), a crucial assumption common to both PM and PP.So what?Why not indeed give up (A6), and, consequently, both (A8) and (A9), i.e., the myth of a fundamental level?It seems that there is nothing wrong with the possibility of a world that is both unky and junky, that is, a world that lacks a fundamental level, either top or bottom.Bohn (2009a) has explored this joint possibility-which he labels "hunk"-in a convincing way.
A hunky world has no minimal parts and is not a maximal whole; it is just many infinite composites. 17Now, if this means that nothing has ultimate ground, so be it.There is nothing inconsistent with this possibility, since it is just the conjunction of two, p ima facie, compati le possibilities: unk and junk.Ca l this view Hunky Pluralism (HP).I cannot explore this alternative in depth, but here is a hint of an ar ument in favour of it: If (A6) is true, then, necessarily, there is a fundamental level, either a maximal whole or many minimal parts, so either unk is impossi le or junk is impossi le.But, as shown before, possi ly, W is unky.And, as also shown before, possi ly, W is junky.Since there seems to be nothing against the joint possibility of unk and junk, except the myth encapsulated by (A6), possi ly, W is both unky and junky (i.e., hunky).That is, possi ly, (A6) is false, and PM and PP are not necessarily true as they are meant to be.Therefore, possi ly, HP is true, and necessarily so.
But can we te l who' s actua ly right, PM, PP, or HP?A certain pattern seems to take place in the ar uments just examined.What is notorious about them is their minimal power to show the fundamental cardinality of W directly, without dogmatic axioms and detours in the modal space.The ar ument from unk does not prove that there is a maximal whole, nor that grounding holds between concrete material objects, nor that grounding must end somewhere.A l these claims are assumed as axioms.Similarly with the ar ument from junk.Neither PM nor PP can go directly for what they want: they need to make heavy loaded assumptions, fix those that are more convenient for their purposes, and then offer a deductive proof that runs, more or less trivia ly, from the possibility of something to the necessity of their own prefer ed alternative.Fo lowing van Fraassen' s analogy, these assumptions play the role of Descartes' s God: they are ecia ly designed to uarantee that what fo lows from them is true (cf.van Fraassen, 2002, p. 1).In fact, the parties just seem to be playing puzzle-solving after agreeing on some posits, constructs or simulacra, namely: W, the ps, (A1)-(A9).The ar ument for HP is less dogmatic, but equa ly uninformative.It i lustrates that if we free ourselves from some dogmas (e.g., (A6)), it might we l be the case that HP is another possibility that we have overlooked.This possibility does not entail by itself the impossibility of PM or PP.This is done by (A1), which makes them incompati le possibilities.This should make us think: haven't we overlooked sti l other possibilities simply because we have wrongly assumed from the start-lin ly fo lowing their sup orters-that the claims of PM, PP, and HP are suita le candidates for being neces a ily true, if true at a l?

The ways of real existence and mereological simplicity
We can have a more clear diagnosis about the reasons of this ap arent metaphysical embar assment if we examine EM and EP.These two views embrace compositional nihilism and they immediately put into question the truth of (A3)-(A9).According to them, no parthood relations ever obtain, so W lacks mereological structure.Therefore, no grounding relations between concrete material objects, map ed in terms of parthood, ever obtain.Concrete material objects do not stand in a hierarchical order based on mereological composition.They live together in an egalitarian level of real existence.Those that really exist are thereby exclusionary: either there are many of them or there is only one of them.Thus, instead of (A3)-(A9), we can say that EP and EM endorse the fo lowing common assumptions: (A10) There is, really, only one concrete material simple, or, alternatively, there are, really, many concrete material simples.Otherwise, there would be no objects at a l.(A11) An object only composes itself.No proper parthood relations obtain. 18(A12) No grounding relations between objects, map ed in terms of parthood, obtain.
Given (A1), (A2), and (A10)-(A12), here is a trivial arument for EM: If EM is true, then it is necessarily so.And if EP is true, then it is necessarily so.It seems that the real existence of one exclusionary concrete material simple is possi le. 19If the real existence of one exclusionary concrete material simple is possi le, then EP is not necessarily true as it is meant to be.In contrast, if EM is true, then the possibility of there being, really, one exclusionary concrete material simple can be accommodated.Since, given the assumptions, EM is the only alternative to EP, then EM is true, and necessarily so.
However, given (A1), (A2), and (A10)-(A12), EP can reply in a trivial way: If EM is true, then it is necessarily so.And if EP is true, then it is necessarily so.It seems that the real existence of many exclusionary concrete material simples is possi le. 20If the real existence of many exclusionary concrete material simples is possi le, then EM is not necessarily true as it is meant to be.In contrast, if EP is true, then the possibility of there being, really, many exclusionary concrete material simples can be accommodated.Since, given the assumptions, EP is the only alternative to EM, then EP is true, and necessarily so.
Infinite cardinality of concrete material atoms might be admitted, but since there are only simples, there is no mereo-logical complexity that can give place to unk or junk.Quantification is either over one or over many simples, but never over composites.Certainly, one can also quantify over sets, but these are abstra a; or over fictions of composition, but these are not real existents.
So the debate between EM and EP cannot be settled unless one, from the very beginning, incorporates one of the fo lowing a ditional assumptions, just like PM and PP do in their own way: (A13) There is, really, only one concrete material simple.
But these assumptions cannot be accepted.Taking both for granted is contradictory.Taking only one of them for granted is to beg the question.Now, in order to re ect the common assumption, (A10), we must remain open and conclude that either EM or EP is true.But which one?!Neither of them is evident or obviously true: Moorean ap earances are not populated neither by one nor by many simples.And we remain clueless if we keep testing them in the modal space.Because if we keep doing so, we can legitimately ask: Why don't we give up (A10), and, therefore, both (A13) and (A14)?This would mean that, possi ly, there are, really, no concrete material objects whatsoever (because, possi ly, neither EM nor EP are true).So what?Is the possibility of there being, really, no concrete material objects whatsoever somehow logica ly inconsistent or inconceiva le?Not at a l.So far, we have presup osed their real existence.Sure, I think, I am, so at least some concrete object exists.But this doesn't mean that some concrete mate ial object exists, let alone that this claim is necessarily true, if true at a l.Firstly, logic and mereology, as formal systems, are sup osed to be neutral on whether the objects of quantification are mate ial or im ate ial. 2118 Horgan and Potrcˇ (2008, ch.7) claim that our world is an object that lacks proper parts.In parallel, they invite us to take it as gunky.This is wrong.If our world lacks parts, then it is an extended simple; if it is gunky, then it is partite all the way down, a vast jello that lacks atomic parts but, nonetheless, has non-atomic parts. 19See the references before when I introduced EM.All those philosophers have conceived it in a logically consistent way (no matter how shocking to common sense); hence, it seems metaphysically possible.E.g., Rea (2001) takes space-time to be one extended simple. 20See the references before when I introduced EP.All those philosophers have conceived it in a logically consistent way (no matter how shocking to common sense); hence, it seems metaphysically possible.E.g., Sider (2013) has argued for a world in which the only concrete material objects are space-time points. 21I understand "formal" as Husserl did (cf.Husserl, 2001, p. 19-20, 39-41).Formal systems (e.g., logic, set theory, mereology) are supposed to be topic-neutral; they attempt to draw laws and general principles that are supposed to hold for a domain of entities, regardless of the nature of these.I am not claiming that logic and mereology are free from metaphysical presuppositions.What I do claim is that we should always keep this in mind and always try to make those presuppositions explicit.For start, we should always keep in mind that all formal systems rest at least on one obvious metaphysical presupposition, which is not always made explicit: that of singular existence.As Leonard puts it, "modern logic tacitly presupposes singular existence for its singular term variables, just as the traditional logic tacitly presupposed general existence for its general term variables" (Leonard, 1956, p. 56).In fact, logic only has a symbol for singular existence.It really presupposes that the "x" in "∃!x" designates something at all.My complaint is that the parties of the debate about the world's cardinality tend to present themselves as arguing for something much more substantial than what they are in fact arguing for.Because most of the weight of the arguments they present for their respective views is really carried by the metaphysical presuppositions embraced by them prior to any of the arguments they deploy.Once you embrace those metaphysical presuppositions, the logical and mereological consequences that follow from them are more or less trivial.
Secon ly, there is nothing that can stop us from conceiving in a logica ly consistent way a world of concrete im ate ial objects; we just need to remember that idealism, in some or other form, was once the dominant metaphysics, and it wasn't defeated for being inconceiva le or logica ly contradictory.Hence, it is metaphysica ly possi le that there are, really, no concrete mate ial objects at a l. 22 In fact, we can give a general ar ument that goes against (A10)-and, therefore, against (A13) and (A14): If EM is true, then it is necessarily true.And if EP is true, then it is necessarily true.But, possi ly, as shown before, EM is false; hence, not necessarily true as it is meant to be.And, possi ly, as shown before, EP is also false; hence, not necessarily true as it is meant to be.So, possi ly, both EM and EP are false; hence, not necessarily true as they are meant to be.Therefore, possi ly, there are, really, no concrete mate ial objects at a l.But since there is, really, at least one concrete object, because at least I exist, then there is, really, at least one concrete but im ate ial object.
According to the last possibility explored, a l real concrete objects are immaterial.A l material objects are fictions, i lusions or the like.This possibility might be realised in five different ways-which cor espond to the idealistic counterparts of the five different ways in which the implicit presupposition that there is, really, at least one concrete material object was realised-, namely: Priority Idealistic Monism (PIM), Priority Idealistic Pluralism (PIP), Existence Idealistic Monism (EIM), Existence Idealistic Pluralism (EIP), and Hunky Idealistic Pluralism (HIP).It does not take too long to think of historical fi ures that have conceived logica ly consistent versions of these alternatives.One just need to take a quick look to Dunham et al. (2011) or to any anthology of Eastern philosophy to find a good number of examples.
What might be wrong?I think we have reached rock bottom.We have been bringing more possi le answers to the light.And we have been capa le of doing so insofar as we have been capa le of giving up some dogmatic assumptions.The fact that the range of metaphysica ly possi le alternatives is wider than what we first thought seems to point to a central pro lem of the debate about fundamental cardinality as it is pra ised today.
The parties explicitly assume that metaphysical claims are necessarily true, if true at a l, and that claims about the number of things are metaphysical claims.But then they, dogmatica ly, restrict the domain of quantification to ou real concrete mate ial objects.This restriction may we l be re ecta le according to the Zeitgeist, but certainly cannot be embraced as a matter of principle, at least not under the standard settled by (A2), which fixes, a lege ly, the limits of rational, thinka le, metaphysical enquiry.If they insist on (A1), then they should make their claims of cardinality under the presup osition that they are quantifying in absolutely unrestricted terms.Otherwise, their claims wi l fail when te ed against other conceiva le and logica ly consistent alternatives.That is what doing first-order ontology amounts to.But they don't do that.They do not say, for instance, that absolutely everything is real, concrete and material.They presup ose the existence of things like these and also want to remain neutral or indifferent on whether there are other things apart from these.So, lou ly, they announce that they are making first-order metaphysical claims, a l of which are meant to be necessarily true, if true at a l; but, whi ering, they restrict the existential quantifier to a domain of objects that doesn't even pretend to be absolutely everything.It should not be surprising, then, that unattended possibilities pose a threat to these views.Once these views are te ed in the modal space, as views that are making absolute/ unrestricted/ necessary claims, they reveal themselves as what they rea ly are: views that are making only relative/ restricted/ contingent claims.So, their sup orters are forced to either admit defeat or admit that they were not, after a l, making metaphysical claims, at least not if metaphysical claims are meant to be necessarily true, if true at a l.The last ar ument shows what has been out of sight.By the very fact of stating it, we can anticipate that the remaining alternatives (PIM, PIP, EIM, EIP, and HIP) could also be cha lenged by putting into question their pretensions of necessity, fo lowing what is now a familiar pattern.This pattern would help us to e a lish that, since it is conceiva le and logica ly consistent, it is metaphysica ly possi le that there are, really, no concrete objects at a l, not even im ate ial ones (after a l, the model has only presu posed the real existence of at least one of these). 23 The likely rejoinder would be the fo lowing: "we l, I think, I am, so at least something concrete exists" .But this claim, in turn, could be refuted by saying: "true, but <I think, I am> is not a necessary truth, since I could have not existed" .
22 I am not saying that the conceivability of there being no concrete material objects entails that it is metaphysically possible that there are no concrete material objects.What I am saying is that the conceivability of such scenario plus the fact that it also seems a contradiction-free scenario are, jointly, very good reasons, perhaps the best expressible, thinkable reasons available to us, for taking it indeed as a metaphysically possible scenario.Recall (A2). 23It is not difficult to entertain such a possibility following what seems to be the perfectly sound and valid route of substraction arguments that toy with the iterated possibility of removing any member from a world that consists in a finite domain of distinct concrete objects that do not depend on each other to exist (cf.Baldwin, 1996;Rodriguez-Pereyra, 2013).But I do not want to limit myself to this single route to defend the metaphysical possibility of there being no concrete objects at all.Because substraction arguments are not available when we are in front of worlds that consist in domains of objects that do depend on each other to exist (e.g., Whiteheadian actual occasions or worlds where there is massive overlapping of objects), or when the domain in question is infinite (e.g., gunky or junky worlds).These worlds do not fit the requirements demanded by substraction arguments for their annihilation.Yet it still seems perfectly conceivable and logically possible that any of those worlds could be wiped out all at once.The fact that <I think, I am> is always true while being entertained or uttered by the thinker does not grant the necessary existence of the thinker.This last reply can only be defused by showing that I am a necessary existent.And I see two ways of doing this: (a) The first alternative is to embrace a form of Spinozism.I am a necessary concrete existent, something whose mere concept or idea entails its concrete existence.I am God.(b) The second alternative is to embrace a metaphysics according to which a l objects are necessary existents, that is, "it is necessary that everything is such that it is necessary that something is identical with it" (Wi liamson, 2013, p. 2).To make sense of a l propositions, the proposal is to distribute existence generously among a l possi le objects, so then we can distin uish between the existence of something qua pos ibilium (which is necessary) and the existence of something qua concretum (which is contingent Nee less to say, none of these alternatives gives us too much hope.First, regarding alternative (a), I know of no sound ontological ar ument that proves the necessary concrete existence of something.And certainly I know of no sound ontological ar ument that shows, in a dition, that I am precisely that thing (!).If there are indeed sound ar uments in this direction, they must show that the fo lowing p ima facie plausi le scenarios are, against the spirit of (A2), metaphysical impossibilities: that there could be no concrete existents; and that, even if there were a necessary concrete existent, I could be a distinct thing, a contingent concrete existent.
Second, as regarding alternative (b), it is obviously less contentious, but its pretensions are also less ambitious.It simply says that a l possi le logical objects, that a l pos ibilia, are necessary existents.This view does not te l us anything about whether concrete existents are many or one, and it assumes that concrete existence is a contingent feature.So far as it goes, the view is consistent with the claim that there could have been nothing concrete at a l; hence, I am not a necessary concrete existent.At the most, the view entails the impossibility of there being absolutely nothing at a l, since, according to it, non-concrete things, such as pos ibilia, are something, and they exist necessarily.
Third, neither of these two forms of necessitism seems a le to exclude the threat of the fo lowing metaphysical pos-sibilities: (i) that what alternative (a) takes to be a necessary concrete existent might not be a real necessary concrete existent, but only a fiction or i lusion of a necessary concrete existent, a purely intentional object; (ii) that what alternative (b) takes to be necessary existent pos ibilia might not be real necessary existent pos ibilia, but only fictions or i lusions of necessary existent pos ibilia, some purely intentional objects; and (iii) that, against the claims of alternatives (a) and (b), it seems that there could ha e been nothing at all.And by "nothing at a l" I mean not even the proposition that there is nothing at a l, not even the mere ap earance, fiction or i lusion of a necessary existent pos ibilium.
Sure, this last possibility is obviously incompati le with (A1) and the fact that I am something that thinks, exists now.But why should we indeed accept (A1)?Why should we expect sin ular existence to be uaranteed in even stronger terms than those that Descartes ever conceived?Semantical considerations as those deployed by Wi liamson (2002,2013) seem not enough to show that a l objects are real necessary existents.What those considerations reveal is p io ontological commitment to those things.This strategy begs the question in favour of those who, like Wi liamson himself, think that ontology is real and necessary.It puts the cart before the horse: why, after a l, should anyone accept a semantics that prejudges in favour of a determinate metaphysics (cf.Alvarado, 2017)?Clearly, no semantical considerations would convince anyone to believe in a certain metaphysics unless that person is alrea y convinced of such metaphysics.Some examples might help.The development of plural logic wi l not convince the Eleatic monist of the real existence of many things.The deployment of distinct quantifiers wi l not convince the Spinozist that they really pick out distinct things, since, according to him, the sentences "some things are F" , "everything is F" , and "there is a unique thing that is F" say, really, just the same.And the development of mereology wi l not convince the compositional nihilist about the real existence of partite wholes and proper parts.In sum, just by changing a few words, we could say against Wi liamson's real necessary existent pos ibilia what Bentham said against natural rights: "a reason for wishing that a certain right were e a lished, is not that right-want is not sup ly-hunger is not bread" (Burton, 1843, p. 67).
What our modal thought and discourse do seem to show is that thinkers like us are committed to intentional objects.But these objects might we l be unreal, that is purely intentional objects, mere theoretical posits, subjective projections, fictions or i lusions.The fact that I can fo low Wi liamson's (2002) ar ument-which starts claiming that "necessarily, if I do not exist then the proposition that I do not exist is true" , and ends up infer ing that I, necessarily, exist, although, possi ly, I am not concrete-certainly cannot uarantee my own real necessary existence.The fact that our modal thought and discourse demands the existence of pos ibilia in order to make sense is not a reason for transforming what seems to be, p ima facie, a conceptual, epistemic or lin uistic necessity into a metaphysical necessity.Sure, if a l propositions are real necessary existents, I can infer my own real necessary existence from a proposition about me.But this holds, again, under an ontological presup osition: if propositions are real necessary existents.Yet I see no compe ling metaphysical considerations to accept the idea that the existence (simplicite ) of something like a proposition about me, or about any other object, can exclude the possibility of my non-existence (simplicite ), or of any other object.The exclusion of this possibility can only be done by answering the perennial question of why there is (simplicite ) something rather than nothing by ap ealing to the fo lowing sufficient reason: because it is impos i le for there to be nothing (simplicite ).However, just like metaphysical nihilism about concreta seems possi le, I am prepared to accept that absolute metaphysical nihilism, that is, that there is nothing at a l whatsoever, not even aletic or propositional reality, also seems possi le.Of course, such a scenario is ineffa le, because from the very minute we attempt to think or express the possibility of absolute metaphysical nihilism we are trap ed in self-contradiction.Just like the thinker that claims <I do not exist> commits a self-refutation, the proposition <there is nothing at a l> is self-contradictory if the scope of "nothing at a l" is meant to be absolutely unrestricted and pretends to cover, among a l other things, itself.But paradoxes involving self-reference affect any thought with global scope (as metaphysical thought aspires to be): one cannot think about the a leged limits of thought without crossing them (cf.Priest, 1995).However, the aporia to which absolute metaphysical nihilism seems to lead us is not dissolved by insisting that, necessarily, everything really exists, or by admitting that reality is contradictory (pace Priest).The aporia simply reveals that we cannot think or say what can only be shown; that when we try to state the ineffa le we end up stating nonsense. 24Now, if absolute metaphysical nihilism is inconceiva le or inexpressi le in contradiction-free terms, then we either have to deny that it is a genuine metaphysical possibility or we need to give up (A2).So what?Why don't we give up (A2) at this point?If we cannot conceive it or express it, perhaps the scenario can be shown or su ge ed by analogy with the status of the cogito.It seems clear that although <I think, I am> is true whenever uttered or thought by me, its truth does not uarantee my necessary existence qua concretum.Similarly, although I exist invariantly whenever there are propositions about myself, this does not uarantee that those propositions, nor their constituents (myself included), are, really, necessary existents.
I do not question that some metaphysical claims, like those that unravel essences, or those that assert the cardinality and composition of some presup osed things, are necessarily true, if true at a l.For instance, gi en the existence (simplicite ) of three non-identical things, e.g., you, I, and the whole [you+I], we can say that a l the fo lowing claims are neces a ily true, if true at a l: I am one thing; you are one thing; you and I are two things; the whole [you+ I] is one thing composed of two things; I am what I am (say, a human being); you are what you are (say, another human being); the whole [you+I] is what it is (say, the whole that has you and I as proper parts). 25But the metaphysical views discussed above, if metaphysical at a l, are first-order views that can only aspire to be contingently true, if true at a l. 26 ). Because, according to Wi liamson's reasoning, if the proposition <I do not exist> is true, then both the proposition and myself as a constituent of it must exist.So, necessarily, if I do not exist, I exist.So, necessarily, I exist.What hap ens is that, possi ly, I am not concrete.If this position is right, then everything is a necessary existent qua pos ibilium, but only a contingent existent qua concretum(cf.Wi liamson, 2002).