1 Introduction

Let an entity x be the fusion of a plurality Y of entities just in case each of the Ys is part of x and every part of x overlaps (that is, has a part in common with) at least one of the YsFootnote 1:

(Fusion):

$$xFuY =_{df} \forall z\left( {z \prec Y \to z \le x} \right) \wedge \forall z\left( {z \le x \to \exists w\left( {w \prec Y \wedge Owz} \right)} \right)$$

According to Strong Composition as Identity (or ‘SCAI’) whenever an entity x is the fusion of a plurality Y of entities, x is literally identical to the YsFootnote 2,Footnote 3:

(SCAI):

$$\forall x\forall Y\left( {xFuY \to x = Y} \right)$$

SCAI-theorists accept Leibniz’s law in its full generality.Footnote 4 However, Leibniz’s law seems to give rise to some serious problems for SCAI. Consider, for instance, the properties ‘being one’ and ‘being many’: the whole is one; yet the parts it fuses are many; so, how could they be identical? For this reason, some SCAI-theorists claim that the instantiation of these problematic properties must be thought of as being relative either to a certain way to count the entities in question, or to a certain way to conceptualize them.Footnote 5 This family of views can be called count-relative SCAI.Footnote 6 In this paper I will focus only on non-count-relative SCAI, that is, the view according to which (i) wholes are literally identical to the parts they fuse, and (ii) problematic properties like ‘being one’ and ‘being many’ are absolutely instantiated, and not relative to a ‘count’ or to a ‘way of conceptualizing’.Footnote 7 Most importantly, non-count-relative SCAI accepts the following principle concerning the ‘one-of’ predicate which I will, thus, take to be valid throughout the paper:

(One-of):

$$\forall X\forall Y\left( {X = Y \to \forall z\left( {z \prec X \leftrightarrow z \prec Y} \right)} \right)$$

The main motivation behind SCAI is its apparent ability to give a straightforward account of the innocence of mereology. In fact, if a whole is identical to the parts it fuses, then insofar as one is committed to the existence of the parts, a commitment to the whole appears to come at no additional ‘ontological cost’.Footnote 8 As it has been argued in the literature, however, non-count-relative SCAI (henceforth, simply ‘SCAI’) gives rise to some serious problems, since it appears to entail the so-called ‘Collapse’ principle, which says that, if an entity x fuses a plurality Y of entities, then something is a part of x if and only if it is one of the Ys (or, more simply, according to which an entity can fuse only the plurality of its parts)Footnote 9:

(Collapse):

$$\forall x\forall Y\left( {xFuY \to \forall z\left( {z \le x \leftrightarrow z \prec Y} \right)} \right)$$

As Sider (2014) and I (Loss 2019) have argued, the culprit behind these problems seems to be the standard plural comprehension principle, according to which if there is at least something that $$\phi$$s, then there is the plurality of the $$\phi$$-ers:

(CMP):

$$\exists x\phi x \to \exists Y\forall x\left( {x \prec Y \leftrightarrow \phi x} \right)$$

Therefore, both Sider (2014) and I (Loss 2019) have suggested that SCAI-theorists should instead employ some ‘weak’ plural comprehension principle entailing, as Sider (2014) puts it, that there are ‘fewer pluralities than one normally expects’ (Sider 2014, p. 213). In particular, I have argued that the best plural comprehension principle for SCAI-theorists is one that entails, jointly with SCAI and the assumption of atomism, that there are only pluralities of mereological atoms (Loss 2019).

The strategy of solving the problems of SCAI by embracing a weak plural comprehension principle may appear to be deeply problematic, at least prima facie. In fact, the kind of ‘restriction’ of the standard plural comprehension principle necessary to avoid all the bad consequences of Collapse appears to be so drastic that it makes plural logic and plural quantification useless for their common purposes such as: the interpretation of second-order logic, the foundation of set theory, or the semantic interpretation of plurals in natural languages. The aim of this paper is to argue that this line of thought can be resisted. As I will argue, SCAI-theorists can simply claim that the plural quantifier by means of which they express their theory is the most ‘joint-carving’ plural quantifier, in the sense of Sider (2011). This, however, appears to be perfectly compatible with the existence of a less-than-maximally-joint-carving plural quantifier obeying the usual, ‘unrestricted’ plural comprehension principle. I will thus conclude that the kind of ontologically innocent mereology promised by SCAI-theorists is not incompatible with plural logic. In the final part of the paper I will also address a couple of objections to this way of understanding SCAI and suggest that, in the same way in which SCAI allows for singular quantification over pluralities of entities (that is, for singular quantifications over things that are identical to pluralities of entities), it should also allow for plural quantification over pluralities of pluralities of entities, or ‘super-pluralities’ (that is, for plural quantifications over pluralities that have proper pluralities as ‘members’).

In what follows I will work against the backdrop of (i) classical mereology, axiomatized by means of the following three principlesFootnote 10:

(Transitivity):

$$\forall x\forall y\forall z\left( {\left( {x \le y \wedge y \le z} \right) \to x \le z} \right)$$

(Weak Supplementation):

$$\forall x\forall y\left( {x < y \to \exists z\left( {z \le y \wedge \sim\,Ozx} \right)} \right)$$

(Universalism):

$$\forall X\exists y\left( {yFuX} \right)$$

and (ii) atomism, thought of as the idea that every entity has atomic parts (‘$$Ax$$’ stands for ‘x is a mereological atom’ and is short for ‘$$\sim\exists y\left( {y < x} \right)$$’)Footnote 11:

(Atomism):

$$\forall x\exists y\left( {y \le x \wedge Ax} \right)$$

2 How to avoid the bad consequences of Collapse

The following plural comprehension principle is usually assumed to be an axiom of plural logicFootnote 12:

(CMP):

$$\exists x\phi x \to \exists Y\forall x\left( {x \prec Y \leftrightarrow \phi x} \right)$$

(CMP) can be used to prove the ‘Covering’ principle, according to which, if x is part of y, there is some plurality of entities W such that y fuses the Ws and x is one of the WsFootnote 13:

(Covering):

$$\forall x\forall y\left( {x \le y \to \exists W\left( {yFuW \wedge x \prec W} \right)} \right)$$

In turn, SCAI and Covering entail the infamous ‘Collapse’ principle, according to which, if x fuses a plurality Y of entities, then something is a part of x if and only if it is one of the Ys (or, more simply, according to which an entity can fuse only the plurality of its parts)Footnote 14,Footnote 15:

(Collapse):

$$\forall x\forall Y\left( {xFuY \to \forall z\left( {z \le x \leftrightarrow z \prec Y} \right)} \right)$$

As it has been shown in the literature, in the presence of the comprehension principle (CMP) the Collapse principle gives rise to some serious problems.Footnote 16 Consider, for instance, the problem that I have called the ‘Walls, Bricks, and Atoms’ problem, or ‘WaBrA’ problem for short (Loss 2019: section 3). Suppose that a certain wall and a certain plurality of bricks are such that every brick is part of the wall and every part of the wall overlaps at least one of the bricks. By the definition of fusion, the wall fuses the plurality of the bricks. However, the wall has also other parts beside the bricks (like, for instance, all of its atomic parts) contra Collapse. Furthermore, Collapse has also been argued to entail mereological nihilism, that is, the claim that nothing has proper parts.Footnote 17 The simplest argument to this effect goes as follows. Take any composite entity x. By (CMP) there is the plurality Y of entities that are identical to x. x clearly fuses the Ys. It follows from Collapse that every part of x is one of the Ys, and therefore, that every part of x is identical to x, so that x has no proper parts. Contradiction! Therefore, nothing has proper parts.Footnote 18

Following Sider (2014), one can take the culprit behind these problems to be (CMP). In fact, on the one hand, (CMP) appears to be necessary to prove Collapse.Footnote 19 On the other hand, it is (CMP) which (implicitly) allows us to infer from the fact that something is a brick that there is the plurality of the bricks, and from the fact that x exists that there is the plurality of the entities that are identical to x. Therefore, the moral of the story in this case seems to be that SCAI-theorists must reject (CMP) and embrace a weaker comprehension principle. Both Sider (2014) and I (Loss 2019) discuss alternative plural comprehension principles on behalf of SCAI-theorists which are formulated by means of the notion of schematic fusion (or ‘S-fusion’):

(S-Fusion):

$$xSFu\phi =_{df} \forall z\left( {\phi z \to z \le x} \right) \wedge \forall z\left( {z \le x \to \exists w\left( {\phi w \wedge Ozw} \right)} \right)$$

According to this definition, an entity x is the S-fusion of everything that $$\phi$$s if and only if, every $$\phi$$-er is a part of x and every part of x overlaps at least some $$\phi$$-er.Footnote 20 Both Sider (2014) and I (Loss 2019) also assume the principle of Unrestricted Schematic Fusion

(USF):

$$\exists y\phi y \to \exists y\left( {ySFu\phi } \right)$$

and formulate our ‘Weak’, ‘Properly Weak’ and ‘Atomic’ comprehension principles as follows (‘$$\phi \left[ {x,y} \right]$$’ is short for ‘$$x \ne y \wedge \phi x \wedge \phi y$$’)Footnote 21

(WCP):

$$\exists x\phi x \to \exists x\exists Y\left( {xSFu\phi \wedge \forall z\left( {z \prec Y \leftrightarrow z \le x} \right)} \right)$$

(PWCP):

$$\exists x\exists y\phi \left[ {x,y} \right] \to \exists z\exists Y\left( {zSFu\phi \wedge \forall x\left( {x \prec Y \leftrightarrow x < z} \right)} \right)$$

(ACP):

$$\exists x\exists y\phi \left[ {x,y} \right] \to \exists z\exists Y\left( {zSFu\phi \wedge \forall x\left( {x \prec Y \leftrightarrow \left( {x < z \wedge Ax} \right)} \right)} \right)$$

As I show in detail (Loss 2019, pp. 7–12), (WCP), (PWCP) and (ACP) all appear to successfully ward off both the WaBrA problem and the threat of mereological nihilism. In fact, on the one hand, the WaBrA problem depends on the possibility that a composite entity fuse different pluralities of entities. However, it is easy to check that (WCP), (PWCP) and (ACP) all entail that a composite entity can fuse at most one plurality of entities: the plurality of its parts, according to (WCP); the plurality of its proper parts, according to (PWCP); the plurality of its atomic parts, according to (ACP). On the other hand, the threat of nihilism stemming from Collapse appears to depend on the possibility of what I call ‘incomplete thick pluralities’ (Loss 2019, p. 5), that is, pluralities of entities containing all the parts of a certain entity except for some of its proper parts (like the ‘improper’ plurality of the entities identical to some composite entity x, or the plurality of entities identical to either x or one of its proper parts y). However, (WCP), (PWCP) and (ACP) all provably entail that there are no incomplete thick pluralities.Footnote 22

It may seem, therefore, that equipped with either (WCP), (PWCP), or (ACP) SCAI is a theory that is at least safe from some of the most serious objections that have been levelled against it. It can be objected, however, that this is but a Pyrrhic victory for SCAI-theorists, as (WCP), (PWCP), and (ACP) all appear to be so weak that they make plural logic useless for most of its common purposes, such as: the interpretation of second-order logic, the foundation of set theory, or the semantic interpretation of plurals in natural languages.Footnote 23 For instance, in the case of the wall and the bricks, it follows from (WCP), (PWCP), and (ACP) that there is no plurality that is the plurality of ‘the bricks’ (that is, the plurality Y such that something is one of the Ys if and only if it is a brick). This means that, there is no plurality available to stand for the property of being a brick, the set of bricks, or indeed for the English phrase ‘the bricks’. (WCP), (PWCP), and (ACP) are, thus, incompatible with the very examples that are put forward in the literature to motivate SCAI, like the idea that a piece of land divided in six parcels is identical to the six parcels taken together, or the idea that a six-pack of orange juice is identical to the six cans of orange juice taken together (Baxter 1988).Footnote 24 It may seem, therefore, that the conclusion one should draw from the arguments put forward by Sider (2014) and me (Loss 2019) is that SCAI ought to be simply rejected as incompatible with plural logic.Footnote 25

As I will argue in what follows, this line of reasoning can be resisted.

3 The most joint-carving plural quantifier

According to Sider (2011), ‘metaphysics, at the bottom, is about the fundamental structure of reality’ (Sider 2011, p. 1), where the idea that reality has a fundamental structure means that there is a ‘privileged description’ of the world—‘an objectively correct way to “write the book of the world” (Sider 2011, p. i). In the same way in which predicates like ‘blue’ and ‘green’ can intuitively be said to be more joint-carving than ‘grue’ and ‘schmue’, for Sider (2011) every expression of a language can be said to be more or less joint-carving and, thus, more or less ‘metaphysically perspicuous’ in this sense: logical connectives, quantifiers, tense and modal operators, et cetera. For instance, Sider embraces ontological realism, understood as the view that there is a perfectly joint-carving singular existential quantifier. However, as he stresses, this is not incompatible with the existence of other, less joint-carving existential quantifiers (like natural language quantifiers, for instance):

Suppose that, fundamentally, there are very few things. Suppose […], for example, that there exist, in the fundamental sense, nothing but subatomic particles. Given such a sparse ontology, the most plausible view about natural language quantifiers might be that they do not carve at the joints. […] So even if there is a joint-carving sort of quantification, the quantifiers of ordinary language might not carve at the joints. (Sider 2011, p. 171)

As Sider (2014) and I (Loss 2019) have argued there appear to be good reasons to take the plural quantifier used by SCAI-theorists to be governed by a comprehension principle that is weaker than (CMP). However, as the existence of a perfectly joint-carving singular quantifier ranging (say) only over subatomic particles doesn’t rule out the existence of other, less joint-carving, singular quantifiers ranging also over tables and chairs, also the existence of a plural quantifier governed by a comprehension principle weaker than (CMP) doesn’t appear to rule out the existence of other plural quantifiers for which (CMP) is valid.Footnote 26

An important question concerning the relation between quantification and the fundamental structure of reality is whether there are any perfectly joint-carving plural quantifiers. Should we take the ‘book of the world’ to be written also by means of plural quantifiers, or should we instead agree with Sider (2011) that, while ‘there is no significant pressure here to admit metaphysically fundamental plural quantification’ (p. 212), the ‘addition of second-order quantifiers to fundamental ideology immensely increases the complexity of one’s fundamental theory’ (p. 209)? Here we don’t need to take a stand on how this issue may be resolved. In fact, whether or not there is any plural quantifier that is perfectly joint-carving,Footnote 27 SCAI-theorists can simply claim (i) that their theory must be understood as employing the most joint-carving plural quantifier, and (ii) that there are other, less joint-carving plural quantifiers which are governed by (CMP) and that can, thus, be adopted for the common applications of plural logic.

I have called ‘Atomic Composition as Identity’ (or ‘ACAI’ for short) the version of SCAI which assumes (ACP) as its plural comprehension principle, entailing, thus, that every composite entity is identical to the plurality of atoms it fuses (Loss 2019). Let us assume in what follows that ACAI is the relevant instance of SCAI. ACAI entails that the plural quantifier by which it is formulated quantifies only over pluralities of atoms.Footnote 28 ACAI-theorists may, thus, embrace a metaontology inspired by Sider (2011) and justify the weakness of (ACP) as follows:

(A1):

There is one singular existential quantifier which is perfectly joint-carving;

(A2):

The most joint-carving singular existential quantifier quantifies only over mereologically atomic entities;Footnote 29

(A3):

The most joint-carving plural quantifier quantifies over all and only pluralities of entities that are quantified over by the most joint-carving singular quantifier;

(A4):

Therefore, the most joint-carving plural quantifier must be thought of as being governed by a comprehension principle weaker than (CMP) and along the lines of (ACP)

(A1) and (A2) are both perfectly legitimate metaphysical positions (even if not completely uncontroversial)Footnote 30 that appear to be perfectly in keeping with ACAI. (A3) has at least the ring of plausibility to it. Quite intuitively, in fact, a plurality of fundamental entitiesFootnote 31 strikes one as being more joint-carving (more natural, less gerrymandered) than a plurality comprising both fundamental and non-fundamental entities. At the same time, it is difficult to see what could possibly justify the idea that only plural quantification over the latter kind of pluralities is possible, and, thus, that the most-joint carving plural existential quantifier does not manage to quantify over pluralities of only fundamental entities.

According to this way of justifying the weakness of (ACP), the central claim of SCAI (according to which every entity that is the fusion of a plurality of entities is identical to them) should be thought of as being formulated by means of a non-perfectly-joint-carving singular quantifier (capable of ranging also over composite entities) and the most joint-carving plural quantifier (ranging only over pluralities of atoms).Footnote 32 Thus understood, SCAI has a kind of ‘reductive’ flavour to it, so to speak, as it identifies composite, metaphysically derivative entities with their most fundamental plural and atomic base.Footnote 33 What is most important, however, is that all this is clearly compatible with the existence of some less joint-carving plural quantifier governed by the ‘unrestricted’ plural comprehension principle (CMP) and quantifying also over pluralities of entities not containing only fundamental atoms. Therefore, in the very same way in which Sider (2011) claims that, although the most joint-carving singular quantifier doesn’t quantify over tables, the ordinary English quantifier does, ACAI-theorists can claim that, although their plural quantifier only quantifies over pluralities of atoms, there is another less joint-carving plural quantifier that is governed by (CMP) and that can, thus, be used for all the other common applications of plural logic. In this sense, although it is true that, fundamentally speaking, there is, for instance, no plurality that is the plurality of the bricks composing the wall (but only the plurality of the atomic parts of the wall), it is nevertheless also true that some plurality of entities are such that something is one of them if and only if it is a brick. It’s just that in the latter case we are not employing the most fundamental plural quantifier, but instead an ‘unrestricted’, less fundamental one. Notice, furthermore, that it also follows from what has been said thus far that there is a sense in which (CMP) is true also for the most joint-carving plural quantifier employed by ACAI-theorists, namely, when (CMP) is formulated also by means of perfectly joint-carving singular quantifiers (‘$${\mathcal{E}}$$’ and ‘$${\mathcal{A}}$$’ stand here for the perfectly joint-carving singular and universal quantifier, respectively)Footnote 34,Footnote 35:

(CMPF):

$${\mathcal{E}}x\phi x \to \exists Y{\mathcal{A}}x\left( {x \prec Y \leftrightarrow \phi x} \right)$$

In other words, the only version of (CMP) that is invalid is the one in which its singular quantifiers are the unrestricted, non-perfectly joint-carving ones, and its plural quantifier is the most joint-carving, restricted one. When both the singular and the plural quantifier used in formulating (CMP) are either unrestricted or the most joint-carving ones, then (CMP) is indeed valid.

Finally, while ACAI-theorists appear to be perfectly in position to justify the weakness of (ACP) by means of an argument like (A1)–(A4), things appear to be more complicated in the case of (WCP) and (PWCP). Let us focus on (WCP) [the reasoning being similar for (PWCP)]. Assuming that the world is endowed with enough mereological structure, (WCP) guarantees the existence of pluralities containing all the parts of some composite entity. Therefore, the counterpart of (A2) in this case would have to allow also for all the parts of some composite entity to be fundamental—a claim that sounds counter-intuitive, at least prima facie. Furthermore, as I have shown (Loss 2019, pp. 12–14), given (WCP) and SCAI the plural quantifier is not guaranteed to be well-behaved. For instance, (WCP) and SCAI taken together do not guarantee that any two pluralities of entities have a plural union.Footnote 36 However, if two pluralities are quantified over by the fundamental quantifier, then—at least on the background of the plausible principle (A3)—the entities belonging to both of them are fundamental entities. So, why shouldn’t the most fundamental plural quantifier be able to plurally quantify over their plural union? It seems, thus, to be more difficult to explain the weakness of (WCP) or (PWCP) in a way that is similar to (A1)–(A4). If this is correct, we may have some additional reason to believe that, as I have suggested (Loss 2019), ACAI is indeed the best version of SCAI.Footnote 37,Footnote 38

4 SCAI, improper pluralities and pluralities of pluralities

Let ‘$$\Sigma$$’ and ‘$$\Pi$$’ stand for the ‘generic’, non-joint-carving plural existential and universal quantifiers governed by (CMP), ‘$$xx$$’, ‘$$yy$$’, ‘$$zz$$’, be ‘generic’ plural variables, and ‘$${ \sqsubset }$$’ stand for the generic ‘one of’ relation. Let’s also keep assuming ACAI as the relevant version of SCAI. ‘$$\Sigma$$’ and ‘$$\Pi$$’ also range over pluralities of atoms. Therefore, ACAI entails in this case the following conditional claim (where ‘$${\mathbb{A}}yy$$’ stands for ‘the yy are a plurality of atoms’ and is short for ‘$$\forall x\left( {x \prec yy \to Ax} \right)$$’; ‘$$xFyy$$’ stands for ‘$$x$$ is a mereological fusion of the $$yy$$’):

(ACAIC):

$$\forall x\Pi yy\left( {\left( {{\mathbb{A}}yy \wedge xFyy} \right) \to x = yy} \right)$$

According to (ACAIC), if an entity x fuses a plurality of atoms, then it is literally identical to them. By using these ‘generic’ plural quantifiers we can, thus, say that, although the wall fuses the bricks of which it is made, it is nevertheless not identical to them. The only plurality of entities the wall is identical to are its ‘atomic footprint’ (Loss 2019, p. 15), that is, the plural union of all the pluralities of entities that are identical to one of the bricks composing the wall. It is, therefore, this fact that accounts for the intuition that the wall is ‘nothing over and above’ the plurality of its bricks (the piece of land is nothing over and above its parcels; the six-pack is nothing over and above the six cans of beer, et cetera; more on this below).

One may object as follows. The generic plural quantifier is thought of as being governed by (CMP) in this sense:

(CMPG):

$$\exists x\phi x \to\Sigma yy\forall x\left( {x{ \sqsubset }yy \leftrightarrow \phi x} \right)$$

(CMPG) entails that there is the improper (generic) plurality having the wall as its only member, (that is, the plurality of the things that are identical to the wall). However, quite independently of SCAI, it appears highly plausible to take an improper plurality of entities to be identical to its only member. After all, contrary to sets, pluralities of entities just are their members: there is nothing ‘more’ to a plurality than its members. It follows, thus, that the wall is identical to both (what we may call) its improper generic plurality and the plurality of its atoms. However, this cannot be the case given the generic version of (One-of) (see Sect. 1):

(One-ofG):

$$\Pi xx\Pi yy\left( {xx = yy \to \forall z\left( {z{ \sqsubset }xx \leftrightarrow z{ \sqsubset }yy} \right)} \right)$$

In fact, both the plurality of atomic parts of the wall and the (improper) plurality of things that are identical to the wall are quantified over by the generic plural quantifier:

(4):

$$\Sigma xx\Sigma yy\left( {\forall z\left( {z{ \sqsubset }xx \leftrightarrow \left( {Az \wedge z \le w} \right)} \right) \wedge \forall z\left( {z{ \sqsubset }yy \leftrightarrow z = w} \right)} \right)$$

Therefore, in the presence of (One-ofG), they cannot be identical, given that they have different members.

What this objection shows is that there is also one kind of generic pluralities whose existence should be rejected by ACAI-theorists, namely, improper generic pluralities. ACAI-theorists should, in other words (i) embrace (CMPG*) instead of (CMPG) as the comprehension principle governing their generic plural quantifier, and (ii) add to their theory an axiom banning improper pluralities from their plural ontology:

(CMPG*):

$$\exists x\exists y\phi \left[ {x,y} \right] \to\Sigma yy\forall x\left( {x{ \sqsubset }yy \leftrightarrow \phi x} \right)$$

(NIPAG):

$$\Pi xx\exists y\exists z\left( {y \ne z \wedge y \prec xx \wedge z \prec xx} \right)$$

Notice, however, that this ‘restriction’ of (CMPG) appears to be harmless, as the pluralities that in this case are ‘missing’ from the plural ontology of ACAI-theorists, so to speak, are still quantified over by their singular quantifier. For instance, according to (NIPAG), there is no plurality xx such that something is one of the xx if and only if it is identical to the wall. However, if such a plurality existed it would be identical to the wall. Yet the wall is still quantified over by the generic singular quantifier of ACAI. Therefore, ACAI-theorists can simply invoke the wall itself anytime the ‘improper plurality of the wall’ is needed by some of the applications of plural logic and plural quantification.

A second important objection to consider is the following one. According to this picture, there is indeed a plurality $$bb$$ of entities (namely, the plurality of the bricks) that are such that a certain brick b is one of them:

(4):

$$\Sigma xx\left( {xx = bb \wedge b{ \sqsubset }bb} \right)$$

At the same time, there is also a certain plurality $$aa$$ of atoms such that b is identical to them:

(5):

$$\Sigma xx\left( {xx = aa \wedge b = aa } \right)$$

We have, thus, that b is not only one of the $$bb$$ but also identical to the $$aa$$:

(6):

$$b{ \sqsubset }bb \wedge b = aa$$

From (6) it follows from Leibniz’s law (which ACAI-theorists, qua SCAI-theorists, accept in its full generality) that the $$aa$$—the atoms to which b is identical—are also one of the $$bb$$—the bricks (where, clearly, the $$aa$$ and the $$bb$$ are two different pluralities, given Leibniz’s law)!

(7):

$$aa{ \sqsubset }bb \wedge aa \ne bb$$

The idea that a plurality of entities may be ‘one of’ another plurality is not new in the literature on higher-order plural quantification.Footnote 39 Notice, however, that strictly speaking (7) doesn’t commit (ACAIC)-theorists to resorting to higher-order plural quantifiers, but only to allowing their (generic) ‘one-of’ relation to admit of plural terms on its left-hand side. ACAI already entails that there is some non-perfectly-joint-carving existential quantifier that can singularly quantify over pluralities of entities, at least in the sense that it can singularly quantify over entities that are identical to (proper) pluralities of entities:

(8):

$$\exists x\exists Y\left( {x = Y \wedge \exists y\exists z\left( {y \ne z \wedge y \prec Y \wedge z \prec Y} \right)} \right)$$

Notice that it follows from (8) and Leibniz’s law that—since x is identical to the Ys and there are two different entities such that each of them is one of the Ys—there is some x and some y such that y is one of x and x is different from y [by (One-ofG)]:

(9):

$$\exists y\exists x\left( {y \prec x \wedge x \ne y} \right)$$

This may come as a surprise, as many authors appear to think that a singular entity can only be one of itself.Footnote 40 However, following my suggestion (Loss 2019: pp. 15–17), ACAI-theorists can simply take the one-of relation to be a proper relation (‘proper’ as in ‘proper parthood’ and ‘proper subset’), and claim that no entity can be (in this sense) one of itself.Footnote 41 At the same time, ACAI-theorists can also insist that, if an entity x is identical to a plurality yy of entities, then, of course, if something is one of the yy it is also one of x (see Loss 2019: pp. 16–17). Therefore, ACAI-theorists can conclude that what (7) shows is only that, in a similar way in which the generic, non-perfectly-joint-carving existential quantifier can singularly quantify over pluralities of entities, so the generic, non-maximally-joint-carving plural quantifier can plurally quantify over (proper)Footnote 42super-pluralities of entities, that is pluralities having (proper) pluralities as members (in what follows ‘$${\mathcal{P}}xx$$’ stands for ‘the xx are a proper plurality of entities’ and is short for ‘$$\exists y\exists z\left( {y \ne z \wedge y{ \sqsubset }xx \wedge z{ \sqsubset }xx} \right)$$’):

(10):

$$\Sigma xx\Sigma yy\Sigma zz\left( {{\mathcal{P}}xx \wedge {\mathcal{P}}yy \wedge xx \ne yy \wedge xx{ \sqsubset }zz \wedge yy{ \sqsubset }zz} \right)$$

Once (8) and (9) are accepted as unproblematic, there appears to be nothing wrong in either (10) or (7).

The introduction of the following notation might help clarify things. Let ‘$$\iota xx.\phi xx$$’ stand for the plural definite description ‘the xx that $$\phi$$Footnote 43 and ‘$$\left[ {x_{1} , \ldots ,x_{n} } \right]$$’ be short for ‘the plurality of things that are identical to either $$x_{1}$$, …, or $$x_{n}$$’:

(TheP):

$$\left[ {x_{1} , \ldots ,x_{n} } \right] =_{df} \iota xx.\forall z\left( {z{ \sqsubset }xx \leftrightarrow \left( {z = x_{1} \vee \ldots \vee z = x_{n} } \right)} \right)$$

Suppose, then, that there are just (i) the three mereological atoms $$a_{1}$$, $$a_{2}$$, and $$a_{3}$$, (ii) the three binary sums of atoms $$b_{1} = a_{1} + a_{2}$$, $$b_{2} = a_{1} + a_{3}$$, and $$b_{3} = a_{2} + a_{3}$$ (the ‘bricks’) and (iii) the fusion c of all the atoms. Given (ACAIC) the following all hold:

(11):

(i) $$b_{1} = \left[ {a_{1} ,a_{2} } \right]$$

(ii) $$b_{2} = \left[ {a_{1} ,a_{3} } \right]$$

(iii) $$b_{3} = \left[ {a_{2} ,a_{3} } \right]$$

(12):

$$c = \left[ {a_{1} ,a_{2} ,a_{3} } \right]$$

It follows from (TheP), (11) and (12) that the plurality of the bricks is identical to the plurality of $$\left[ {a_{1} ,a_{2} } \right]$$, $$\left[ {a_{1} ,a_{3} } \right]$$, and $$\left[ {a_{2} ,a_{3} } \right]$$ taken together:

(14):

$$\left[ {b_{1} ,b_{2} ,b_{3} } \right] = \left[ {\left[ {a_{1} ,a_{2} } \right],\left[ {a_{1} ,a_{3} } \right],\left[ {a_{2} ,a_{3} } \right]} \right]$$

In fact, on the one hand we have that $$\left[ {b_{1} ,b_{2} ,b_{3} } \right]$$ is the plurality of things that are identical to either $$b_{1}$$, $$b_{2}$$, or $$b_{3}$$:

(15):

$$\left[ {b_{1} ,b_{2} ,b_{3} } \right] = \iota xx.\forall z\left( {z{ \sqsubset }xx \leftrightarrow \left( {z = b_{1} \vee z = b_{2} \vee z = b_{3} } \right)} \right)$$

on the other hand, from (11) we have that $$b_{1}$$, $$b_{2}$$, and $$b_{3}$$ are identical to $$\left[ {a_{1} ,a_{2} } \right]$$, $$\left[ {a_{1} ,a_{3} } \right]$$, and $$\left[ {a_{2} ,a_{3} } \right]$$, respectively:

(16):

$$b_{1} = \left[ {a_{1} ,a_{2} } \right] \wedge b_{2} = \left[ {a_{1} ,a_{3} } \right] \wedge b_{3} = \left[ {a_{2} ,a_{3} } \right]$$

(14) can be simply derived by (15) and (16) by Leibniz’s Law, which ACAI-theorists (qua SCAI-theorists) accept in its full generality.Footnote 44 Therefore, (7) can be simply seen as belonging to the family of the prima facie odd claims (ACAIC) are committed to due to Leibniz’s Law, like the claim [expressed by (14)] that $$\left[ {b_{1} ,b_{2} ,b_{3} } \right]$$ are identical to $$\left[ {\left[ {a_{1} ,a_{2} } \right],\left[ {a_{1} ,a_{3} } \right],\left[ {a_{2} ,a_{3} } \right]} \right]$$, or the claim that, since $$a_{1}$$ is part of $$b_{1}$$ and $$b_{1}$$ is identical to $$\left[ {a_{1} ,a_{2} } \right]$$, then $$a_{1}$$ is not only one of $$\left[ {a_{1} ,a_{2} } \right]$$, but also part of $$\left[ {a_{1} ,a_{2} } \right]$$:

(17):

$$a_{1} \le \left[ {a_{1} ,a_{2} } \right]$$

All these results might seem startling at first. At a closer look, however, they shouldn’t come as a surprise once (ACAIC) is accepted: if $$\left[ {a_{1} ,a_{2} } \right]$$ are identical to $$b_{1}$$, then, of course, $$\left[ {a_{1} ,a_{2} } \right]$$ are also one of $$\left[ {b_{1} ,b_{2} ,b_{3} } \right]$$; if $$b_{1}$$, $$b_{2}$$, and $$b_{3}$$ are identical to $$\left[ {a_{1} ,a_{2} } \right]$$, $$\left[ {a_{1} ,a_{3} } \right]$$, and $$\left[ {a_{2} ,a_{3} } \right]$$, respectively, then, of course, $$\left[ {b_{1} ,b_{2} ,b_{3} } \right]$$ (that is, the plurality of the things identical to either $$b_{1}$$, $$b_{2}$$, or $$b_{3}$$) are also identical to $$\left[ {\left[ {a_{1} ,a_{2} } \right],\left[ {a_{1} ,a_{3} } \right],\left[ {a_{2} ,a_{3} } \right]} \right]$$ (that is, to the plurality of the things identical to either $$\left[ {a_{1} ,a_{2} } \right]$$, $$\left[ {a_{1} ,a_{3} } \right]$$, or $$\left[ {a_{2} ,a_{3} } \right]$$); if $$a_{1}$$ is part of $$b_{1}$$ and $$b_{1}$$ is identical to $$\left[ {a_{1} ,a_{2} } \right]$$, then, of course, $$a_{1}$$ is also a part of $$\left[ {a_{1} ,a_{2} } \right]$$.

It can also be straightforwardly proved from (TheP) that the plurality of the bricks is different from the plurality of the atoms:

(18):

$$\left[ {b_{1} ,b_{2} ,b_{3} } \right] \ne [a_{1} ,a_{2} ,a_{3} ]$$

In fact, none of the atoms is identical to a brick and none of the bricks is identical to an atom:

(19):

$$\sim\exists x\left( {x \prec \left[ {a_{1} ,a_{2} ,a_{3} } \right] \wedge \left( {x = b_{1} \vee x = b_{2} \vee x = b_{3} } \right)} \right)$$

(20):

$$\sim\exists x\left( {x \prec \left[ {b_{1} ,b_{2} ,b_{3} } \right] \wedge \left( {x = a_{1} \vee x = a_{2} \vee x = a_{3} } \right)} \right)$$

The idea that the plurality of the bricks is different from the plurality of the atoms might raise two worries: (i) How can the two pluralities be different given that each of the bricks is identical to a plurality of atoms? (ii) Isn’t this result incompatible with the very intuition that lies behind SCAI-theories in general?

As for the first worry, the question boils down to what it means for two pluralities to be identical. In Sect. 1 we encountered the (One-of) principle, of which (One-ofG) is the generic version. (One-of) says that identical pluralities have the same members. However, (ACAIC)-theorists can simply claim that (One-ofG) follows from the very definition of plural identityFootnote 45:

(PI-def):

$$xx = yy =_{df} \forall z\left( {z{ \sqsubset }xx \leftrightarrow z{ \sqsubset }yy} \right)$$

According to (PI-def), what it is for the xx and the yy to be identical is to have the same members. Therefore, once (PI-def) is on board there should be no mystery as to why the plurality of the atoms and the plurality of the bricks are different: they contain different members.Footnote 46

As for the second worry, the wall, the bricks, and the atoms can be said to be the same ‘portion of reality’ in virtue of having the same ‘atomic footprint’ (Loss 2019, p. 15). Letting ‘$${\mathbb{P}}^{x}$$’ stand for ‘the plurality of atoms x is identical to’, the notion of atomic footprint of a singular entity x (‘$$AF\left( x \right)$$’) and the notion of atomic footprint of a plurality $$\left[ {x_{1} , \ldots ,x_{n} } \right]$$ of entities (‘$$AF\left( {\left[ {x_{1} , \ldots ,x_{n} } \right]} \right)$$’) can be defined as follows (where ‘$$\alpha_{1} \cup \cdots \cup \alpha_{n}$$’ stands for the plurality containing, for each $$\alpha_{m}$$ ($$1 \le m \le n$$), either $$\alpha_{m}$$, if $$\alpha_{m}$$ is an atom, or all the members of $$\alpha_{m}$$, if $$\alpha_{m}$$ is a plurality of entities, and nothing else):

(AF(x)-def):

\begin{aligned} & {\text{if}}\;x\;{\text{is}}\;{\text{a}}\;{\text{composite}}\;{\text{entity}},\;AF\left( x \right) =_{df} {\mathbb{P}}^{x} \\ & {\text{if}}\;x\;{\text{is}}\;{\text{an}}\;{\text{atom,}}\;AF\left( x \right) =_{df} x \\ \end{aligned}

(AF(xx)-def):

$$AF\left( {\left[ {x_{1} , \ldots ,x_{n} } \right]} \right) =_{df} AF\left( {x_{1} } \right) \cup \ldots \cup AF\left( {x_{n} } \right)$$

It can be easily checked that the following holds:

(21):

$$AF\left( c \right) = AF\left( {\left[ {b_{1} ,b_{2} ,b_{3} } \right]} \right) = AF\left( {\left[ {a_{1} ,a_{2} ,a_{3} } \right]} \right)$$

Therefore, also (ACAIC) can straightforwardly account for the innocence of mereology. The wall is ‘nothing over and above’ both the atoms and the bricks because, in virtue of being identical to the atoms, the atomic footprint of the wall is identical to the atomic footprint of both the bricks and the atoms.

Notice, finally, that even if (ACAIC)-theorists are committed to the existence of super-pluralities (in the sense specified above) they are not committed to the existence of super-super-pluralities, that is, pluralities having (proper) super-pluralities as members. For instance, the wall c is different from the super-plurality $$\left[ {\left[ {a_{1} ,a_{2} } \right],\left[ {a_{1} ,a_{3} } \right],\left[ {a_{2} ,a_{3} } \right]} \right]$$. Therefore, one cannot consider a plurality having the wall c and a different wall d (having $$\left[ {e_{1} ,e_{2} ,e_{3} } \right]$$ as its atomic footprint) as members and argue that the plurality $$\left[ {c,d} \right]$$ is identical to the super-super-plurality $$\left[ {\left[ {\left[ {a_{1} ,a_{2} } \right],\left[ {a_{1} ,a_{3} } \right],\left[ {a_{2} ,a_{3} } \right]} \right],\left[ {\left[ {e_{1} ,e_{2} } \right],\left[ {e_{1} ,e_{3} } \right],\left[ {e_{2} ,e_{3} } \right]} \right]} \right]$$. More in general, it only follows from (ACAIC) that the only pluralities that are identical to a singular entity are plurality of atoms, which are clearly not super-pluralities. Therefore, although it does follow from (ACAIC) (and the fact that there are composite entities) that there are pluralities containing pluralities of atoms, it doesn’t follow that there are pluralities containing pluralities of pluralities of atoms.

5 Conclusion

According to Sider (2014) and I (Loss 2019), SCAI-theorists must uphold some weak form of the plural comprehension principle. As I have suggested, it is wrong to consider this option to be damning for SCAI-theorists. In fact, at least SCAI-theorists embracing (ACP) as their plural comprehension principle appear to be perfectly in position to claim that, while their theory is cast by means of the most joint-carving plural quantifier (which quantifies only over pluralities of fundamental entities), the standard ‘unrestricted’ plural comprehension principle (CMP) is expressed by means of a less joint-carving plural quantifier. Finally, as I have argued in Sect. 4, ACAI-theorists appear to be forced to allow the existence of ‘super-pluralities’ in their theory, that is, pluralities having proper pluralities as ‘members’. However, this seems to be a price that may be well worth paying in order to have both an innocent mereology and the full power of plural quantification.Footnote 47