The Main Thesis

Both Reference LewisLewis (1991) and Reference LewisLewis (1993) target a theory of sets and classes that is generally known as Morse–Kelley set theory (MK), one whose variables range over individuals and classes. Lewis uses the term ‘class’ to apply to classes with members and construes the empty set as a chosen individual. Sets with members are special cases of classes, which are contrasted with proper classes. In addition to this, he is a compositional monist for whom there is no significant difference between the way in which a subclass is part of a class and the way in which a motherboard, say, is part of a computer. There is just one basic relation of part to whole, which applies across ontological categories in accordance with the axioms of classical mereology.

Reference LewisLewis (1991, 7) defends a succinct answer to the question of what exactly are the parts of a class:

That is, for something to be a part of a class is for it to be a subclass of it. David Lewis arrives at the thesis that the parts of a class are exactly its subclasses in two steps.

There is, first, the First Thesis:

Even if classes are members of other classes, they will not constitute a part of another class unless they are a subclass of it. That is, while the class $\left\{a\right\}$ is a member of the class $\left\{\left\{a\right\},b\right\}$ , the former is not a part of the latter because it is not one of its subclasses. Given the stipulation that classes have members, and the subsequent decision to classify the empty set as an individual rather than a class, notice that nothing in the letter of the thesis requires the empty set to be itself part of a class.

The First Thesis leaves open whether a class may include further parts as well. Lewis’ Second Thesis settles that question:

We list all the parts of a given class once we list all of its subclasses: $\left\{\left\{a\right\}\right\}$ , $\left\{\right\{b\left\}\right\}$ , and $\left\{\left\{a\right\},\left\{b\right\}\right\}$ exhaust the parts of $\left\{\left\{a\right\},\left\{b\right\}\right\}$ . Notice that neither $\left\{a\right\}$ nor the empty set are parts of that class: $\left\{a\right\}$ is not part of it because it is not a subclass but rather a member of $\left\{\left\{a\right\},\left\{b\right\}\right\}$ . The empty set, on the other hand, is not a part of that class because it is not, in fact, a class, given our stipulation that all classes have members.

Lewis derives the Main Thesis from the two theses, but he is careful to mount an argument for the Second Thesis on the basis of more basic mereological hypotheses (on which more later).

Whatever we think of Lewis’ arguments for the Main Thesis, we should acknowledge one important limitation. For given the Main Thesis, we are not in a position to characterize membership in terms of part. The Main Thesis tells us that a class has a decomposition into singletons, which are mereologically simple. To the extent to which they lack further mereological structure, they remain a black box from a strictly mereological standpoint: they are mereologically indiscernible, which means that mereological structure alone will not help us uncover which objects are members of which singletons. One could in principle exchange the contents of these singletons conceived as black boxes from a mereological perspective without making a difference with respect to the relation of part to whole. The exchange would, however, interfere with the relation a member bears to a class.

These informal considerations translate into a more precise observation Joel Hamkins and Makoto Kikuchi record in Reference Hamkins and KikuchiHamkins and Kikuchi (2016, theorem 2). It is clear that $\subseteq$ is definable in terms of $\in$ . If $〈V,\in 〉$ is a model of ZFC, we define the relation $\subseteq$ on $V$ as usual: $x\subseteq y:=\forall z\left(z\in x\to z\in y\right).$ But the question now is whether we may conversely define $\in$ in terms of $\subseteq$ . To refute this claim, we may rely on the general observation that an automorphism on a structure $〈V,\subseteq 〉$ will preserve relations that are definable in $〈V,\subseteq 〉$ . By an automorphism $\tau$ of $〈V,\subseteq 〉$ , we mean, as usual, an isomorphism of the structure onto itself. That is, $\tau$ is a one-to-one map from $V$ to $V$ that preserves $\subseteq$ , that is, $\tau \left(x\right)\subseteq \tau \left(y\right)$ whenever $x\subseteq y$ . It is routine to check that given such an automorphism $\tau$ of $〈V,\subseteq 〉$ , if a binary relation $R$ is definable in $〈V,\subseteq 〉$ , then we have that for all $x,y\in V$ , $xRy$ if, and only if, $\tau \left(x\right)R\tau \left(y\right)$ .Footnote ¹¹ But given a model of ZFC $〈V,\in 〉$ , we may produce an automorphism $\tau$ of $〈V,\subseteq 〉$ which does not preserve $\in$ . That is, there are $x,y\in V$ such that $x\in y$ even though $\tau \left(x\right)\notin \tau \left(y\right)$ . Unfortunately, given a model of ZFC $〈V,\in 〉$ , we may produce an automorphism $\tau$ of $〈V,\subseteq 〉$ which does not preserve $\in$ . That is, there are $x,y\in V$ such that $x\in y$ even though $\tau \left(x\right)\notin \tau \left(y\right)$ .

Proof Outline Given a model of ZFC of the form $〈V,\in 〉$ , we produce an automorphism $\tau$ of $〈V,\subseteq 〉$ , which is not an automorphism of $〈V,\in 〉$ . Let $\theta$ be a permutation of $V$ that exchanges $\varnothing$ with $\{\varnothing \}$ but leaves every other set unchanged. Define $\tau :V\to V$ such that $\tau \left(x\right)=\left\{\theta \left(y\right):y\in x\right\}$ . Since $\theta$ is a permutation of $V$ , $\tau$ is an automorphism on $〈V,\subseteq 〉$ : $\begin{array}{ccccc}x\subseteq y& \text{if and only if}& \left\{\theta \left(z\right):z\in x\right\}\subseteq \left\{\theta \left(z\right):z\in y\right\}& \text{if and only if}& \tau \left(x\right)\subseteq \tau \left(y\right).\end{array}$ But τ does not preserve $\in$ . For notice that although $\varnothing \in \{\varnothing \}$ , $\tau (\varnothing )\notin \tau \left(\{\varnothing \}\right)$ . On the one hand, $\tau (\varnothing )=\varnothing$ , which is the set of $\theta$ -images of members of $\varnothing$ . But on the other hand, $\tau \left(\{\varnothing \}\right)=\left\{\theta (\varnothing )\right\}$ , which is none other than $\left\{\{\varnothing \}\right\}$ . Since $\varnothing \notin \left\{\{\varnothing \}\right\}$ , we conclude that $\tau (\varnothing )\notin \tau \left(\{\varnothing \}\right)$ . It follows that $\tau$ is an automorphism of $〈V,\subseteq 〉$ that does not preserve membership.

More generally, Hamkins and Kikuchi explain how to make changes in the relation $\in$ of a model of ZFC $〈V,\in 〉$ while preserving the same subclass relation. Given a definable nontrivial permutation $\theta$ of $V$ , one may define $x{\in }^{\theta }y$ as $\theta \left(x\right)\in y$ and note that $x{\subseteq }^{\theta }y$ if, and only if, $x\subseteq y$ –even though it is not the case that $x{\in }^{\theta }y$ if, and only if, $x\in y$ . There is nothing special about ZFC, and it is not difficult to verify that the observation generalizes to cover models of theories of classes such as NBG and MK.

The moral seems inescapable: if the Main Thesis is true, then neither set theory nor the theory of classes reduces to mereology alone. We do better to supplement the language of mereology with a primitive singleton operation governed by its own distinctive axioms. Hamkins and Kikuchi acknowledge, for example, that the member relation remains definable in terms of subclass and singleton. They prove, in particular, that a model of set theory $〈V,\in 〉$ is interdefinable with the model $〈V,\subseteq ,\sigma 〉$ , where $\sigma$ is the singleton operator $\sigma :a\mapsto \left\{a\right\}$ , which maps an object to its singleton.Footnote ¹² This observation opens the door to a broad research program, whose goal is to reduce class theory to the combination of mereology and a theory of singletons.

David Lewis had implemented the program in Reference LewisLewis (1991), but the extent to which the framework provides a foundation for class theory is sensitive to one’s attitude toward the singleton operation. One option is to attempt to identify singletons with more familiar objects to which we may have independent epistemic access. David Armstrong, for example, identifies singletons with certain states of affairs in Reference ArmstrongArmstrong (1991), and John Bigelow similarly identifies them with haecceities in Reference BigelowBigelow (1993).Footnote ¹³ Other candidates are not difficult to isolate: self-identity tropes in the manner of D. C. Williams, tropes (Reference ForrestForrest 2002a), and rigid embodiments (Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder 2010).Footnote ¹⁴ In contrast to this, David Lewis dismisses the prospects of a reduction of singletons to more familiar objects on the grounds that they either involve what he regards as unmereological modes of composition or remain profoundly mysterious. Instead, he posits the existence of an operation, which satisfies certain structural axioms from which we can deduce the basic principles of Morse–Kelley class theory.

Hierarchical Composition

One apparently untoward consequence of the Main Thesis is that classes are not composed of their members, which is how we often describe them. We often talk of the members as contained in the class to which they belong, which again suggests that they are regarded as parts. Subscribers to the Main Thesis will presumably construe such talk as nonliteral or as a figure of speech, one that is perhaps grounded in an analogy between member and the relation some atoms bear to the sum they compose. Suggestive as the analogy may be, they may continue, we shouldn’t read too much into it.

One more argument against the hypothesis that classes are composed of members is that unlike part, the member relation is neither reflexive nor transitive: $a$ is a member of $\left\{a\right\}$ , which is a member of $\left\{\left\{a\right\}\right\}$ , but $a$ is not a member of $\left\{\left\{a\right\}\right\}$ . To the extent to which the part-to-whole relation should form a partial order, that is, be reflexive, antisymmetric, and transitive, we have reason to reject the identification of part and member. That isn’t, however, a decisive consideration against the hypothesis that classes have their members as parts. For the suggestion is not that we identify part and member, but rather that we regard cases of membership as special cases of part. Classes exhibit, in fact, a hierarchical mereological structure: they include its members as immediate parts, but they may include other parts as well, for example, members of their members, members of the members of their members, and so on. In other words, the proper parts of a class would include whatever stands in the ancestral of the relation of immediate part to it. These may include classes and individuals, which are objects without members, and while the immediate parts of classes are members, the immediate parts of individuals are not.Footnote ¹⁵

Classes are not the exception but rather the rule. Our body is a complex material object composed of limbs and organs, but these are complex material objects themselves composed of bones and tissue. Even if we count limbs and organs as immediate parts of the body, the bones and tissues that compose these are not immediate parts of the body but rather are mere parts of it. The parts of a body would include not only its immediate parts, but the immediate parts of those, the immediate parts of the immediate parts of those, and so on. While the relation of immediate part is not reflexive or transitive, the reflexive closure of the ancestral of that relation is in fact a partial order, for example, reflexive, antisymmetric, and transitive. So, the reflexive closure of the ancestral of immediate part is a candidate relation of part, for example, Reference FineFine (1999, p. 563).Footnote ¹⁶ This is not to say we should expect this relation to abide by the axioms of classical mereology. Proponents of the hierarchical perspective will depart from classical mereology at crucial junctures, which result in a rather different account of the relation of part to whole. Different variations on the hierarchical outlook are outlined in Reference FineFine (1999), Reference JohnstonJohnston (2006), and Reference KoslickiKoslicki (2008).Footnote ¹⁷

Classes are particularly amenable to the hierarchical perspective. To be sure, each class is part of itself, but in addition to this, each class contains each of its members as an immediate part. Since immediate parts are parts, and the part–whole relation is transitive, each class contains the members of its members as further parts. In other words, the parts of a class include that very class, its members, the members of its members, the members of the members of its members, and so on. Furthermore, if some of these parts are individuals, then the class includes the parts of these individuals as well.

We have implicitly restricted attention to the hierarchical mereological structure of impure classes generated from a given domain of individuals, but it is not obvious how to model pure classes on this approach. For whatever we make of the empty set, it should not have members as immediate parts. One option at this point is to follow Reference LewisLewis (1991) and conceive of the empty set as a chosen individual without immediate parts. The choice would be arbitrary: there is no reason to prefer one individual to another to encode the existence of a set without members. While that is our preference in this Element, there are other alternatives in the market: Reference JohnstonJohnston (2006) suggests the identification of the empty set with the singleton of an arbitrary item, one that is no object in particular. The thought is that there would be no item in particular that would be a member of the empty set as desired. On the other hand, Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder (2010) propose to identify the empty set with a certain attribute, one they take to unify the immediate parts of a class. So, the empty set is not a class but rather an attribute, which, for them, is part of each and every class.

Before we continue the discussion of the hierarchical approach, let us codify the first pass at the hierarchical perspective on classes:

The parts of a class include the class itself, its members, the members of its members, the members of the members of its members, and so on. If some of those members are nonclasses, their immediate parts will be parts of the class as well.

The letter of Hierarchical Composition is neutral with respect to whether immediate part should be taken as basic or whether it should be explained in terms of a more basic relation of part. We have already intimated one reason to prefer the latter option: to the extent to which one may expect the most basic relation of part to form a partial order, one has reason to rule immediate part as the more basic relation of part. While members are parts of classes, being a member is not a basic way of being a part. We will eventually consider an alternative outlook on which we would do better if we adopted immediate part as basic and explained part in terms of it.

One may be tempted to take the further step to regard a class as a sum of its members. It is important to note, however, that the further step is not compulsory: in line with Reference ForrestForrest (2002b), one may subscribe to the letter of Hierarchical Composition and nonetheless deny that a singleton $\left\{a\right\}$ , for example, is a sum of its sole member on the grounds that $a$ itself – as opposed to the singleton – is the join of the sole proper part of $\left\{a\right\}$ . They will, in fact, note that a class is more than the sum of its proper parts.

Others may operate with a conception of sum on which classes are indeed sums of their members. But notice that it is part of the hierarchical layout that some individuals may in fact compose more than one sum. The molecules that compose a body to the extent to which they exemplify a certain pattern of organization compose a class of molecules, which exists to the extent to which they do and regardless of how they are organized. For a simpler example, three individuals $a$ , $b$ , and $c$ may be the immediate parts of the class $\left\{a,b,c\right\}$ as they exist, but they may as well be the immediate parts of a queue as they exemplify a certain spatial arrangement. The difference between the class $\left\{a,b,c\right\}$ and the queue $a$ , $b$ and $c$ lies in the fact that the very same three individuals are unified by different relations in the words of Reference FineFine (1999) or different principles of unity in the words of Reference JohnstonJohnston (2006). One crucial difference between a class of $a$ , $b$ , and $c$ and a queue made out of $a$ , $b$ and $c$ lies in the fact that unlike the latter, the former is guaranteed to exist if the immediate parts exist.

Hierarchical Composition departs from the Main Thesis at least twice over. On the one hand, classes have parts other than its subclasses: in the case of impure classes, they may even include individuals as parts. But on the other hand, and no less importantly, Hierarchical Composition appears to exclude proper subclasses as parts: if a subclass is not an ancestral member of a given class, then it is not part of it, for example, the singleton $\left\{a\right\}$ is ruled out as a part of the class $\left\{a,b\right\}$ . Maybe we should have expected that. Classes are formed out of its members rather than its subclasses, and we should not read too much into the undeniable formal analogy between subclass and part.Footnote ¹⁸ This is the view Reference JohnstonJohnston (2006) appears to endorse.

One of the limitations of the Main Thesis, you may recall, is that there is no definition of member exclusively in terms of part. Hierarchical Composition is subject to a similar limitative constraint. To the extent to which one may be inclined to take immediate part as derivative on a more basic relation of part, one may hope to explain membership in terms of that relation of part. Indeed, if we could define immediate part in terms of part, we would be able to define membership in terms of part and class; to be a member would just be to be an immediate part of a class.

There is, unfortunately, no optimal characterization of immediate part in terms of part. Reference SimonsSimons (1987:108) and Reference Cotnoir and VarziCotnoir and Varzi (2021, sec. 3.3.2) propose to make do with the relation of maximal proper part, where one object is a maximal proper part of another if the former is a proper part of the second, and there is no intermediate proper part between them, that is, no object that includes the former as a proper part is a proper part of the latter. For purposes of illustration, on the strict hierarchical view, $\left\{a\right\}$ , for example, would count as a maximal proper part of $\left\{\left\{a\right\},\left\{b,c\right\}\right\}$ , while $a$ would not on the grounds that there is an intermediate proper part between $a$ and $\left\{\left\{a\right\},\left\{b,c\right\}\right\}$ , namely, $\left\{a\right\}$ . So, the tentative definition of immediate part in terms of maximal proper part would allow us to rule out $a$ as an immediate part of $\left\{\left\{a\right\},\left\{b,c\right\}\right\}$ . Unfortunately, the characterization of immediate part in terms of maximal proper parts is not without problems. Reference FineFine (1992, footnote 16) observes that while $a$ is admittedly an immediate part of $\left\{a,\left\{a\right\}\right\}$ , it is not a maximal proper part of $\left\{a,\left\{a\right\}\right\}$ : $a$ is an immediate part of $\left\{a\right\}$ , which is in turn an immediate part of $\left\{a,\left\{a\right\}\right\}$ .

Kit Fine turned that observation into a formal argument for the thesis that there is no definition of member, $\in$ , in terms of its ancestral, which we write ${\in }^{\infty }$ .Footnote ¹⁹ It is clear that ${\in }^{\infty }$ is definable in terms of $\in$ , since to be an ancestral member of a set $x$ is to be a member of its transitive closure, which is the $\subseteq$ -least set, which contains $x$ and contains every member of every member it contains. But since the reflexive closure of ${\in }^{\infty }$ corresponds to the more basic relation of part on the view under consideration, we would like to know whether we may conversely define $\in$ in terms of ${\in }^{\infty }$ . The style of argument we find in Reference FineFine (1992, footnote 16) is parallel to the one we used to establish that there is no definition of $\in$ in terms of $\subseteq$ . Given a model of ZFC $〈V,\in 〉$ , we outline an automorphism of $〈V,{\in }^{\infty }〉$ that does not preserve $\in$ . That is, there will be $x,y\in V$ such that $x\in y$ even though $\tau \left(x\right)\notin \tau \left(y\right)$ .

Proof Outline Given a model of ZFC of the form $〈V,\in 〉$ , we seek an automorphism $\tau$ of $〈V,{\in }^{\infty }〉$ which is not an automorphism on $〈V,\in 〉$ . Suppose $\tau$ is a permutation of $V$ that exchanges $\left\{\varnothing ,\{\varnothing \}\right\}$ with $\left\{\{\varnothing \}\right\}$ in all sets in which they occur.Footnote ²⁰ Because $\tau$ exchanges each set with another set which has exactly the same ancestral members, the result is, in fact, an automorphism of $〈V,{\in }^{\infty }〉$ . But $\tau$ does not preserve $\in$ . On the one hand, $\varnothing \in \left\{\varnothing ,\{\varnothing \}\right\}$ , but, on the other hand, $\tau (\varnothing )\notin \tau \left(\left\{\varnothing ,\{\varnothing \}\right\}\right)$ , since $\varnothing \notin \left\{\{\varnothing \}\right\}$ .

Proponents of Hierarchical Composition face a dilemma. One horn to cope with the indefinability of member in terms of the basic relation of part is to make do with a surrogate for that relation. That would be in line with the direction of travel Reference ForrestForrest (2002b) outlines, even though he hints at a more liberal view of the mereology of classes. The other horn is to invert the order of explanation and to adopt the relation of immediate part as the more basic relation and to explain part in terms of it. Admittedly, immediate part is neither reflexive nor transitive, which many take to disqualify it as a basic way of being a part. But that is no decisive objection for the development of a hierarchical mereology in terms of immediate part within which part is treated as a less basic relation, one which corresponds to the reflexive closure of the ancestral of immediate part. That is at least one interpretation of the project Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder (2010) undertake. They outline a reduction of ZFC against the background of a mereological framework closely related to the theory of embodiments Reference FineFine (1999) develops, except that unlike Reference FineFine (1999), Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder (2010) explicitly adopt the relation of immediate part as a basic mereological relation.

Hierarchical Composition takes at face value the judgment that classes contain their members as parts, but it ignores the judgment that classes include subclasses as parts. Its proponents are not moved by the structural analogy between the subclass relation and the relation of part to whole, and they recommend to construe the latter judgment as nonliteral or as a figure of speech. That is, however, not a compulsory feature of the hierarchical outlook.

One option at this point is to draw a distinction between two different modes of composition in the spirit of compositional pluralism and to suggest that Hierarchical Composition is concerned with just one of them. Once we do this, we have reason to expand the range of parts we ascribe to a class.

Liberal Hierarchical Composition accommodates the hypothesis that a class may be regarded both as a sum of its members and as a sum of its proper subclasses. Thus $\left\{\left\{a\right\},b\right\}$ is, on the one hand, composed of $\left\{a\right\}$ and $b$ and, on the other, composed of $\left\{\left\{a\right\}\right\}$ and $\left\{b\right\}$ . One difference is that $\left\{a\right\}$ and $b$ are immediate parts of the class, whereas $\left\{\left\{a\right\}\right\}$ and $\left\{b\right\}$ are mere parts of it. These modes of composition behave differently: one generates a class as the union of its proper subclasses, whereas the other generates it directly from its members. While Reference FineFine (2010) takes the first mode of composition as basic and the second as derived, the point remains that there is no unsurmountable obstacle to regarding both the members and the subclasses of a class as parts of that class.

The observation that $\left\{\left\{a\right\},b\right\}$ is, on the one hand, composed of $\left\{a\right\}$ and $b$ and, on the other, composed of $\left\{\left\{a\right\}\right\}$ and $\left\{b\right\}$ is of a piece with the more general observation that sums are often subject to more than one decomposition into proper parts. Since the relation of proper part is transitive, the picture that emerges is one on which the relation of proper part subsumes chains of member and proper subclass: $x$ would be a proper part of a class $y$ if there is a chain $x_1,\cdots ,x_n$ , where $x=x_1$ , and for each $i<n$ , $x_i\in x_{i+1}$ or $x_i\subset x_{i+1}$ , and $x_n=y$ . The class $\left\{a\right\}$ , for example, would be a proper part of $\left\{\left\{a,b\right\},\left\{c\right\}\right\}$ , even if the former is itself neither a member nor a subclass of the latter. It is not difficult to verify that the reflexive closure of this relation, which we write $x⪯y$ , is reflexive and transitive.Footnote ²¹ Reference FineFine (2010) notes that it takes more work to convince ourselves that the hybrid relation is, in fact, antisymmetric.Footnote ²²

The prospects of a definition of member in terms of the hybrid of member and proper subclass don’t look too bright. For the crucial observation stands that given an individual $a$ , the classes $\left\{a,\left\{a\right\}\right\}$ and $\left\{\left\{a\right\}\right\}$ share $a$ and $\left\{a\right\}$ as their sole proper parts and, in fact, remain indistinguishable from the standpoint of chains of member and proper part. In view of this, Reference ForrestForrest (2002b), who appears to conceive of proper part in those terms, proposes to make do with a surrogate for the relation a member bears to a class. One advantage of the surrogate in question is that it admits of a purely mereological characterization against the background of Heyting mereology, on which more later. Reference ForrestForrest (2002b) points out how given the existence of a sufficiently rich and varied domain of individuals, one may, in fact, provide surrogates for pure classes and mimic the theory of classes. In fact, the framework that results provides a hospitable environment for much of mathematics.

Classical Extensional Mereology

Classical Extensional Mereology (CEM) emerges as a natural extension of two weaker systems.Footnote ²⁵ There is, first, Core Mereology (M), which consists of three partial order axioms:

$\begin{array}{cccc}\text{Reflexivity}:& & x\le x& \\ \text{Antisymmetry}:& & x\le y\wedge y\le x\to x=y& \\ \text{Transitivity}:& & x\le y\wedge y\le z\to x\le z& \end{array}$

Reflexivity requires objects to be parts of themselves; antisymmetry bars distinct objects from being mutual parts; and transitivity requires the part-to-whole relation to be transitive: if one object is part of a second, and the second part of a third, then the first object is part of the third.

Different axiomatizations of CEM correspond to different extensions of the axioms of core mereology with further axioms concerned with the question of composition and decomposition, respectively. While some axiomatizations are more elegant than others, we want to highlight the range of options available to a theorist who is prepared to relax the axioms of CEM in response to certain conflicts.

The system M makes sure that part is a partial order but remains neutral as to how sums decompose into further parts. We list three candidate axioms in order of strength:

$\begin{array}{cccc}\text{Weak Supplementation}:& & x<y\to \exists z\left(z\le y\wedge \neg z\circ x\right)& \\ \text{Strong Supplementation}:& & x\nleqq y\to \exists z\left(z\le x\wedge \neg z\circ y\right)& \\ \text{Remainder}:& & x\nleqq y\to \exists z\forall u\left(u\le z\leftrightarrow \left(u\le x\wedge \neg u\circ y\right)\right)& \end{array}$

Weak supplementation tells us that if a whole has a proper part, then the whole has another part disjoint from the first proper part. If a torso is a proper part of a statue, then the statue has some part with no parts in common with the torso, for example, a head. Strong supplementation is similar: if the statue is not part of the torso, then it has some part with no parts in common with the torso. Finally, remainder requires the existence of a unique maximal part of $x$ which is disjoint from $y$ . This remainder may be conceived as the relative complement of $y$ in $x$ , namely, what would remain of $x$ if we deleted $y$ . Neither Weak nor Strong Supplementation guarantees the existence of such a remainder on their own, but they do in the presence of appropriate composition principles, on which more in the following discussion.Footnote ²⁶

Minimal (MM) and Extensional Mereology (EM) are the systems that extend Core Mereology with Weak and Strong Supplementation, respectively.

Given the distinction between join, fusion, and Goodman fusion, there is more than one candidate axiom schema for composition. One of them asserts the existence of a join for each nonempty condition but leaves open whether the join is a fusion, whereas the other two assert the existence of a fusion for each nonempty condition and one of them explicitly requires the fusion to be unique.

$\begin{array}{cccc}\text{Join}:& & \exists x\phi \left(x\right)\to \exists yy=\vee \phi \left(x\right)& \\ \text{Fusion}:& & \exists x\phi \left(x\right)\to \exists yF{u}_{\phi \left(x\right)}y& \\ \text{Goodman Fusion}:& & \exists x\phi \left(x\right)\to \exists yFu{\text{'}}_{\phi \left(x\right)}y& \\ \text{Unique Fusion}:& & \exists x\phi \left(x\right)\to \exists !yF{u}_{\phi \left(x\right)} y& \end{array}$

There is an array of mereological systems, which supplement the axioms of Core Mereology with different axioms for composition and decomposition, respectively. One of them stands out as a candidate standard for mereology, which is, in fact, CEM.

There are at least four different but equivalent characterizations of CEM:

• Core Mereology + Weak Supplementation + Fusion (Reference HovdaHovda 2009)

• Core Mereology + Strong Supplementation + Goodman Fusion (Reference EberleEberle 1970; Reference Casati and VarziCasati and Varzi 1999)

• Core Mereology + Remainder + Join (Reference Cotnoir and VarziCotnoir and Varzi 2019; Reference Cotnoir and VarziCotnoir and Varzi 2021)

• Transitivity + Unique Fusion (Reference Tarski and Corcoran (Ed.)Tarski 1983; Reference LewisLewis 1991)

What these axiomatizations have in common is that they require part to form a complete Boolean algebra (without the zero element).Footnote ²⁷ The distinction between these axiomatizations will eventually be important as they will each suggest different responses to the first limitative constraint we will discuss in the last part of this section.

Classical Extensional Mereology remains neutral with respect to the existence of mereological atoms conceived as objects without a decomposition into proper parts. More precisely, we may tentatively define an object is a mereological atom if, and only if, it is not a sum of proper parts. This is equivalent to the claim that the object lacks proper parts against the backdrop of CEM but not relative to weaker mereological systems introduced in the following discussion.Footnote ²⁸ Atomism is often phrased as the thesis that everything has some atom as a part, which, as Reference Cotnoir and VarziCotnoir and Varzi (2019) report, is in the presence of transitivity and strong supplementation, enough to secure the thesis that everything is a fusion of atoms. One salient atomless alternative is to insist that everything is gunk understood as something whose parts divide forever into further proper parts.

CEM is the framework David Lewis takes for granted in Reference LewisLewis (1991) and Reference LewisLewis (1993). Since the Main Thesis requires singletons to be classes without parts other than themselves, they automatically become mereological atoms.

Heyting Mereology

Hierarchical Composition endows classes with a hierarchical mereological structure: they include members as immediate parts, which may, in turn, come with further immediate parts. On the strict formulation of the hypothesis, there is a distinction between the relation of immediate part that a member bears to a class, and the relation of part corresponding to the reflexive closure of the ancestral of immediate part. On the more liberal formulation of the hypothesis, they include subclasses as further parts. There is a distinction between the relation of immediate part that a member bears to a class, and the relation corresponding to the reflexive closure of a hybrid of the relations of member and proper subclass.

As mentioned earlier, one important question is whether to take immediate part as basic and treat part as derivative or whether to take part as a basic and to derive immediate part from it. One reason to prefer the latter option is that immediate part is not transitive: to be an immediate part of an immediate part of a class, for example, is not sufficient in order to be an immediate part of the class. It is not uncommon to think that a nonnegotiable constraint on a basic relation of part is that it should be transitive.Footnote ²⁹ There are dissenters, of course, but the case for the transitivity of part seems more robust than the case for other mereological principles.Footnote ³⁰

Heyting mereology provides a framework for a transitive relation of part to whole in line with Hierarchical Composition. This is an extension of Core Mereology that has recently been discussed in connection to this and related projects, for example, Reference ForrestForrest (2002b), Reference MormannMormann (2012) and Reference Mormann(2013), and Reference RussellRussell (2016). What makes Heyting mereology distinctive is the rejection of many supplementation principles available in classical mereology. One reason to make do without supplementation principles stems from cases of mereological coincidence: to the extent to which a statue and some clay are made of the same matter, something overlaps one if, and only if, it overlaps the other. If we make the further assumption that the clay is a proper part of the statue, we obtain a counterexample to the principle of weak supplementation: The clay is a proper part of the statue, but the clay overlaps everything the statue overlaps. That is the moral Reference GoodmanGoodman (2022), for example, extracts from the case. Not all friends of coincidence construe the case to be a counterexample to weak supplementation, for example, Reference CotnoirCotnoir (2010) presents the case as a counterexample to antisymmetry: the clay and the statue are mutual proper parts.

Hierarchical Composition motivates a much more clear-cut counterexample to weak supplementation. For consider a singleton $\left\{a\right\}$ of which $a$ is a proper part according to Hierarchical Composition. Since the parts of $\left\{a\right\}$ include $a$ and any parts thereof, every part of $\left\{a\right\}$ must overlap $a$ , pace weak supplementation.

Heyting mereology is a salient fallback for subscribers to Hierarchical Composition. One axiomatization of the system results from Cotnoir and Varzi’s axiomatization of classical mereology when we replace the axiom of remainder with a complete distributivity axiom, which connects the meet of an object and a join with the join of certain meets. The meet of two objects $x$ and $y$ , which we symbolize $x\wedge y$ , is the greatest lower bound under part, which is guaranteed to be unique in the presence of antisymmetry. The complete distributivity axiom now reads:

$x\wedge \vee \phi \left(x\right)=\vee \exists z\left(y=\left(x\wedge z\right)\wedge \phi \left(z\right)\right).$

The models of the system form a Heyting algebra (without a zero element) in which the meet of an object $x$ with a join of the instances of a condition $\phi \left(x\right)$ is the join of the meets of $x$ with each of the instances of $\phi \left(x\right)$ . All complete Boolean algebras satisfy the constraint, but not all Heyting algebras are complete Boolean algebras.

To summarize, Heyting mereology is the following system:

• Core Mereology + Complete Distributivity + Join (Reference ForrestForrest 2002b)

We motivated Heyting mereology as a congenial framework for proponents of the hierarchical outlook who remain reluctant to take immediate part as basic on the grounds that it is neither reflexive nor transitive. It will now be helpful to highlight some of its distinctive features for present purposes.

One important difference between Heyting and CEM lies in the behavior of the complement operation. Let us define complement in terms of join, as usual:

$\begin{array}{ccc}\overline{a}& :=& \vee \neg x\circ a.\end{array}$

CEM proves the identity $\overline{\overline{a}}=a$ , which may fail in Heyting mereology. To be sure, the system proves that $a\le \overline{\overline{a}}$ , but it leaves open whether $\overline{\overline{a}}$ may include parts, which are themselves not part of $a$ . Given Hierarchical Composition, singletons provide a case in point. Since $a\circ \left\{a\right\}$ , the complement $\overline{\left\{a\right\}}$ of a singleton $\left\{a\right\}$ includes neither $a$ nor $\left\{a\right\}$ as parts. It follows that the join corresponding to the condition $\neg x\circ \overline{a}$ must, in fact, include $\left\{a\right\}$ as a part. Since $\left\{a\right\}\nleqq a$ , we conclude $\overline{\overline{a}}\nleqq a$ . One way to put it is that Heyting mereology is to Classical mereology what intuitionistic logic is to classical logic. Much like intuitionistic logic rejects the equivalence between a proposition and the negation of its negation, Heyting mereology rejects the identification of an object with the complement of its complement. In intuitionistic logic, we may not simply assume that the negation of the negation of a proposition entails that proposition; similarly, in Heyting mereology, we may not simply assume that the complement of the complement of a given object includes that object as a part.

One feature of singletons in this framework is that they are more than the sum of their proper parts. For the join of the proper parts of the singleton $\left\{a\right\}$ is just $a$ , which is itself a proper part of $\left\{a\right\}$ . In fact, $a$ is a maximal proper part of $\left\{a\right\}$ , while a proper part of the latter, the former is not a proper part of a proper part of the latter. In line with Reference SimonsSimons (1987: 108) and Reference Cotnoir and VarziCotnoir and Varzi (2021, sec. 3.3.2), we may consider a preliminary characterization of immediate part in terms of maximal proper part:

$x⊲y:=x<y\wedge \neg \exists z\left(x<z\wedge z<y\right).$

Given Hierarchical Composition, each of two individuals $a$ and $b$ would be maximal proper parts of the class $\left\{a,b\right\}$ : they each would be proper parts of $\left\{a,b\right\}$ , and none of them would be proper parts of further proper parts of $\left\{a,b\right\}$ . But whatever its merits, that would still not be the relation of immediate part at play in Hierarchical Composition. For as Reference FineFine (1992) observes, we would like to count $a$ as an immediate part of $\left\{a,\left\{a\right\}\right\}$ even after we acknowledge that it is not a maximal proper part of that class. Given Hierarchical Composition, $a$ is a proper part of $\left\{a\right\}$ , which is, in turn, a proper part of $\left\{a,\left\{a\right\}\right\}$ . Therefore, $a$ is not a maximal proper part of $\left\{a,\left\{a\right\}\right\}$ .

Other relations in the vicinity may approximate the relation of member more closely than that of maximal proper part. One may, for example, take a page from Forrest (2002b) and consider

$xRy:=x⊲y\wedge \neg \exists z\left(x⊲z\wedge z⊲y\right).$

That relation seems closer to immediate part than the relation of maximal proper part. For notice that $\neg aR\left\{a,\left\{a\right\}\right\}$ , since $a⊲\left\{a\right\}$ and $\left\{a\right\}⊲\left\{\left\{a\right\}\right\}$ . One research program in the area becomes to investigate the question of whether such a relation provides us with a mathematically fruitful surrogate for the member relation.

Reference ForrestForrest (2002b) embarks in a similar project. One difference is that his point of departure is closer to Liberal Hierarchical Composition, which means that the first pass at a preliminary characterization of member is the relation of maximal proper part of a part:

$x⊴y:=\exists z\left(x⊲z\wedge z\le y\right).$

Given Liberal Hierarchical Composition, two individuals $a$ and $b$ would be maximal proper parts of parts of the class $\left\{a,b\right\}$ : they each would be maximal proper parts of $\left\{a\right\}$ and $\left\{b\right\}$ , respectively, which are themselves proper parts of $\left\{a,b\right\}$ . But notice that $a$ is, in fact, a maximal proper part of a part of $\left\{\left\{a\right\}\right\}$ , that is, $a$ is a maximal proper part of $\left\{a\right\}$ , which is part of $\left\{\right\{a\left]\right\}$ , even though a is not a member of $\left\{\left\{a\right\}\right\}$ .

One reaction to this is to attempt to make do with another surrogate for the relation of member:

$xEy:=x⊴y\wedge \neg \exists z\left(x⊴z\wedge z⊴y\right).$

Indeed, Reference ForrestForrest (2002b) observes that this relation is in fact mathematically fruitful and supports a framework within which to interpret much of pure set theory against the background of the existence of a sufficiently rich and varied domain of individuals.

Hierarchical Mereology

We described Heyting mereology as a theory of the relation of part to whole within which to mimic a relation of immediate part. The fact that the system does not support a faithful characterization of immediate part in terms of part suggests a different tack. That is to seek an axiomatization of hierarchical mereology as a theory of immediate part within which to characterize part as a derived mereological relation. The plan now, that is, is to take the relation of immediate part as basic and to characterize part in terms of it. That will allow us to do justice to the hierarchical mereological structure of complex objects as formed out of some immediate parts as they are bound by a relation or a principle of unity.Footnote ³¹

What is perhaps the most developed framework for hierarchical mereology comes from Reference FineFine (1999). Fine originally operates in an interpreted first-order language with predicates of different adicities and a primitive predicate $\le$ for part. Fortunately, as Reference Jacinto and CotnoirJacinto and Cotnoir (2019) observe, it is not difficult to reformulate the theory with a primitive predicate for immediate part for which we use the symbol $\ll$ .Footnote ³² The operation of rigid embodiment combines certain objects into a mereological complex, which Fine calls a rigid embodiment whose immediate parts stand in certain relations to each other.

The informal core of the theory of rigid embodiments is this. Given some objects $a,b,c,\dots$ related by a relation $R$ , there is a rigid embodiment $a,b,c,\dots /R$ composed of $a,b,c,\dots$ and $R$ . The objects $a,b,c,\dots$ and the relation $R$ are its immediate parts. The objects $a,b,c,\dots$ are material parts of the embodiment, and $R$ is the principle of embodiment.

A water molecule would be a paradigmatic example of a rigid embodiment, which consists of two hydrogen atoms linked with an oxygen atom by certain covalent bonds. The two hydrogen atoms and the oxygen atom are its material parts, and the link between them is its principle of embodiment.

Reference FineFine (1999) lays down six postulates for rigid embodiments. Two specify their existence and identity conditions, one governs their interaction with location, and three more postulates govern their interaction with the relation of part to whole. Here is a reformulation of the postulates in terms of immediate part.

We may now characterize the relation of part as the reflexive closure of the ancestral of immediate part: $x$ is part of $y$ if, and only if, $x$ is an ancestral immediate part of $y$ or $x=y$ .

One further constraint makes sure that a rigid embodiment has no parts that are not themselves parts of either its immediate material parts or its principle of embodiment.

Some clarification is in order. Notice first that postulates R2 and R5 will play a minimal role in what follows. The purpose of R2 is to secure appropriate locations for rigid embodiments, which are inherited from its immediate parts. Postulate R5 gives expression to the thought that the form that unifies certain objects into an embodiment is itself an immediate part of the embodiment. While that may strike some as an attractive thought, it is not universally accepted by subscribers to hierarchical perspective, for example, Reference JohnstonJohnston (2006). More importantly for our purposes, it seems to be in conflict with the letter of Hierarchical Composition.

The existence and identity postulates should presumably be understood against the background of an independent theory of relations, which constraints the interpretation of the predicates of the language. They are, for example, sensitive to the individuation conditions for relations. To use Fine’s own example, suppose you make a distinction between the relation one object bears to another when the first is placed above the second and the relation one bears to another when the second is placed below the first. That would impose very fine-grained identity conditions for rigid embodiments: the rigid embodiment whose members are $a$ and $b$ when $a$ is above $b$ would be different from the rigid embodiment whose members are $a$ and $b$ when $b$ is below $a$ .Footnote ³³ There is a similar question for the combination of existence and identity, which may be thought to collapse into an inconsistent principle on a theory of relations on which no matter what objects may be, there are related by at least one relation.Footnote ³⁴ Indeed, Reference FairchildFairchild (2017) raises a similar difficulty for the assumption that no matter what an object may be, for every property $F$ the object exemplifies, there is a rigid embodiment of the form $a/F$ .

The theory of rigid embodiments is supplemented with an account of variable embodiments. The core of the theory of variable embodiments is this. Given an individual concept $F$ , there is a variable embodiment $/F/$ , which is manifested by whatever objects are $F$ at different times. Fine takes a car to be a prototypical example of a variable embodiment, one which is at a given time manifested by a rigid embodiment whose immediate members are different car parts such as the chasis, the engine, the wheel, and so on, and whose form is a certain relation in which the different immediate parts stand to each other. He later gives a series of postulates for variable embodiments, which would take us too far afield from our focus. Since hierarchical composition invites the identification of classes of rigid embodiments of a certain sort, and variable embodiments will play no role in the implementation of the proposal.

In order to formulate a perfectly general theory of embodiments, we ascend to a second-order language with a binary relation symbol $\ll$ for immediate part and a range of second-order variables $X^n$ of the same syntactic category as $n$ -place predicates. The ascent to a second-order language is part of what will allow us to offer an explicit definition of part in terms of immediate part in line with Reference Jacinto and CotnoirJacinto and Cotnoir (2019: 933):

$\begin{array}{cc}x\le y:=& \forall X(\left(\forall u\left(x\ll u\to Xu\right)\wedge \forall u\forall t\left(\left(Xu\wedge u\ll t\to Xt\right)\right)\to Xy\right)\vee x=y\end{array}.$

That is, $x$ is part of $y$ if, and only if, $x$ stands in the ancestral of immediate part to $y$ or $x=y$ . Or, equivalently, $x$ stands in the weak ancestral of immediate part to $y$ .

We now regiment the rest of the principles in line with Reference Jacinto and CotnoirJacinto and Cotnoir (2019). We use ${\overrightarrow{x}}_n$ as abbreviations for finite sequences of variables $x_1,\cdots ,x_n$ .Footnote ³⁵ We have omitted the initial quantifiers in order to enhance readability, but notice that we have the resources to express each axiom as a universal generalization.

${\begin{array}{cc}R1& \exists xx={\overrightarrow{y}}_n/R^n\leftrightarrow R^n{\overrightarrow{y}}_n\\ R3& \exists xx={\overrightarrow{y}}_n/R^n\to \left({\overrightarrow{x}}_n/R^n={\overrightarrow{y}}_m/S^m\leftrightarrow \left(\underset{1\le i\le m}{\wedge }{x}_i=y_i\wedge R^n=S^m\right)\right)\\ R4& \exists xx={\overrightarrow{y}}_n/R^n\to \underset{1\le i\le m}{\wedge }{x}_i\ll {\overrightarrow{x}}_n/R^n\\ R5& \exists xx={\overrightarrow{y}}_n/R^n\to R^n\ll {\overrightarrow{x}}_n/R^n\\ R6& \left(x\le {\overrightarrow{y}}_n/R^n\to \underset{1\le i\le m}{\vee }x\le y_i\right)\wedge \left(S\le {\overrightarrow{y}}_n/R^n\to \underset{1\le i\le m}{\vee }{S}^m\le y_i\text{V}{S}^{\text{m}}{R}^n\right)\end{array}}_{}$

This provides a candidate framework for an articulation of Hierarchical Composition similar to the one Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder (2010) undertake.Footnote ³⁶ One apparent limitation of the provisional framework we have outlined is that we have by default restricted attention to rigid embodiments composed of finitely many immediate parts as they exemplify a finitary relation, and one may want to allow for a more liberal form of existence for rigid embodiments. The resources of plural logic will eventually allow us to do just that.

We have considered first-order formulations of each mereological framework, but they receive a more perspicuous formulation against the background of a plural extension of the language.

Classical Extensional Mereology

One important difference between pluralities and fusions is that fusions need not come with a unique decomposition into parts: a fusion of statues is a fusion of torsos, heads, and limbs as much as it is a fusion of statues even if most of those parts are not themselves statues. Matters are different for pluralities, since a plurality of statues consists exclusively of statues and not of torsos, heads, or limbs. In this respect, pluralities behave less like fusions and more like classes.

On occasion, we are able to use fusions to mimic the behavior of classes. Reference LewisLewis (1970) observed that in favorable circumstances, if we use a condition $\theta$ to distinguish some parts of a fusion, we may find that the fusion comes with a unique decomposition into $\theta$ -parts. Fusions of $\theta$ -parts behave much like pluralities of $\theta$ -objects, and the relation a $\theta$ -part bears to a fusion of $\theta$ -parts is perfectly parallel to the relation a $\theta$ -object bears to a plurality of $\theta$ -objects. For ease of exposition, we will write that a condition $\theta$ is a filter if, and only if, a fusion of $\theta$ -parts has a unique decomposition into $\theta$ -parts.

The condition $\forall y\left(y\le x\to x\le y\right)$ , which corresponds to atomicity, is a filter. The fusion of some atoms has a unique decomposition into atoms.Footnote ³⁸ This is because no atom is part of another atom, and no atom is part of a fusion of atoms that do not include it. More generally, since no two atoms overlap, given the definition of fusion, a fusion of some atoms is one and the same as a fusion of others if, and only if, each of the former is one of the latter and vice versa.

Reference LewisLewis (1970) noticed that when the language is expanded to formulate the condition of being a maximal spatiotemporal connected object, we find that maximal spatiotemporal connected parts behaved much like classes of maximal spatiotemporal connected objects and used this observation to explain how to exploit it to mimic the behavior of classes in a purely mereological framework. One limitation of the method is that there is no reason whatever to think that all objects are in fact $\theta$ -objects, whether mereological atoms or maximal spatiotemporal connected parts or what have you.

One may hope to do better if one is able to find a $\theta$ -code for each and every object. That is, one would do better if one could a $\theta$ -code to each object and treat fusions of $\theta$ -parts as surrogates for classes of the objects they encode. That is, in fact, part and parcel of the approach Lewis takes in Reference LewisLewis (1991) where the language of mereology is expanded to include a singleton operation. Since, for Lewis, singletons are atoms, a fusion of singletons has a unique decomposition into singletons and the relation an object bears to a fusion of singletons if its singleton is part of it is akin to the member relation. The singleton operation provides a code for each object, which allows us to use fusions of such codes to play the role of classes.

There is a limitative result closely related to plural Cantor. Given a filter $\theta$ , we write that $\rho$ is a total $\theta$ -coding if, and only if, $\rho$ is an operation that satisfies three conditions:

$\begin{array}{ccc}\text{Totality}:& \exists yy=\rho \left(x\right)& \\ \text{Injectivity}:& \rho \left(x\right)=\rho \left(y\right)\to x=y& \\ \theta \text{-coding}:& \theta \left(\rho \left(x\right)\right)& \end{array}$

Totality makes sure that $\rho$ provides a code for each object, whereas Injectivity ensures that different objects are assigned different codes. The third, and last condition, requires that the codes be $\theta$ -objects. We are now in a position to state the first limitative result:

Proof Outline If $\theta$ is a filter, then a total $\theta$ -coding would enable us to define a binary relation $R$ for which

$\forall xx\exists x\forall y\left(Rxy\leftrightarrow y\prec xx\right).$

Given some objects $xx$ , we would let $x$ be the fusion of their $\theta$ -codes, and we would define:

$Rxy:=\rho \left(y\right)\le x.$

If $y\le xx$ , the $\rho \left(y\right)$ , which is a $\theta$ -code for $y$ would be a $\theta$ -part of $x$ , which is the fusion of $\theta$ -codes of each of $xx$ . On the other hand, if $y$ is a $\theta$ -part of $x$ , which is the fusion of $\theta$ -codes of each of $xx$ , then given the fact that $\theta$ is a filter, we would have that $y$ is one of $xx$ .

The existence of such a binary relation is now in direct conflict with Cantor’s theorem for plural logic.

One corollary of this observation is the observation Gideon Rosen made in Reference RosenRosen (1995), and recently rehearsed in Reference McCarthyMcCarthy (2015), where $\theta \left(u\right)$ is $\forall x\left(x\le u\to x=u\right)$ , that is, $u$ is a mereological atom. One immediate consequence of the Main Thesis is that singletons are atoms, which means that as a corollary, we find that if the Main Thesis is true, then there is no total coding of objects by singletons. The problem is more general, and it generalizes to other approaches on which being a singleton is in fact a filter. We do not evade the difficulty if we simply reject atomicity on the grounds, for example, that all singletons share a common part in line with the suggestions Reference ArmstrongArmstrong (1991) and Reference Caplan, Tillman and ReederCaplan, Tillman, and Reeder (2010) make. The former suggest that singletons include the null set as a further part, whereas the latter suggests that singletons have parts other than classes. But the threat of inconsistency persists provided that we maintain that the condition of being a singleton is a filter and a fusion of singletons continues to have a unique decomposition into singletons.Footnote ³⁹

Hierarchical Mereology

We now look at a plural variation on the theory of rigid embodiments introduced earlier. We expand a first-order language with a primitive predicate $\ll$ for immediate part into a plural language supplemented with a stock of plural predicates $P,Q,\dots$ with a plural argument. These plural predicates express attributes some objects may exemplify. One may now reformulate some of Fine’s postulates in the enriched plural language. The plural formulation of the existence postulate posits a rigid embodiment whenever some objects $xx$ exemplify some attribute $X$ :

PR1. (Plural Existence): $xx/P$ exists at $w$ if, and only if, $Pxx$ at $w$ .

PR2. (Location): If $xx/P$ exists at $w$ , then $xx/P$ is located at a point $p$ in $w$ if, and only if, at least one of $xx$ is located at $p$ in $w$ .

PR3. (Plural Identity): $xx/P=yy/Q$ if, and only if, $xx\approx yy$ , and $P=Q$ .

This last postulate is, of course, sensitive to what we take to be the identity conditions for attributes. One option is to deem $P$ and $Q$ to be identical if they are necessarily coextensive in a suitably expanded language with a modal operator $\square$ for metaphysical necessity:

$\square \forall xx\left(Pxx\leftrightarrow Qxx\right)\to P=Q.$

More fine-grained views of the identity conditions for attributes would require yet a different interpretation of the postulate, but the difference is not important for present purposes.

The next principle specifies the immediate parts of a rigid embodiment:

PR4. (Immediate Part): Each of $xx$ are immediate parts of $xx/P$ .

We do not offer a plural formulation of the fifth postulate because nothing in the letter of Hierarchical Composition requires a plural attribute $P$ to be an immediate part of the rigid embodiment $xx/P$ .

We continue to understand part as the weak ancestral of part, which will now be couched in plural terms. Armed with a definition of part in terms of proper part, we formulate one final constraint on the natura of rigid embodiments according to which the parts of the immediate parts of a rigid embodiment exhaust the range of its parts:

PR6. (Part) A part of $xx/P$ is a part of one of $xx$ .

Here is the official definition of part in terms of immediate part:

$\begin{array}{cc}x\le y:=& \forall xx(\left(\forall u\left(x\ll u\to u\prec xx\right)\wedge \forall u\forall t\left(\left(u\prec xx\wedge u\ll t\to t\prec xx\right)\right)\to y\prec xx\right)\vee x=y\end{array}.$

What follows is the plural formulation of the four postulates we adopt as part of our official formulation of hierarchical mereology:

$\begin{array}{ccc}PR1& & \exists xx=xx/P\leftrightarrow Pxx\\ PR3& & \exists xx=xx/P\to \left(xx/P=yy/Q\leftrightarrow xx\approx yy\wedge P=Q\right)\\ PR4& & \exists xx=xx/P\to \forall x\left(x\prec xx\to x\ll xx/P\right)\\ PR6& & x\le xx/P\to \exists y\left(y\prec xx\wedge x\le y\right)\end{array}$

The combination of the existence and identity postulates now places us on the brink of inconsistency on abundantist views of attributes. For let $E$ be a plural attribute some objects exemplify just in case each of them exists:

$\square \left(Exx\leftrightarrow \forall x\left(x\prec xx\to \exists yx=y\right)\right).$

No matter what some objects may be, such a plural attribute will bind them into a rigid embodiment $xx/E$ , which, given PR1, exists whenever the objects in question exist. But given PR2, unless some objects $xx$ coincide with some objects $yy$ , the rigid embodiments $xx/E$ and $yy/E$ , respectively, will differ from each other, $xx/E\ne yy/E$ .

Proof Outline The plural formulation of the existence and identity postulates enable us to define a binary relation $R$ for which