## Abstract

In this paper I address two important objections to the theory called ‘(Strong) Composition as Identity’ (‘CAI’): the ‘wall-bricks-and-atoms problem’ (‘WaBrA problem’), and the claim that CAI entails mereological nihilism. I aim to argue that the best version of CAI capable of addressing both problems is the theory I will call ‘Atomic Composition as Identity’ (‘ACAI’) which consists in taking the plural quantifier to range only over proper pluralities of mereological atoms and every non-atomic entity to be identical to the (proper) plurality of atoms it fuses. I will proceed in three main steps. First, I will defend Sider’s (in: Baxter D, Cotnoir A (eds) Composition as identity. Oxford University Press, Oxford, pp 211–221, 2014) idea of weakening the comprehension principle for pluralities and I will show that (*pace* Calosi in Philos Q 66(263):219–235, 2016a) it can ward off both the WaBrA problem and the threat of mereological nihilism. Second, I will argue that CAI-theorists should uphold an ‘atomic comprehension principle’ which, jointly with CAI, entails that there are only proper pluralities of mereological atoms. Finally, I will present a novel reading of the ‘one of’ relation that not only avoids the problems presented by Yi (J Philos 95:163–190, 1999a, in: Baxter D, Cotnoir A (eds) Composition as identity. Oxford University Press, Oxford, pp 169–191, 2014) and Calosi (Log Log Philos 25(3):429–443, 2016b, Am Philos Q 55(3):281–292, 2018) but can also help ACAI-theorists to make sense of the idea that a composite entity is both one and many.

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## Change history

### 05 March 2020

(PWCP) and (ACP) in Sect. 5 should be reformulated as follows.

## Notes

- 1.
‘[…] perhaps the major motivation for CAI is that it implies the ‘ontological innocence’ of classical mereology’ (Cotnoir 2014: p. 7). ‘If Lewis’s claim were that the fusion is

*literally identical*to the cats that compose it, he would clearly be entitled to ontological innocence’ (Bennett 2015: p. 256). ‘[…] the thought that a fusion is numerically identical to the things that compose it*taken together*[…] would vindicate the intuition that such double countenancing is ultimately redundant, hence the innocence thesis’ (Varzi 2014: p. 49). ‘But why think that mereology is ontologically innocent? If composition is identity, then ontological innocence is secured’ (Hawley 2014: p. 72). - 2.
Notice that, for the ease of expression, I will sometimes use ‘plurality’ and plural terms like ‘

*X*’, ‘*Y*’ and ‘*Z*’ as grammatically singular and say things like ‘there is a plurality*X*such that*it*is such-and-such’. - 3.
This definition of fusion appears to be the most common in the debate that is relevant to this paper. See, for instance, Yi (1999a: p. 143; 2014: p. 183), Sider (2007: p. 52; 2014: p. 212), Calosi (2016a: p. 221; b: p. 3; 2018: p. 282), and Loss (2018: p. 370). For a discussion of alternative definitions of mereological fusion see,

*inter alia*, Hovda (2009) and Varzi (2016: Section 4). - 4.
See Hovda (2009) for alternative axiomatizations of Classical Mereology.

- 5.
- 6.
See Linnebo (2017: Sect. 1.2). Without this assumption the mereological axiom of universalism (see p. 3) would have to be in conditional form: \(\forall X(\exists z(z \prec X) \to \exists y(yFuX)).\)

- 7.
- 8.
See, for instance, Linnebo (2017: Sect. 1.2).

- 9.
Suppose that, for some

*x*and*y*, \(x\) is part of \(y\). It follows that that there is something that is identical to either*x*or \(y\). By (CMP) we have, thus, that there is a plurality \(W\) of entities such that something is one of the*W*s if and only if it is identical to either*x*or*y*. Therefore,*x*is one of the*W*s. From the definition of fusion it follows that*y*fuses the*W*s.*□* - 10.
Suppose that

*x*fuses the*Y*s. By the definition of fusion we have that if an entity*z*is one of the \(Y\)s, then*z*is part of*x*. Conversely, if*z*is part of*x*, it follows from (PC) that there is some plurality \(W\) such that*x*fuses the*W*s and*z*is one of the*W*s. By CAI,*x*is identical to both the*W*s and the*Y*s. Therefore,*W*and*Y*are identical. By (LLI), they have the same members, so that*z*is also one of*Y*s.*□* - 11.
- 12.
- 13.
I take here a

*thick plurality*to be any plurality*Y*of entities such that, for some*x*, (i)*x*is one of the*Y*s and (ii)*x*fuses the*Y*s. A*complete*thick plurality can thus be taken to be any plurality*Y*of entities such that, for some*x*, (i)*x*is one of the*Y*s, (ii)*x*fuses the*Y*s, and (iii) for every*z*,*z*is one of the*Y*s if and only if*z*is a part of*x*. - 14.
- 15.
Suppose

*y*is a proper part of*x*and consider the property*P*= ‘being a proper part of*x*’.*y*clearly has*P*. However,*x*doesn’t have*P*(nothing is a proper part of itself), thus contradicting Duplication. - 16.
Notice that for the second case Calosi’s argument relies on the existence of different pluralities fused by the same composite entity, which makes it relevantly similar to the WaBrA scenario: ‘[…] assume that there is a property

*P*that at least one of the proper parts of*x*, let’s say*y*, has but*x*has not. Consider the plurality \(W_{2}\) of things that have the following property: “being part of*x*and having a*P*-part”. […] Hence \(xFuW_{2}\). […]. Now consider the plurality \(W_{3}\) of*P*-parts of*x*. Once again, \(xFuW_{3}\) […]’ (Calosi 2016a, b: p. 227). Here \(W_{2}\) and \(W_{3}\) are different pluralities, since*x*is one of the \(W_{2}\)s but not of the \(W_{3}\)s, and yet*x*fuses both. - 17.
- 18.
See Loss (2018: p. 372)

- 19.
In fact, without (CMP) there is no guarantee in the proof of Plural Covering that there is a plurality

*W*of entities such that something is one of the*W*s if and only if it is identical to either*x*or*y*(see footnote 9). - 20.
Notice that there are two (non-exclusive) ways in which the assumption of a weaker comprehension principle may block the problems of Collapse: (i) by invalidating Collapse; (ii) by blocking the arguments from Collapse. Sider’s weak comprehension principle (see below) actually validates Collapse, and yet it blocks the argument from Collapse by excluding the existence of the problematic pluralities.

- 21.
*Proof*. Consider an arbitrary entity*x*. Let \(\phi\) e \(v\)‘part of*x*’. By (WCP) there is a plurality*Y*such that the*Y*s are all the parts of*x*. By the definition of fusion,*x*fuses the*Y*s. By CAI,*x*identical to the*Y*s. Suppose that, for some plurality*W*,*x*fuses the*W*s. By CAI*x*is identical to the*W*s. Hence, the*Y*s are identical to the*W*s.*□* - 22.
To be clear, what does

*not*follow from (i) and (ii) is the claim (where ‘\(R\)’ stands for ‘is a red atom’):\(\left( * \right)\quad \;\exists X\forall y\left( {y \prec X \leftrightarrow Ry} \right)\)

- 23.
*Proof*. Suppose that*Y*is an incomplete thick plurality. This means that, for some*x*, (*a*)*x*is one of the*Y*s, (*b*)*x*fuses the*Y*s, and for some*z*, (*c*)*z*is a proper part of*x*and yet*z*is not one of the*Y*s (see Sect. 3). Consider, now the set*s*of things that are one of the*Ys*.*Y*clearly corresponds to*s*. Since*x*is one of the*Y*s, (i)*x*is a member of*s*. From the fact that*x*fuses the*Y*s it follows that each of the*Y*s is part of*x*. Therefore, (ii) every member of*s*is part of*x*. Some proper part of*x*is not one of the*Y*s, so (iii) it is also not a member of*s*.*s*is, thus, an incomplete thick set.*□* - 24.
Notice, that if

*s*is an incomplete thick set and*x*complies with (i)–(iii), then*x*‘set-fuses’*s*:- (Set-Fusion):
\(xSetFs =_{df} \forall y\left( {y \in s \to y \le x} \right) \wedge \forall y\left( {y \le x \to \exists z\left( {z \in s \wedge Ozy} \right)} \right)\)

- 25.
The proof is very similar to the proof that (WCP) + CAI entail (SR1) and is, thus, left to the reader.

- 26.
Alternatively, one may \(\Sigma\)-weaken the plural quantifier so that it quantifies only over pluralities corresponding to the atomic proper

*and improper*parts of a certain entity, thus admitting also improper pluralities of atoms in one’s ontology. The reason I am not pursuing this strategy will be clear in Sect. 6 (footnote 37). - 27.
The proof is very similar to the proof that (WCP) + CAI entail (SR1) and is, thus, left to the reader.

- 28.
(ATI-2) immediately excludes the possibility of different pluralities fused by the same entity, while from CAI + (PWCP) it follows that every composite entity is the fusion of only the plurality of its proper parts. From (SR2) it follows that there are only pluralities containing all the proper parts of certain entities, thus ruling out the existence of incomplete thick pluralities. Instead, (SR3) excludes the existence of incomplete thick pluralities by entailing that no composite entity can be a member of a plurality.

- 29.
Instead, (PQ3) appears to be validated by CAI + (WCP) but not by CAI + (PWCP).

*Proof*. Suppose that \(X \triangleright s\), \(Y \triangleright r\), and \(C\left( {s\mathop \cap \nolimits r} \right)\). Letting \(\phi\) be ‘\(v \in s\mathop \cap \nolimits r\)’, it follows by (USF) that some entity*k*S-fuses everything that is a member of \(s\mathop \cap \nolimits r\). It can be proved that, for every*z*,*z*is part of*k*if and only if it is a member of \(s\mathop \cap \nolimits r\) (The proof is left to the reader [*Hint*: if \(s = r\), the proof is trivial; if \(s \ne r\), then both (SR1) and (SR2) entail that*k*is the mereological product—see below—of the fusion of the*X*s with the fusion of the*Y*s]). Assuming (WCP), there is a plurality*W*such that*W*is the plurality of the*proper and improper*parts of*k*. By (COR),*W*corresponds, thus, to \(s\mathop \cap \nolimits r\).*Instead*, from (SR2) we have that there is a plurality corresponding to \(s\mathop \cap \nolimits r\)*only if*there is some entity*z*such that \(s\mathop \cap \nolimits r\) is the set of all the*proper*parts of*z*. Suppose such an entity exists.*z*is not a proper part of*z*and is, thus, not a member of \(s\mathop \cap \nolimits r\). It follows, therefore, that either*z*is not a part of*x*, or*z*is not a part of*y*. However, every part of*z*overlaps both*x*and*y*so that, by Strong Supplementation (see the “Appendix”),*z*is part of both*x*and*y*. Hence,*z*is a member of \(s\mathop \cap \nolimits r\).*Contradiction*! Therefore, if (PWCP) is assumed, there is no plurality corresponding to \(s\mathop \cap \nolimits r\).*□* - 30.
I assume in what follows that both singular and plural (see below) definite descriptions are eliminable using Russell’s theory of descriptions:

- (RT1):
\(F(\iota x.\phi x) =_{df} \exists x\left( {\phi x \wedge \forall y\left( {\phi y \to y = x} \right) \wedge Fx} \right)\)

- (RT2):
\(P\left( {IX.\phi X} \right) =_{df} \exists X\left( {\phi X \wedge \forall Y\left( {\phi Y \to Y = X} \right) \wedge P\left( X \right)} \right)\)

- 31.
Notice that it follows directly from ACAI that, for every non-atomic

*x*, something is one of the \({\mathbb{P}}^{x}\)s if*and only if*it is an atomic part of*x*or, in other words, that \({\mathbb{P}}^{x}\) is the plurality of the*atomic parts*of*x*:\(\left( {{\mathbb{P}}{\text{A}}} \right)\quad \;\forall x\left( { \sim Ax \to \forall z\left( {{\mathbb{P}}^{x} z \leftrightarrow \left( {Az \wedge z \le x} \right)} \right)} \right)\) - 32.
*Proof*. See the “Appendix”. - 33.
*Proof*. See the “Appendix”. - 34.
(MP2) and (MP3) are validated by CAI + (WCP) but not by CAI + (PWCP). The proof is left to the reader (see, however, footnote 29 for (MP3):

*k*is the product of*x*and*y*, yet only according to CAI + (WCP) the plural intersection of \({\mathbb{P}}^{x}\) and \({\mathbb{P}}^{y}\) is the plurality to which*k*is identical). - 35.
The proof is straightforward and is, thus, left to the reader.

- 36.
See below on the (admittedly awkward sounding) phrase ‘

*x*is one of*y*’. - 37.
Since, according to ACAI, there are no improper pluralities it follows that atoms are not fusions of any plurality. Notice that an atomic \(\Sigma\)-weakening allowing also the existence of improper pluralities of atoms would be incompatible with (OF4-5) given that—quite independently from CAI—it is highly plausible to think that an improper plurality is identical to its only member. In this case, one may reformulate (OF4-5) as follows and allow atoms to be one of themselves:

- (OF4*):
\(\forall x\forall \alpha \left( {x \prec \alpha \to \left( {x \ne \alpha \vee A\alpha } \right)} \right)\)

- (OF5*):
\(\forall x\left( {x \prec x \to Ax} \right)\)

Although this route seems to be viable (at least

*prima facie*) the asymmetry it introduces between entities that can be one of themselves (atoms) and entities that cannot be one of themselves (composite entities) strikes me as unnatural. It appears to be more natural to think of the notion of being properly one-of by analogy to other familiar ‘proper’ notions, like that of proper part and proper subset, which are irreflexive across the board, so to speak (see below on the possibility to define an improper one-of notion on the basis of the proper one). - 38.
Yi (1999b: p. 147; 2014: p. 178) advances also a version of the second argument presented in this section based on a relation \({\mathbf{H'}}\)‘’Yi 2014: p. 178) defined on the basis of the ‘is one of’ relation as follows (‘\(\left[ {x_{1} , \ldots ,x_{n} } \right]\)’ stands for ‘the plurality of entities that are identical to either \(x_{1}\), ..., or \(x_{n}\)):

- (H′-df):
\(x{\mathbf{H}}'\alpha =_{df} \forall y(x \prec \left[ {y,\alpha } \right]\)

However, since for ACAI no plurality contains a composite entity, there is no composite entity

*x*that for ACAI is such that \(x{\mathbf{H}}'x\) so that Yi’s second version of the second argument presented in this section is also blocked. - 39.
According to (OF5), nothing can be one of itself. Therefore, \(a\) cannot be

*one*in the sense of (O1). - 40.
In fact, \(\exists y\forall z\left( {z \prec a \leftrightarrow z = y} \right)\) contradicts (OF4), while \(\exists y\forall z\left( {z \prec W \leftrightarrow z = y} \right)\) would entail that

*W*is an improper plurality,*contra*ACAI. - 41.
To be fair, Cotnoir leaves it actually open whether the ultimate entities in his theory are atoms: ‘You may think of them as atoms, but one

*needn’t*think of them as atoms. […] Another possible interpretation—to be explored in future work—is to think of them merely as propertied spacetime points. I largely leave the underlying metaphysics open, since the semantic approach endorsed here is compatible with a number of metaphysical views.’ (Cotnoir 2013: p. 302). - 42.
See footnote 31.

## References

Bennett, K. (2015). Perfectly understood, unproblematic, and certain: Lewis on mereology. In B. Loewer & J. Schaffer (Eds.),

*A companion to David Lewis*(pp. 250–261). Hoboken: Wiley.Calosi, C. (2016a). Composition is identity and mereological nihilism.

*The Philosophical Quarterly,**66*(263), 219–235.Calosi, C. (2016b). Composition, identity, and emergence.

*Logic and Logical Philosophy,**25*(3), 429–443.Calosi, C. (2018). Failure or boredom: The pendulum of composition as identity.

*American Philosophical Quarterly,**55*(3), 281–292.Cotnoir, A. (2013). Composition as general identity. In K. Bennett & D. Zimmerman (Eds.),

*Oxford studies in metaphysics*(Vol. 8, pp. 295–322). Oxford: Oxford University Press.Cotnoir, A. (2014). Composition as identity: Framing the debate. In D. Baxter & A. Cotnoir (Eds.),

*Composition as identity*(pp. 3–23). Oxford: Oxford University Press.Gruszczyński, R. (2015). On mereological counterparts of some principle[s] for sets.

*Logique et Analyse,**232,*535–546.Hawley, K. (2014). Ontological innocence. In D. Baxter & A. Cotnoir (Eds.),

*Composition as identity*(pp. 70–89). Oxford: Oxford University Press.Hovda, P. (2009). What is classical mereology?

*Journal of Philosophical Logic,**38*(1), 55–82.Hovda, P. (2014). Logical considerations on composition as identity. In D. Baxter & A. Cotnoir (Eds.),

*Composition as identity*(pp. 192–210). Oxford: Oxford University Press.Lewis, D. (1991).

*Parts of classes*. Oxford: Blackwell.Linnebo, Ø. (2017). Plural quantification. In E. Zalta (Ed.),

*The Stanford encyclopedia of philosophy*(Summer 2017 Edition). URL https://plato.stanford.edu/archives/sum2017/entries/plural-quant/. Accessed April 2018.Loss, R. (2018). A sudden collapse to nihilism.

*The Philosophical Quarterly,**68,*370–375.Sider, T. (2007). Parthood.

*Philosophical Review,**116,*51–91.Sider, T. (2013). Against parthood. In

*Oxford studies in metaphysics: volume 8*(pp. 237–293). Oxford: Oxford University Press.Sider, T. (2014). Consequences of collapse. In K. Bennet & D. Zimmerman (Eds.),

*Composition as identity*(pp. 211–221). Oxford: Oxford University Press.Varzi, A. (2014). Counting and countenancing. In D. Baxter & A. Cotnoir (Eds.),

*Composition as identity*(pp. 47–69). Oxford: Oxford University Press.Varzi, A. (2016). Mereology, In E. Zalta (Ed.),

*The Stanford encyclopedia of philosophy*(Winter 2016 Edition)., URL https://plato.stanford.edu/archives/win2016/entries/mereology/. Accessed April 2018.Yi, B. (1999a). Is two a property?

*Journal of Philosophy,**95,*163–190.Yi, B. (1999b). Is mereology ontologically innocent?

*Philosophical Studies,**93,*141–160.Yi, B. (2014). Is there a plural object? In Donal Baxter & Aaron Cotnoir (Eds.),

*Composition as identity*(pp. 169–191). Oxford: Oxford University Press.

## Acknowledgements

I am very grateful to four anonymous referees for this journal for very useful comments that greatly improved the paper. Special thanks to Claudio Calosi for discussions on this and related topics.

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## Appendix

### Appendix

### ACAI entails (MP1-3)

ACAI entails (L1):

- (L1):
\(\forall x\left( \left( { \sim Ax \wedge \forall z\left( {z \le x \leftrightarrow \phi z} \right)} \right) \to {\mathbb{P}}^{x} = IW.\forall z\left( {Wz \leftrightarrow \left( {Az \wedge \phi z} \right)} \right) \right)\)

###
*Proof*

Suppose that *x* is not an atom and that something is a \(\phi\)-er if and only if it is a part of *x*. By (AT), every \(\phi\)-er has some atomic part. By (ACP), there is, thus, a plurality *W* such that something is one of the *W*s if and only if it is an atomic \(\phi\)-er. Each of the *W*s is a \(\phi\)-er and is, thus, part of *x*. Therefore, every atomic \(\phi\)-er is part of *x*. By (AT), every part of *x* has atomic parts. By transitivity, every atomic part of a part of *x* is an atomic part of *x* and is, thus, an atomic \(\phi\)-er. Therefore, every part of *x* overlaps an atomic \(\phi\)-er. By the definition of fusion, *x* fuses the *W*s. By ACAI, *x* is identical to the *W*s. By (ATI-2), \({\mathbb{P}}^{x}\) is identical to *W*. *□*

(T1) is a theorem of classical mereology that follows from the definition of mereological sum and Strong Supplementation (which is itself a theorem of classical mereology):

- (Strong Supplementation):
\(\forall x\forall y\left( {x{ \nleqslant }y \to \exists z\left( {z \le x \wedge \sim Ozy} \right)} \right)\)

- (T1):
\(\forall x\forall y\forall z\left( {x = y + z \to \forall w\left( {w \le x \leftrightarrow \forall k\left( {k \le w \to \left( {Oky \vee Okz} \right)} \right)} \right)} \right)\)

###
*Proof*

Suppose that *x* is the sum of *y* and *z*. (*a*) Let *w* be an arbitrary part of *x*. By the definition of sum, every part of *x* overlaps either *y* or *z*. By the transitivity of parthood, every part of *w* overlaps either *y* or *z*. (*b*) Let *w* be an entity such that every part of *w* overlaps either *y* or *z* and suppose that *w* is not part of *x*. Since *y* and *z* are parts of *x* it follows that every part of *w* overlaps *x*. By Strong Supplementation, *w* is part of *x*. *Contradiction*! Therefore, something is part of *x* if and only if all of its parts overlap either *y* or *z*. *□*

The left-to-right directions of (MP1-3) can be, thus, proved as follows (*x*, *y* and *z* are thought of as ranging over *non*-*atomic* entities):

- (MP1-lr):
\(x = y + z \to {\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\)

###
*Proof*

Suppose that *x* is the sum of *y* and *z*. By (T1), something is part of *x* if and only if all of its parts overlap either *y* or *z*. Letting \(\phi\) be ‘\(\forall w\left( {w \le v \to \left( {Owy \vee Owz} \right)} \right)\)’ it follows from (L1) and the definition of mereological sum that \({\mathbb{P}}^{x}\) is the plurality of atoms overlapping either *y* or *z*, and thus the plurality of the atomic parts of either *y* or *z*. Therefore, by the definitions of ‘\({\mathbb{P}}^{x}\)’ and ‘\(\mathop \cup \nolimits\)’, it follows that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\). *□*

- (MP2-lr):
\(x = y - z \to {\mathbb{P}}^{x} = {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\)

###
*Proof*

Suppose that *x* is identical to \(y - z\) and let \(\phi\) be ‘\(v \le y \wedge \sim Ovz\)’. It follows from (L1) and the definition of mereological difference that \({\mathbb{P}}^{x}\) is the plurality of the atomic parts of *y* that don’t overlap *z*. Therefore, \({\mathbb{P}}^{x}\) is the plurality of the atomic parts of *y* that are not part of *z*. By the definitions of ‘\({\mathbb{P}}^{x}\)’ and ‘\(-\)’, it follows, thus, that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\). *□*

- (MP3-lr):
\(x = y \times z \to {\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z}\)

###
*Proof*

Suppose that *x* is identical to \(y \times z\) and let \(\phi\) be ‘\(v \le y \wedge v \le z\)’. It follows from (L1) and the definition of mereological product that \({\mathbb{P}}^{x}\) is the plurality of atoms that are parts of both *y* and *z*. Therefore, by the definitions of ‘\({\mathbb{P}}^{x}\)’ and ‘\(\mathop \cap \nolimits\)’, it follows that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z}\). *□*

From the definition of ‘\({\mathbb{P}}^{x}\)’ and (\({\mathbb{P}}\)A)^{Footnote 42}

- (\({\mathbb{P}}\)A):
\(\forall x\left( { \sim Ax \to \forall z\left( {{\mathbb{P}}^{x} z \leftrightarrow \left( {Az \wedge z \le x} \right)} \right)} \right)\)

it is possible to derive (L2) which says that, for every *non*-*atomic x* and *y*, *x* is part of *y* if and only if the \({\mathbb{P}}^{x}\)s are among (‘\(\subseteq\)’) the \({\mathbb{P}}^{y}\)s:

- (\(\subseteq\)):
\(X \subseteq Y =_{\text{df}} \forall z\left( {Xz \to Yz} \right)\)

- (L2):
\(\forall x\forall y\left( {\left( { \sim Ax \wedge \sim Ay} \right) \to \left( {y \le x \leftrightarrow {\mathbb{P}}^{y} \subseteq {\mathbb{P}}^{x} } \right)} \right)\)

###
*Proof*

*Left*-*to*-*right.* Suppose that (i) *x* and *y* are not atomic, (ii) *y* is part of *x*, and (iii) *w* is one of the \({\mathbb{P}}^{y}\)s. By (\({\mathbb{P}}\)A), *w* is an atomic part of *y*. Therefore, since *y* is part of *x*, *w* is also an atomic part of *x*. By generalisation, the \({\mathbb{P}}^{y}\)s are among the \({\mathbb{P}}^{x}\)s. *Right*-*to*-*left*. Suppose that (i) *x* and *y* are both not atomic, (ii) \({\mathbb{P}}^{y} \subseteq {\mathbb{P}}^{x}\), and (iii) *y* is not part of *x*. By Strong Supplementation, there is a part *w* of *y* that doesn’t overlap *x*. Therefore, since, by (AT), *w* is either an atom or has atomic parts, there must be an atomic part *v* of *y* that doesn’t overlap *x* and is, thus, not part of *x*. Since *v* is an atomic part of *y*, it is one of the \({\mathbb{P}}^{y}\)s. But we are assuming that each of the \({\mathbb{P}}^{y}\)s is also one of the \({\mathbb{P}}^{x}\)s. *Contradiction*! Therefore, *y* is part of *x*. *□*

The right-to-left directions of (MP1-3) can be proved by means of (L2) (*x*, *y* and *z* are thought of as ranging over *non*-*atomic* entities):

- (MP1-rl):
\({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z} \to x = y + z\)

###
*Proof*

Suppose that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\). (*a*) Clearly, \({\mathbb{P}}^{y} \subseteq {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\) and \({\mathbb{P}}^{z} \subseteq {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\). By (L2), both *y* and *z* are parts of *x*. (*b*) Suppose that *w* is part of *x*. If *w* is an atom, then *w* is one of the \({\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\) and so it clearly overlaps either *y* or *z*. If *w* is not atomic, we have by (L2) that \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} \mathop \cup \nolimits {\mathbb{P}}^{z}\), so that all the atomic parts of *w* are atomic parts of either *y* or *z*. Therefore, *w* overlaps either *y* or *z*. Either way, *w* and thus, by generalization, every part of *x* overlaps either *y* or *z*. Therefore, *x* is the sum of *y* and *z*. *□*

- (MP2-rl):
\({\mathbb{P}}^{x} = {\mathbb{P}}^{y} - {\mathbb{P}}^{z} \to x = y - z\)

###
*Proof*

Suppose that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\).

*Part I*. Let *w* be part of *x*. (*a*) Suppose that *w* is not atomic. Then, by (L2), \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{x}\) and, so \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\). Clearly, \({\mathbb{P}}^{y} - {\mathbb{P}}^{z} \subseteq {\mathbb{P}}^{y}\). Therefore, \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y}\), and by (L2), *w* is part of *y*. Suppose *w* overlaps *z*. Then some *v* is part of both *w* and *z*. By (L2), we have that \({\mathbb{P}}^{v} \subseteq {\mathbb{P}}^{w}\) and \({\mathbb{P}}^{v} \subseteq {\mathbb{P}}^{z}\). Therefore, each of the \({\mathbb{P}}^{v}\)s is both one of the \({\mathbb{P}}^{w}\)s and one of the \({\mathbb{P}}^{z}\) s, thus contradicting \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\) (by the definition of ‘\(-\)’). Therefore, every non-atomic part of *x* is a part of *y* that doesn’t overlap *z*. (*b*) Suppose that *w* is an atom. Then *w* is one of the \({\mathbb{P}}^{x}\)s and, by the definition of plural difference, also one of the \({\mathbb{P}}^{y}\)s without being one of the \({\mathbb{P}}^{z}\)s. Suppose *w* overlaps *z*. Since *w* is an atom, *w* is an atomic part of *z* and, thus, one of the \({\mathbb{P}}^{z}\)s. *Contradiction*! Therefore, also every atomic part of *x* is a part of *y* that doesn’t overlap *z*.

*Part II*. Let *w* be a part of *y* that doesn’t overlap *z*. (*a*) Suppose that *w* is not an atom. By (L2), we have, thus, that \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y}\) and that none of the \({\mathbb{P}}^{w}\)s is one of the \({\mathbb{P}}^{z}\)s. Therefore, \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} - {\mathbb{P}}^{z}\) and thus \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{x}\). By (L2), *w* is part of *x*. (*b*) Suppose that *w* is an atom. *w* is, thus, an atomic part of *y* and, thus, one of the \({\mathbb{P}}^{y}\)s. Since *w* is an atom that doesn’t overlap *z*, *w* is not an atomic part of *z* and is, thus, not one of the \({\mathbb{P}}^{z}\)s. Therefore, *w* is a member of \({\mathbb{P}}^{y} - {\mathbb{P}}^{z}\) and, thus, one of the \({\mathbb{P}}^{x}\)s. By (L2), *w* is part of *x*. Every part of *y* that doesn’t overlap *z* is, thus, part of *x*.

*Therefore, x* is the mereological difference between *y* and *z*. *□*

- (MP3-rl):
\({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z} \to x = y \times z\)

###
*Proof*

Suppose that \({\mathbb{P}}^{x} = {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z}\).

*Part I*. Let *w* be part of *x*. (*a*) Suppose that *w* is not an atom. By (L2), \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z}\). By the definition of ‘\(\mathop \cap \nolimits\)’, we have both \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y}\) and \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{z}\). By (L2), *w* is, thus, part of both *y* and *z*. (*b*) Suppose that *w* is an atom. *w* is, thus, one of the \({\mathbb{P}}^{x}\)s. By the definition of ‘\(\mathop \cap \nolimits\)’, *w* is both one of the \({\mathbb{P}}^{y}\)s and one of the \({\mathbb{P}}^{z}\)s. Therefore, *w* is part of both *y* and *z*.

*Part II*. Let *w* be part of both *y* and *z*. (*a*) Suppose that *w* is not an atom. By (L2), we have both \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y}\) and \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{z}\) and, thus, that \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{y} \mathop \cap \nolimits {\mathbb{P}}^{z}\). By (L2), *w* is part of *x*. (*b*) Suppose that *w* is an atom. *w* is, thus, both one of the \({\mathbb{P}}^{y}\)s and one of the \({\mathbb{P}}^{z}\)s. By the definition of ‘\(\mathop \cap \nolimits\)’, *w* is also one of the \({\mathbb{P}}^{x}\)s. *w* is, thus, part of *x*.

*Therefore*, *x* is the product of *y* and *z*. *□*

### ACAI entails (PQ-SF)

In what follows *x* is thought of as ranging over *non*-*atomic* entities:

- (PQ-SF-lr):
\(xSFu\phi \to \forall z\left( {{\mathbb{P}}^{x} z \leftrightarrow \left( {\left( {Az \wedge \phi z} \right) \vee \exists w\left( {\phi w \wedge {\mathbb{P}}^{w} z} \right)} \right)} \right)\)

###
*Proof*

Suppose that *x* S-fuses everything that \(\phi\)s.

*Left*-*to*-*right*. Let *z* be one of the \({\mathbb{P}}^{x}\)s. *z* is, thus, an atomic part of *x*. Since *x* S-fuses everything that \(\phi\)s, it follows there is some *w* such that *w* is a \(\phi\)-er and *z* overlaps *w*. *z* is, thus, an atomic part of *w*. Therefore, if *w* is a composite entity, *z* is one of the \({\mathbb{P}}^{w}\)s. If, instead, *w* is an atom, then *z* is identical to *w* and is, thus, a \(\phi\)-er. In either case, it follows that *z*, and thus, by generalization, each of the \({\mathbb{P}}^{x}\)s is either an atomic \(\phi\)-er or it is one of the \({\mathbb{P}}^{w}\)s for some *w* that \(\phi\)s.

*Right*-*to*-*left*. (*a*) Suppose that *w* is a \(\phi\)-er and that *z* is one of the \({\mathbb{P}}^{w}\)s. Since *x* S-fuses everything that \(\phi\)s, *w* is part of *x*. By (L2), \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{x}\), so that *z* is also one of the \({\mathbb{P}}^{x}\)s. (*b*) Suppose instead that *z* is an atomic \(\phi\)-er. Then, since *x* S-fuses everything that \(\phi\)s, it follows that *z* is an atomic part of *x* and, thus, that *z* is one of the \({\mathbb{P}}^{x}\)s. *□*

- (PQ-SF-rl):
\(\forall z\left( {{\mathbb{P}}^{x} z \leftrightarrow \left( {\left( {Az \wedge \phi z} \right) \vee \exists w\left( {\phi w \wedge {\mathbb{P}}^{w} z} \right)} \right)} \right) \to xSFu\phi\)

###
*Proof*

Suppose that, for every *z*, *z* is one of the \({\mathbb{P}}^{x}\)s if and only if, either *z* is an atomic \(\phi\)-er or there is some *w* such that *w* is a \(\phi\)-er and *z* is one of the \({\mathbb{P}}^{w}\)s.

*Part I*. Let *w* be a \(\phi\)-er. Suppose that *w* is a composite entity. Then each of the \({\mathbb{P}}^{w}\)s is one of the \({\mathbb{P}}^{x}\)s. It follows by (L2) that *w* is part of *x*. Suppose that *w* is, instead, an atom. Therefore, *w* is an atomic \(\phi\)-er and, thus, one of the \({\mathbb{P}}^{x}\)s. It follows that *w* is part of *x*. Therefore, every \(\phi\)-er is part of *x*.

*Part II*. Let *w* be part of *x*. Suppose that *w* is a composite entity. By (L2), \({\mathbb{P}}^{w} \subseteq {\mathbb{P}}^{x}\). Suppose that *z* is one of the \({\mathbb{P}}^{w}\)s. *z* is, thus, also one of the \({\mathbb{P}}^{x}\)s and, therefore, either an atomic \(\phi\)-er or one of the atomic parts of a \(\phi\)-er. Either way, *z* is also part of a \(\phi\)-er. Therefore, *w* overlaps a \(\phi\)-er. Suppose, instead, that *w* is an atom. Then, *w* is one of the \({\mathbb{P}}^{x}\)s, and so either an atomic \(\phi\)-er or one of the atomic parts of a \(\phi\)-er. In both cases, *w* overlaps a \(\phi\)-er. Therefore, every part of *w* overlaps a \(\phi\)-er.

We have, thus, proved that (i) every \(\phi\)-er is part of *x*, and that (ii) every part of *x* overlaps a \(\phi\)-er, and so that *x* S-fuses everything that \(\phi\)s. *□*

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### Cite this article

Loss, R. On atomic composition as identity.
*Synthese* (2019). https://doi.org/10.1007/s11229-019-02295-6

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### Keywords

- Mereology
- Composition as identity
- Collapse
- Mereological nihilism