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Is composition identity?

  • S.I.: Mereology and Identity
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Abstract

Say that some things compose something, if the latter is a whole, fusion, or mereological sum of the former. Then the thesis that composition is identity holds that the composition relation is a kind of identity relation, a plural cousin of singular identity. On this thesis, any things that compose a whole (taken together) are identical with the whole. This article argues that the thesis is incoherent. To do so, the article formulates the thesis in a plural language, a symbolic language that includes counterparts of plural constructions of natural languages, and shows that it implies that nothing has a proper part. Then the article argues that the thesis, as its proponents take it, is incoherent because they take it to imply or presuppose that some things have proper parts.

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Notes

  1. This is meant to suggest the notion of composition. For a precise definition, see below: Sects. 1 and 2.1 (Definition 2).

  2. In the dialogue, the main character, Socrates, states and assumes this thesis to argue that “a syllable is a single form, indivisible into parts” (Theaet. 205c), and uses this to argue against the view that “the complex is by nature knowable, and the element unknowable” (ibid. 206b). Note that he also says, “in the case of a thing that has parts, both the whole and the sum will be all the parts” (ibid. 205a). (Some might object to attributing this thesis to Plato himself, but it would be hard to argue that the argument he has Socrates present in the dialogue does not reflect his own view.) (All quotations from Plato’s Theaetetus and Parmenides are from Plato (1992) and (1996), respectively.)

  3. I use the parenthetical ‘(viz., is identical with)’ (or ‘(viz., are identical with)’) to highlight that ‘is’ (or ‘are’) is used not as a copula or predicate for existence but as a predicate for a kind of identity. (I use ‘predicate’ not just for one-place predicates but also for multi-place predicates, including verbs (e.g., ‘love’, the ‘be’ as used for identity).)

  4. An anonymous referee suggests that the statement by Plato (or Socrates) might just mean that “it is necessary (analytically, de facto) that a whole must have the proper parts it has”, and argues that “this still says nothing about sufficiency, which would be required for the thesis that composition is identity (i.e. the extensionalist conception of wholes).” Proponents of the composition as identity thesis might also hold the sufficiency thesis or the extensionalist conception by assuming a version of the necessity of identity (e.g., M3 in “Appendix”). But the composition as identity thesis must be distinguished from both the sufficiency thesis and its converse (the necessity thesis). It is hard to take (a) ‘when a whole exists, it is its proper parts’ to just mean (b) ‘when a whole exists, all its proper parts exist.’ More importantly, note that Plato (or Socrates) uses (a) to conclude that a complex or whole (e.g., the syllable ‘SO’) has no parts from the thesis that a whole is not identical with its parts or elements (e.g., the two letters ‘S’ and ‘O’). This is a valid argument (on my reading), but replacing (a) with (b) yields an invalid argument. (Note that the same problem arises for the reading that takes (a) to hold the extensionalist conception.)

  5. For ‘Some things (taken together) are identical with something if and only if the latter is identical with the former (taken together)’ is a logical truth (symmetry of identity).

  6. For CAI-2 is logically equivalent to ‘Everything composes itself’, which is equivalent to reflexivity of parthood. See PL7(a) in Sect. 2.3.

  7. What I call CAI in Yi (2014) is CAI-1. Clearly, however, the view that the composition relation is a version of identity yields CAI-2 as well.

  8. The singular and plural ‘is’ and ‘are’ are used in the same way in (a) ‘He is Cicero’, ‘Cicero is Cicero’, and ‘Tully is Cicero’ and (b) ‘They are Russell and Whitehead’, ‘Cicero and Russell are Cicero and Russell’, and ‘Cicero and Russell are Tully and Russell’, respectively. (The ‘are’ in the latter sentences can be replaced with ‘are the same things (or persons) as’. For more about this, see Definition 3 in Sect. 2.1) Although some might take ‘is’ not to be used as a predicate for identity to account for the semantic or cognitive difference between, e.g., ‘Cicero is Tully’ and ‘Cicero is Cicero’, both the Fregean and Russellian accounts of the difference take it to be used for identity in the sentences. See, e.g., Kripke (1972: p. 107f).

  9. See also Baxter (1988a, b, 1989). Lewis (1991) uses the weaker thesis to argue that mereology is ontologically innocent. I argue that the argument fails in Yi (1999a).

  10. For discussions of various versions of the weak composition thesis, see Cotnoir (2014).

  11. This view dates back to Plato (Parm. 129c–d). See Yi (2014: Sect. 1).

  12. See PL11 in Sect. 2.5. See also Yi (2014).

  13. We can see that the objection is implicit in my argument against CAI in Yi (1999a: Sect. 2). And it is further developed in Yi (2014), where I prove that CAI-1 and reflexivity of parthood (which is equivalent to CAI-2) imply the part-whole triviality thesis. See note 43.

  14. We can prove that it is logically equivalent to the thesis that nothing has a proper part. See note 47.

  15. Thus he implicitly invokes the thesis to conclude that a person (or his body) is both one and many (Parm. 129c–d). See note 11.

  16. The presentation of plural languages and logic in Sects. 2.1 and 2.2 is a sketch of the account of logic and semantics of plural constructions presented in Yi (2002: Chapter 2, 2005, 2006). See also Linnebo (2017), McKay (2006), Oliver and Smiley (2016), and Rayo (2002) for similar accounts.

  17. Superscripts on predicates indicate arities. They are omitted when the arities are clear from the contexts.

  18. While singular variables relate to any one thing, plural variables relate to any one or more things. And plural quantifiers are semantically neutral about number. They amount to ‘Any one or more things are such that…’ and ‘There are some one or more things such that…’. But the plain ‘some (things)’, for example, amounts to ‘some one or more (things)’, not to ‘some two or more (things)’, for ‘Some Romans are orators’ logically follows from ‘Cicero is a Roman orator and Tully is a Roman orator’, which is compatible with ‘Cicero is Tully’ and does not imply ‘There are two Roman orators.’

  19. Note that ‘They are one’ occurs in the negation ‘They are not one’ and that ‘Cicero and Tully are one’ is true.

  20. I call plural argument places that can also admit singular terms neutral (and those that do not exclusively plural). See Yi (2005: p. 479f). I think most argument places of natural language predicates (not their singular or plural forms) have neutral argument places. And all the plural argument places of plural predicates introduced in plural languages in this article are meant to be neutral argument places, although we might introduce plural language predicates with exclusively plural argument places.

  21. While ‘Η2’ is a plural predicate because its second argument place is plural, its first argument place is singular, and it amounts to the singular form ‘is one of’, rather than ‘be one of’.

  22. Plural languages that result from adding plural variables, quantifiers, and predicates to elementary languages are first-order languages, and they are sufficient for considering the logic of CAI. For higher-order plural languages, see Yi (2013: Sect. 6.1, Forthcoming: Sect. 3.3).

  23. Note that ‘Bob is one of the boys surrounding the piano’ is not logically equivalent to ‘Bob is a boy surrounding the piano’ while ‘Russell is one of the authors of Principia Mathematica’ is logically equivalent to ‘Russell is an author of Principia Mathematica.’ Similarly, ‘Russell is one of the logicians who wrote Principia Mathematica’ is not logically equivalent to ‘Russell is a logician who wrote Principia Mathematica.’

  24. In English glosses on definitions, I use ‘this’, ‘that’, ‘these’, and ‘those’ (as well as ‘it’ and ‘they’) as counterparts of variables. (Thus the two occurrences of ‘this’ in (a), like those of ‘x’, are meant to coordinate with each other.)

  25. For example, ‘z < x’, ‘wΗxs’, and ‘wOz’ are used as shorts for ‘< (z, x)’, ‘Η(w, xs)’, and ‘O(w, z)’, respectively.

  26. The listed symbolic sentences are meant to have implicit frontal universal quantifiers that bind apparent free variables (singular or plural). In some cases, the frontal universal quantifiers are made explicit for clarity of exposition.

  27. Plural logic gives an account of the logic of basic plural constructions of natural languages by relating them to their counterparts in regimented plural languages. This does not mean that the logic is applicable only to languages that, like English, distinguish between singular and plural constructions. Some languages (e.g., Chinese, Japanese, Korean) do not have a grammatical number system and draw no syntactic distinction between singular and plural constructions (thus they have no singular or plural forms of nouns or verbs). Nevertheless, they have counterparts of both singular and plural constructions of English: ‘Cicero wrote Academica’, ‘A boy lifted the piano’, ‘Russell and Whitehead wrote Principia Mathematica’, ‘Some boys lifted the piano’, etc. Regimented plural languages have counterparts of the number-neutral cousins of these sentences in languages without grammatical systems, and plural logic applies to such languages as well. See, e.g., the discussions in Yi (2018a, b).

  28. See Yi (1999b: p. 181, 2006: pp. 257–262) about the non-axiomatizability of plural logic. For model-theoretic characterizations of the logic, see Yi (2002: Sect. 2.2.2, 2006: Sect. 6). (Plural logic is a conservative extension of elementary logic.) See also note 75.

  29. SPL is for the logic of first-order plural languages, which is sufficient for formulating CAI. See note 22.

  30. A variable is said to be suitable for a quantifier, if they are both singular or both plural. Greek letters are used as metavariables for ℒ: ‘φ’ and ‘ψ’ for sentences (open or closed); ‘π’ for predicates, and ‘πn’ for n-place predicates; ‘τ’ and ‘μ’ for terms of any kind; ‘ς’ and ‘σ’ for singular terms; and ‘υ’ for singular variables, ‘ω’ for plural variables, and ‘ν’ for variables of any kind. (Note that the metavariables for plural variables (e.g., ‘ω’), unlike the variables themselves (e.g., ‘xs’), do not have the italicized ‘s’.) And ‘Q’ is used as a metavariable for quantifiers. In addition, results of adding numerical subscripts to ‘Q’ or the Greek letters are used as metavariables of the same kind.

  31. The axioms assume the usual definitions of universal quantifiers in terms of existential quantifiers (or vice versa) and the definition of plural identity (Definition 3).

  32. φ(μ/τ) is the result of properly substituting τ for μ in φ.

  33. These axioms differ from the axioms of SPL in earlier formulations of the system in having Axiom 10 instead of the singular cousin of Axiom 11. Given Axiom 11, however, these are equivalent.

  34. For plural existential and universal quantifiers are semantically neutral about number. See note 18.

  35. In subsequent proofs, I usually make implicit use of quantifier rules.

  36. The statements in the PL series are theorems of SPL and thus logical truths of plural languages.

  37. Note that PL3 implies PL2. One might render PL3 as ‘A single thing is one of a single thing if and only if the former is the latter’, but this rendering presupposes a consequence of PL3: ‘Everything is a single thing (i.e., one thing) (∀x One(x)).’ See Definition 5 in Sect. 2.5 and the penultimate paragraph of Sect. 3.4.

  38. In < ν: φ >, ν must be singular but may occur in a plural argument place of φ, as in ‘One(x)’. (Note that the English rendering of this (‘it is one’) has the singular form of ‘be one’.)

  39. Use of plural definite descriptions rests on implicit specification of their scopes. In plural language sentences discussed in this article, they take as their scopes the smallest sentences that contain them.

  40. The operator introduced above is meant to capture one kind of plural definite descriptions of English. There are other kinds of plural definite descriptions (e.g., ‘the two logicians who wrote Principia Mathematica’ and ‘the boys surrounding the piano’), but it is not necessary to consider them for the present purpose. For more on plural definite descriptions, see Yi (2006: p. 244f, 2016: Sect. 5.1).

  41. We can define the existence predicate for plural language terms as follows:

    Exist(xs) ≡ : Σys ys ≈ xs. (They exist ≡ : there are some things that are identical with them.)

    (In modal languages, it is necessary to modify this definition by refining Definition 3. See note 94.) Using this predicate, we can formulate PL5(d) as follows:

    PL5.

    dʹ.

    Exist(< x: φ >) ↔ ∃x φ. (The so-and sos exist if and only if there is something that is a so-and-so.)

  42. Triviality and PL3 imply (3a). Triviality implies (3b) (Definition 1), and (3b) implies (3c).

  43. See Yi (2014: Sect. 4), where I prove PL7(b)–(c) and argue that this means CAI-1 implies Triviality because reflexivity of parthood (T2) is an analytic truth.

  44. Statements in the T series are logical consequences of CAI.

  45. In the above formulations of T2–T5, frontal universal quantifiers are made explicit for clarity of exposition.

  46. For ‘y < x’ imply both ‘H(y, < y: y < x >)’ and ‘yOy’ by T4 and T3.

  47. By following the derivation of Triviality from T6, we can prove that ‘< x: x < y ∧ y ≠ x > ≈ y’ is inconsistent. So Plato’s Thesis (∃y[y < x ∧ y ≠ x] → < x: x < y ∧ y ≠ x > ≈ y) is logically equivalent to the thesis that nothing has a proper part (~ ∃x∃y[y < x ∧ y ≠ x]).

  48. UC together with T10(c) and T11, below, yields an axiomatization of classical mereology in plural languages that imply all theses of classical mereology. See Tarski (1929) and Lewis (1991: p. 72ff).

  49. This is logically equivalent to ‘∀x ∀y x = y (Everything is identical with everything).’

  50. I use henceforth ‘Reflexivity’ for reflexivity of parthood.

  51. He proves that CAI implies T10(a), T10(c), T14, and a consequence of T12: ‘x < y ∧ x ≠ y → ∃z(z < y ∧ ~ zOx)’ (Weak Supplementation) (ibid.: p. 60). (Note that he does not accept CAI in Sider (2007). See Sect. 4.)

  52. In a later work, however, he says “I argued in Sider (2007: Sect. 3.2) that composition as identity implies all of classical mereology except for unrestricted composition” (2013: p. 212; my italics).

  53. Note that we can use PL9(a) to prove that CAI and UC imply Monism (cf. PL8), for it is straightforward to see that they imply Dodge.

  54. Note that PL9(a) is a corollary of Theorem 16 (Plural Cantor) in Yi (2006: p. 271). The proof of this theorem in Yi (ibid., “Appendix 2”) uses Plural Comprehension but does not use PL3 (or Axiom 8–9). While using PL3, the above proof of PL9(a) uses only an instance of the weak version of Plural Comprehension that Sider (2014) proposes (see Sect. 3.3). Incidentally, Hovda notes that Dodge (or “UMOID”) implies Monism (2014: p. 203). But he does not give a proof. One cannot give the above proof in his system of plural logic (“NPL”), for the system has no counterpart of Axiom 9 or PL3—NPL is equivalent to the system one can get from SPL by dropping Axioms 8–9 while adding reflexivity of singular identity (PL4(a)) (ibid., 199–203). (Note that the system does not have Axiom 8, either.) But one can use the full power of Plural Comprehension to prove PL9(a) in NPL, as in the proof of Plural Cantor in Yi (2006).

  55. Definition 3 yields ‘xs ≈ xs’, and this yields PL10 (plural existential and universal generalizations).

  56. These definitions involve only logical expressions, and ‘One’ and ‘Many’ are logical predicates.

  57. Note that PL11 is based on the elementary logic part of plural logic: ‘∃y∀z(P(x) ↔ z = y)’ and ‘∃y∃z(y ≠ z ∧ P(y) ∧ P(z))’ are inconsistent.

  58. Sider gives another argument from CAI to UC (ibid., 61f). Instead of Dodge, the argument uses some modal theses, including a modal cousin of UC: ‘Πxs ◊ ∃y Comp(xs, y)’. The argument is not valid, for the modal cousin of UC is much weaker than the thesis that it is possible for any (actually existing) things to compose something while they all exist together. For more about the argument, see “Appendix”.

  59. This thesis is opposite to the Part-Whole Triviality Thesis (TRV), which implies that nothing has a proper part (see Sect. 2).

  60. For more on the relation between CAI and the Many-One Identity Thesis, see Yi (2014).

  61. The converse does not hold. Assuming Reflexivity (which CAI implies), however, NTP implies NTC.

  62. He says, “I no longer find that argument [i.e., the argument against CAI in Sider (2007)] convincing since I now doubt that Boolos’s logic should be taken as metaphysically basic” (2014: p. 211).

  63. He says, “Collapse is so-called because it in effect identifies mereologically equivalent pluralities” (ibid., 211). But the identification is stated by a direct consequence of CAI: ‘Comp(xs, y) ∧ Comp(zs, y) → xs ≈ zs’ (PL1). Collapse results from combining this with T5: ‘Comp(< y: y < x > , x)’.

  64. Another way to see this is to use T8 (which figures in the derivation of TRV from CAI): ‘x < y ↔ xΗy’. Using this consequence of CAI, we can get Collapse from T14(a) and CAI.

  65. Monism and Reflexivity imply CAI (PL8). And Monism implies Plural Comprehension:

    Proof ∀y w = y and ∃x φ imply ∀x(x = w ↔ φ), which implies ∀x(xHw ↔ φ) (PL3). And this implies Σxs ∀x (yHxs ↔  φ) (plural existential generalization).

  66. He also says, “Collapse prohibits there being things such that something is one of them iff it is a human being” (ibid., 214).

  67. This is used in the step from T2 and T4: T2 implies the antecedent of PL14, and T4 is essentially its consequent (Definition 4).

  68. Compare this with Definition 2. Unlike ‘Comp(< x: φ(x) > , s))’, ‘φ(x)-Fusion(s)’ does not imply ‘Σxs xs ≈ < x: φ(x) >’.

  69. Instead of the full CAI, Sider gives the conditional cousin (i.e., CAI-1). But he adds Reflexivity (T2a or T10a), which is equivalent to the converse (i.e., CAI-2). See PL7(a). See also note 76.

  70. Sider lists both S2 and ST1 (but not S3) to formulate his system. Given ST1, however, S2 is redundant. Instead of removing S2, it is useful to separate ST1 into two parts: S2 and S3. (Given CAI, which implies Reflexivity, ST1 implies both S2 and S3.)

  71. For PL14 and CAI imply T4 (Σxs xs ≈ < x: x < y >) and thus T6. See the proofs of PL7(a)–(b) in Sect. 2.3.

  72. So S3 (together with Reflexivity) implies T6, and we can use S3 instead of ST1 (which is an axiom in Sider’s own formulation of S) to prove Collapse in S. Note that T4 implies PL14.

  73. This is used in the step from Dodge to (6a): ‘∃y < x: x = x > ≈ y’.

  74. For recent systems of plural logic, see, e.g., Linnebo (2017), McKay (2006), Oliver and Smiley (2016), Rayo (2002), and Yi (2002: Chapter 2, 2005, 2006).

  75. Moreover, the so-called Boolos’s logic, which results from adding Plural Comprehension (or its second-order analogue) to elementary logic, cannot be taken to capture even the logic of Boolos’s languages, which include (singular) two-place predicates (e.g., ‘admires’ in the Geach-Kaplan sentence ‘Some critics admire only one another’). Their logic is not axiomatizable, and the addition of Plural Comprehension is not meant to yield a complete axiomatization of the logic.

  76. ‘∀x x(x)’ is a theorem of Sider’s system (as formulated above). It follows from singular reflexivity of composition (T1) (which follows from CAI-2), reflexivity of parthood (T2), and Collapse. But this does not mean that it is not a logical truth but depends on CAI and other mereological truths. Nor does it mean that logical systems of plural languages do not need axioms that imply it, for the logic of plural languages with no mereological predicates must also include it. (Note also that axioms of Sider’s formulation of the system do not include CAI but CAI-1 and T2 and that one cannot show that T2 implies CAI-2 (which implies T1) without assuming ‘∀x x(x)’. See note 69.)

  77. For more on this, see Yi (1999b: p. 185f). ‘∀x(xHxs ↔ xHys) ∧ G(xs) → G(ys)’ is a theorem of SPL (Definition 3, PL1). (Note that it implies ‘∀x(xHy ↔  xHz) ∧ G(y) → G(z)’.) Sider’s system, which includes CAI, includes it. Again, however, this does not mean that it is not a logical but a mereological truth that rests on CAI among others.

  78. For more on Boolos’s treatment of plural constructions, see Yi (2006: “Appendix 1”).

  79. This shows that ‘∀x One(x)’ is a theorem of SPL. The proof uses PL2 but Sider cannot reject it (see two paragraphs back, esp. note 76).

  80. In arguing that CAI implies transitivity of parthood (T10(c)), Sider (2007) uses a thesis that one cannot prove without assuming theses that imply PL3: ‘[(x and y) and z] are identical with [x and (y and z)]’ (ibid., 60).

  81. See the proof of PL7(c) in Yi (2014: Sect. 4).

  82. See, e.g., Lewis (1991: p. 82ff). In formulating CAI, he takes the composition relation to be a “many-one relation”, one that can hold between many things taken together, on the one hand, and one thing, on the other (ibid., 84). (Although he does not in the end accept CAI, he takes those who hold it to presuppose NTC as he does in holding a weak cousin of CAI.)

  83. He says the thesis defended in the article is “what Peter van Inwagen calls nihilism: composite entities (entities with proper parts) do not exist” (2013: p. 237). This is the negation of NTP. van Inwagen (1990: p. 72f) officially defines nihilism as the thesis that some things compose something if and only if they are one: Comp(xs, x) ↔ One(xs). Assuming Reflexivity, this thesis is equivalent to the negation of NTC and thus to the negation of NTP.

  84. He says, “it is difficult to believe it” (2001: p. 160). In the same paragraph, however, he also says “It is difficult to argue against it” (ibid., 160). But I see no special difficulty in arguing against S-CAI or CAI*. They are inconsistent or incoherent, as we have seen.

  85. (11b) and (11d) would draw parallels with (10b) and (10d), respectively, but they are ill-formed. We can modify the definition of ‘Comp’ to make them well-formed, but it is not necessary to do so for the present purpose.

  86. He objects to Lewis’s analogical version because it fails to explain the necessity of UC (ibid., 160).

  87. See also note 52.

  88. He says, “by yielding unrestricted composition, it justifies the plausible asymmetry noted at the outset” (ibid., 88).

  89. He says, “we saw in Sect. 3.2… the relation between strong composition as identity and unrestricted composition is not straightforward” (ibid., 72).

  90. He attributes the argument to Karen Bennett and says that it is “compelling though hard to evaluate” (ibid., 72). It is hard to see how a thesis hard to evaluate can be compelling.

  91. Defenders of Sider might attempt to avoid this objection by replacing (13) with its weak cousin:

    1. (13ʹ)

      If it is in general an open question whether some things compose something and yet some things compose something that is not one of them, the latter (the composed) is something over and above the former (the parts).

    I am inclined to take this to be true, for I doubt that there is a plausible notion of “over and above” that does not trivialize Sider-CAI on which ‘Socrates is nothing over and above the so-and-sos’, for example, does not imply ‘Socrates is (literally) identical with some of the so-and-sos.’ But the view that the former implies the latter yields a stronger cousin of (13ʹ):

    1. (13ʺ)

      If some things compose something that is not one of them, the latter (the composed) is something over and above the former (the parts).

    (The view yields a thesis even stronger: Anything that is not one of some things (e.g., Socrates) is something over and above them (e.g., all his proper parts).) But (13ʺ) conflicts with Sider-CAI (and thus CAI). They imply ‘Anything that some things compose is one of them (Comp(xs, y) → yHxs)’, and this together with Reflexivity implies Triviality. So defenders of Sider-CAI must reject the defense of (13ʹ) suggested above. And I see no reason that those who reject the defense would still have to accept (13ʹ).

  92. David Nicolas suggests that this is a plausible view (private conversation).

  93. This might be a trivial statement. It would be no surprise if one could not make adequate assessments of statements involving plurals without resorting to an account of their logic. It might sometimes be useful, however, to remind oneself of some trivialities.

  94. Accordingly, one might replace ‘Exist(xs)’ in M3 and M5 with ‘Exist*(xs)’. We can define ‘Exist*’ using the modification of Definition 3 that results from replacing the actualist quantifier ‘∀z’ with a quantifier ranging over all possible objects (see note 41 in Sect. 2.3).

  95. Note that the above derivation of Monism does not use M4.

  96. As Sider points out (ibid., 62), M3 states a version of the thesis of necessity of plural identity. I do not think one can reject it to deny Monism while accepting both M1* and M2.

  97. So he argues that the defender cannot hold “two people could not possibly have composed something, while allowing that their subatomic particles could” (ibid., 16; original italics).

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Acknowledgements

The work for this article was supported in part by a SSHRC Insight Grant [Grant No. 435-2014-0592], which is hereby gratefully acknowledged. I presented its previous versions in the Mereology and Identity Workshop and at Nihon University. I would like to thank G. Lando, M. Carrara, and T. Iida for the invitations and the audiences for their comments and discussions. I would also like to thank two anonymous referees for Synthese for helpful comments on the penultimate version, P. Hovda for discussions on topics of this article, and Y. El Gebali and E. Darnell for editorial assistance. The penultimate version of this article was written while I was visiting Hokkaido University as a visiting scholar in 2018. I am grateful to T. Yamada and K. Sano for their invitation and hospitality during the visit. Needless to say, I am solely responsible for any errors and infelicities in this article.

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  1. Department of Philosophy, University of Toronto, Toronto, ON, M5R 2M8, Canada

    Byeong-uk Yi

  2. Department of Philosophy, Kyung Hee University, Seoul, 130-701, Republic of Korea

    Byeong-uk Yi

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Appendix

Appendix

After giving an argument that rests on the Dodge Thesis (see Sect. 2.5), Sider gives an “alternative argument from strong composition as identity to unrestricted composition” (2007: p. 61). This argument rests on five theses, three of which involve modality. Using ‘Exist’ as a plural language predicate for existence, we can formulate the theses as follows:

M1.:

Πxs ◊ ∃y Comp(xs, y).

M2.:

□ Πxs ∀y (Comp(xs, y) → xs ≈ y).

M3.:

□ Πxs ∀y [xs ≈ y → □ (Exist(xs) → Exist(y) ∧ xs ≈ y)].

M4.:

Πxs ∀y (xs ≈ y → Comp(xs, y)).

M5.:

Πxs Exist(xs).

Sider argues that these imply Unrestricted Composition (UC):

UC.:

Πxs ∃y Comp(xs, y).

I think there are two major problems with this argument. One of them is specific to the argument. The other is an instance of the general problem with arguing that proponents of CAI must accept UC by deriving UC from CAI together with additional theses (see the last paragraph of Sect. 2.5).

(M1)–(M5) do not imply UC, for M1 does not mean that it is possible for any (actually existing) things to compose something while they all exist together. Unpacking the thesis using Definitions 1 and 2 (Sect. 2.1) yields M1ʹ:

M1ʹ.:

Πxs ◊ ∃y [∀z(zΗxs → z < y) ∧ ∀z(z < y → ∃w(wΗxs ∧ wOz))].

Here the singular quantifier phrases (e.g., ‘∀z’) occur in the scope of the possibility operator and are restricted, for each possible world, to the things that exist in that possible world. Thus M1 does not require that there be a world in which something is composed by, e.g., all the things that actually exist, but only that there be a world with something composed by all those among them that exist in that world. This is satisfied if there is a world that has just one of the many actual existents and where the one object composes itself. Now, consider a two-world model satisfying ‘□ ∀x x < x’ that includes (a) the actual world whose only existents are two atomic objects and (b) the world whose only existent is one of those atomic objects. This model does not satisfy UC but satisfies all of M1–M5.

Defenders of Sider might propose a modification of the argument. One might add additional theses or replace M1 with a thesis that Sider might have in mind:

M1*.:

Πxs ◊ (Exist*(xs) ∧ ∃y Comp(xs, y)),

where ‘Exist*’ is an undefined plural language predicate amounting to the English ‘exist together’ used above.Footnote 94 But modifying the argument to reach a valid argument has a general problem noted above. M2 (a modal cousin of CAI-1) and M4 (i.e., CAI-2) imply CAI. So if M2 and M4 together with some additional theses (e.g., M1*, M3, M5) imply UC, this means that all of them (taken together) imply Monism: there is only one thing (in the actual world). So some of them must be rejected by those who hold CAI without accepting Monism.

Now, it might be useful to consider how M1* and M2–M5 imply Monism. Consider all the things that exist in the actual world (i.e., < x: x = x >). They exist together (in the actual world) by M5. And by M1*, there is a possible world, Wʹ, in which they all exist together and compose something, A. So all of them (taken together) are identical with A (in Wʹ) by M2. By M3 (and M5), A exists and is identical with them in the actual world. So any one of them must be identical with A in the actual word (PL3). This implies Monism.Footnote 95 The culprits in this case are M1* and M2. Thus proponents of CAI who reject Monism must reject M1*.

Sider argues that M1 “is harder than one might think for defenders of strong composition as identity to deny” (ibid.: p. 61). I agree, but for a different reason: M1 (which is much weaker than M1*) does not imply that there is any possible object with a proper part. But the same does not hold for M1*, which I think Sider has in mind in talking of M1. M2 is a consequence of the view that the composition relation is literally the plural identity relation. Thus those who hold this view cannot accept M1* without committing themselves to Monism.Footnote 96 Sider argues that “the defender” of CAI must accept M1* because he or she “identifies the people with the subatomic particles” in their bodies.Footnote 97 I agree that usual proponents of CAI hold that people (or their bodies) have proper parts. But one cannot consistently combine this view with CAI, for CAI implies the Part-whole Triviality Thesis as we have seen (PL6 in Sect. 2.3).

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Yi, Bu. Is composition identity?. Synthese 198 (Suppl 18), 4467–4501 (2021). https://doi.org/10.1007/s11229-018-02000-z

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