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Composition as identity, now with all the pluralities you could want

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Abstract

According to ‘composition as identity’ (CAI), a composite object is identical to all its parts taken together. Thus, a plurality of composite objects is identical to the plurality of those objects’ parts. This has the consequence that, e.g., the bricks which compose a brick wall are identical to the atoms which compose those bricks, and hence that the plurality of bricks must include each of those atoms. This consequence of CAI is in direct conflict with the standard analysis of plural definite descriptions (and hence with the standard plural comprehension schema which uses it). According to that analysis, the denotation of ‘the bricks’ can include only bricks. It seems, then, that if CAI is true, ‘the bricks’ doesn’t denote anything; more generally, if CAI is true, there are fewer pluralities than we ordinarily think. I respond to this argument by developing an alternative analysis of plural descriptions (and an alternative comprehension schema) which allows the denotation of ‘the bricks’ to include non-bricks. Thus, we can accept CAI, while still believing in all the pluralities we could want. As a bonus, my approach to plural descriptions and plural comprehension blocks recent arguments to the effect that CAI entails compositional nihilism.

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Notes

  1. See McKay (2006), Oliver and Smiley (2016), and Yi (2005, 2006) for more detailed discussion of plural languages.

  2. On one view, if ‘a@b’ denotes a and b, then it denotes each of them individually. On a competing view, there’s no such implication, and indeed, if ‘a@b’ denotes a and b then it doesn’t denote either of them individually. I remain neutral on this dispute; see Oliver and Smiley (2016: ch. 6) for discussion.

  3. Some distinguish more sharply between ‘is one of’, which only takes singular terms at its first argument-place, and ‘are among’, which only takes plural terms at its first argument-place (McKay, 2006: ch. 6; Yi, 2005: pp. 485–486). See Oliver and Smiley (2016: pp. 108–109) for discussion.

  4. a overlaps b just in case they have a part in common.

  5. CAI is often understood to imply the converse conditional: if x = yy, x fuses yy—see, e.g., Bohn (2014) and Wallace (2011). That conditional won’t matter for our purposes.

  6. I say ‘so to speak’ because, as we’ll see, we can introduce the description ‘the Fs’ in cases where ‘F’ applies collectively to pluralities of individuals, but not to any individual.

  7. Notice, it’s equivalent to: ∃xx(∃x(x = xx) & φxx) ⊃ ∃yy∀zz(zz ≼ yy ≡ (∃z(z = zz) & φzz)).

  8. If x fuses xx and x fuses yy, then by CAI, x = xx and x = yy; by the transitivity of identity, xx = yy.

  9. Variants of these arguments appear in Sider (2007: pp. 57–59, pp. 63–66; 2014: pp. 215–216) and are discussed in Calosi (2018: pp. 282–287), Carrara and Lando (2017: pp. 510–511), Cotnoir (2013: pp. 313–317), and Loss (2019: p. 4). Sider’s presentation uses a principle, Collapse, which I haven’t discussed (although see Sect. 9), and a principle of unrestricted fusion. My presentation avoids these complications.

  10. Objection: (4) is true on the distributive reading; ‘a wrote musicals’ doesn’t imply that a wrote them on their own.

    Reply: Other sentences more clearly illustrate the phenomenon in which I’m interested – e.g. ‘Three architects designed four buildings’, which can be true even if (i) there are no four buildings such that the three architects worked collectively on each of them and (ii) none of the three architects worked on four buildings, either individually or in a team. But these involve additional complexities – e.g. the interplay between the determiner phrases ‘three architects’ and ‘four buildings’ – so I’ll assume, if only for simplicity, that (4) is false on the distributive reading.

  11. See Oliver and Smiley (2016: pp. 4–7) for discussion of multivalued functions.

  12. Question: Instead of multi-valued functions, could we use sets? E.g. could we say that a cover δi maps aa to a set, {bb1, …, bbn}, and that ‘Faa’ is true w.r.t. δi just in case ‘Fbbi’ is true for each member of that set?

    Answer: As set theory is typically understood, each member of a set is an individual. This raises a problem for (4). ‘Wrote musicals’ needs to distribute over Rogers and Hart, on one hand, and Rogers and Hammerstein, on the other. But while Rogers and Hammerstein can be members of a set individually, they can’t be members of a set collectively (and likewise for Rogers and Hart).

    We could introduce pairs, understood as individuals distinct from the individuals that make them up. Then, δi could map aa to {< r, h1 > , < r, h2 >}, and ‘wrote musicals’ could distribute over the members of this set. But if we already adopt the resources of plural languages, we should be skeptical of this move. Defenders of plural languages are traditionally opposed to views on which a collective predicate—like ‘wrote Principia Mathematica’ in ‘Russell and Whitehead wrote Principia Mathematica’—is true, not of some things—i.e., Russell and Whitehead—but of some thing, typically a set with those things as members (Boolos, 1984; McKay, 2006: pp. 22–32; Oliver and Smiley, 2016: ch. 3; Yi, 2005: pp. 463–476). But if we deny that ‘wrote Principia Mathematica’ is true of a set, I don’t see why we’d accept that ‘wrote musicals’ is true of a pair (so understood).

  13. There’s still a sense in which nothing ‘other than’ Fs are included in the Fs. If bb1,…,bbn are all the Fs, then nothing outside bb1@…@bbn gets included in the Fs. Thus, we can circumscribe the things which are F and use ‘the Fs’ to refer to that plurality and nothing outside it.

  14. Objection: Alice and Beth don’t themselves exemplify the property being a team. Rather, they constitute an individual which does exemplify that property.

    Reply: This view has been argued against elsewhere – see, e.g., Horden and López de Sa (2020) and Payton (2019: pp. 14–18). Here, I assume it’s false.

  15. On an alternative approach, talk about ‘pluralities of pluralities’ is captured using ‘higher-level’ plural terms and variables, which stand to ordinary plural ones as the latter stand to singular ones (Linnebo and Nicolas, 2008; Rayo, 2006). However, such ‘higher-level’ plural resources are highly controversial (Ben-Yami, 2013; McKay, 2006: pp. 137–139; Uzquiano, 2004: pp. 438–440), and my approach doesn’t require them.

  16. Remember, plural variables are inclusive, so each of Alice and Beth can be a value for ‘zz’.

  17. Note: since plural variables are inclusive, Comprehension 3 still holds when ‘φ’ is true only of individuals.

  18. Objection: Your approach doesn’t get us all the pluralities we could want. E.g., there’s no plurality which includes all and only the bricks.

    Reply: All we should want, or expect, is for there to be such a plurality as the bricks. What’s pre-theoretically clear is that we can (in principle) single out the bricks. What’s not pre-theoretically clear is that doing so requires that we single out a plurality which includes all and only bricks (and I’ve argued in Sect. 5.1 that this assumption about how plural descriptions work is false, for reasons independent of CAI).

  19. Sider (2014: pp. 214–215). To derive an equivalent of the original definition of ‘Fu(a, bb)’, substitute ‘λx.(x ≼ bb)’ for ‘φ’.

  20. In keeping with CAI, I allow that many yy can collectively be a part of x.

  21. Sider (2007: p. 58) is well aware of the problems with combining CAI and the approach to distributivity from Sect. 3. I suspect he would reject the Sideresque analysis for that reason.

  22. See, e.g., Hirsch (2013), Thomasson (2014: pp. 308–317), and Warren (2016) for concerns.

  23. Since parthood is transitive, we can’t always carve a whole into all its parts at once. E.g., we can’t have both the bricks which compose the wall and the atoms which compose the bricks in our domain of quantification.

  24. See also Baxter (1988: p. 193).

  25. For further discussion of Cotnoir’s view, see Carrara and Lando (2016) and Hawley (2013).

  26. Proof: First, ∀x(x ≼ cc ⊃ x ≤ a) follows trivially from C4. Second, we have ∀y(y ≤ a ⊃ ∃z(z ≼ cc & y ∘ z)): since we’ve assumed Pa, and parthood is reflexive, we have ‘a ≤ a & Pa’; by C4, a ≼ cc; but then, every part of a overlaps one of a’s P-parts, since every part of a overlaps a.

  27. Proof: First, we have ∀x(x ≼ aa ⊃ x ≤ a), since the only thing included in aa is a itself, and parthood is reflexive. Second, we have ∀y(y ≤ a ⊃ ∃z(z ≼ aa & y ∘ z)): by the definition of ‘overlap’ (n. 4), any part of a overlaps a. Since a is the only thing included in aa, it follows that any part of a overlaps one of those things, namely a.

  28. Loss (2018: pp. 372–373) considers a different kind of skeptic about improper pluralities, who grants that everything has an improper plurality, but rejects the inference from L1 to L2. By contrast, I reject L1 outright. See Loss (2019: p. 10) for his own way of blocking the road to Nihilism.

  29. Proof: First, we have ∀x(x ≼ cc ⊃ x ≤ a): by L9, the only things included in cc are a and b, and a is part a by reflexivity while b is part of a by L8. Second, we have ∀y(y ≤ a ⊃ ∃z(z ≼ cc & y ∘ z)): since a is included in cc, every part of a overlaps something included in cc, namely a.

  30. (Let a be a composite object and b and c its proper parts. By CAI, aa is identical to b@c, and so c included in aa, even though it’s distinct from both a and b.).

  31. Calosi (2018: pp. 287–288) and Yi (2014: pp. 183–184; 2019: pp. 13–15) give other arguments from CAI to Nihilism, which don’t obviously hinge on issues discussed in this paper. But they do rely on assumptions a defender of CAI should reject, e.g. that x is included in y only if x is identical to y (Payton 2019: pp. 18–19).

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Acknowledgements

Thanks to Noa Latham, David Liebesman, and three anonymous referees for feedback on earlier versions of this paper. Research was funded by a Postdoctoral Fellowship from the Social Sciences and Humanities Research Council of Canada.

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  1. Department of Philosophy, Faculty of Humanities and Letters, Bilkent University, 06800, Bilkent, Ankara, Turkey

    Jonathan D. Payton

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Payton, J.D. Composition as identity, now with all the pluralities you could want. Synthese 199, 8047–8068 (2021). https://doi.org/10.1007/s11229-021-03152-1

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