[S]ome things can be one and many at the same time, but of course in different respects. This is not a contradiction, but rather a kind of gestaltwechsel which is supported by a solid mathematical correspondence.
– Godehard Link.
Abstract
The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting three necessary and independently motivated restrictions on the metalinguistic predicates ‘atom’ and ‘non-atom’: both are sort, property, and context-relative. With these restrictions in place, Schein’s problematic assumption becomes evident: his presumed singularist analysis of ‘non-atom’ conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage.
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Notes
See Carrara et al. (2016) for a helpful discussion.
There are two parthood relations in Link’s semantics, one for the count domain and another for the mass domain. Since we are only be concerned with count nouns here, we will drop subscripts distinguishing these, hoping no confusion results.
See fn. 18 below.
By “non-trivial sum”, we mean a sum having at least two distinct atomic parts (the sum-operation is idempotent, i.e. \(x = x \sqcup x\) for any x, including atoms).
The following is a straightforward corollary of the comprehension scheme (5) for mereology: \((\exists x\phi (x) \wedge \texttt {ATOM}(x))\rightarrow \exists y \forall z[z \sqsubseteq _{At} y \leftrightarrow (\texttt {ATOM}(z) \wedge \phi (z))]\). But the result of substituting \(\lnot \texttt {ATOM}(x)\) for \(\phi (x)\) in this scheme merely results in a conditional with a contradictory antecedent. Again, classical mereology, with its comprehension principle, is consistent. Yet (6) apparently is not.
Link (1998) calls this “the Counting Fallacy”.
As far as we know, the only serious attempt to debunk Schein’s argument is Link (1998, Ch. 13), where he says: “The air of absurdity is engendered by the fact that an inherently relative concept is taken as absolute” (p. 326). The analysis developed here can perhaps be viewed as a sustained, detailed defense of this brief remark.
An anonymous referee observes that (ia) has a distributive interpretation (ib).
-
(i)
a. Russell and Whitehead weighed 140 kg.
-
(ii)
b. Russell and Whitehead each weighed 140 kg.
Plausibly, this is because measure phrases such as ‘140 kg’ are quantized in the sense of Krifka (1989), defined by Krifka as (ii):
-
(ii)
\(\forall P[\texttt {QUA}_S(P) \leftrightarrow \forall x \forall y[P(x) \wedge y \sqsubseteq _S x \rightarrow \lnot P(y)]]\)
Thus, no proper part of something weighing 140 kg weighs 140 kg. Atomicity, as defined in (17a), is a special case of quantization, as no P-atom has proper P-parts. Whether measure phrases denote predicates true of atoms is contentious within linguistic semantics (see e.g. Schwarzchild (2002) and Rothstein (2017) for differing views). Regardless, most semanticists would agree that measure phrases determine a sort, namely that of degrees, or abstract representations of measurement.
-
(i)
To be sure, Rothstein’s analysis is not the only version of mereological singularism on which the denotations of count nouns is context-sensitive. In fact, a similar view is found in Krifka (1989).
Following Chierchia (2010), Rothstein understands the atoms of M as “vague”, “unspecified”, or “unstable”, thus presumably explaining why mass nouns are not countable. However, we see no reason to insist that M be completely atomic, as it might contain “gunk”. In that case, some mass terms will not denote sets or sums of atoms. We hope to address this in future work.
In the language of mereology, count atoms, in a given context, need not be discrete.
This contrasts importantly with Link (1983)’s original analysis, where there is a homomorphism between the lattice of mass quantities and the single lattice that is used to interpret all count nouns. Fingers and hands are composed of materially of overlapping stuff, which, in Rothstein’s semantics, refers to elements of M. This implies that the ordering of the material “stuff” via \(\sqsubseteq _{M}\) will not generally be preserved once we have have partitioned that “stuff” into countable entities in a context. The finger “stuff” is indeed part of the hand “stuff”, but as an atom in a given context, the hand has no proper parts.
Of course, in many cases it is possible to preserve the part-whole ordering on elements in M even after partitioning the “stuff” into countable atoms. Specifically, this will happen when the contextually-determined atoms in question do not share parts, as is presumably the case with examples like (i):
-
(i)
I can move the trash can and the five boxes.
Since the trash can and the five boxes do not share overlapping parts, presumably, the different contextually-relativized atoms will correspond to different elements of M. Thus, in such cases there will be a homomorphism between the structures determined by \(\sqsubseteq _{M}\) and the atomic semilattice resulting from the context.
-
(i)
Incidentally, these meta-metalinguistic atoms correspond structurally to groups in the sense of Link (1984, 1998). Roughly, the distinction between pluralities and groups resembles Russell (1919)’s well-known distinction between collections as many and collections as one. Whereas sums have no identity apart from the atoms comprising them, groups are collections viewed as entities in their own right, namely as special sorts of atoms. In fact, Link invokes this distinction to debunk what he calls “the Counting Fallacy”, similar to the Argument from Counting. Since groups are atoms, this is consistent with the general solution proposed here. However, we intentionally avoid identifying the meta-metalinguistic atoms in Fig. 1 with groups, for two reasons. The first is linguistic: the prototypical cases of group-denoting nouns are definite plurals such as ‘the cards’ or ‘the deck of cards’, not singular count nouns (Rothstein (2017)). The second is theoretical: unlike mereological sums, groups really do lead to a version of Russell’s Paradox. This unnecessarily invites the suspicion that the solution invoked to solve Schein’s paradox only relocates the problem at the level of groups.
In contrast, Link (1998, Ch. 13) replaces the comprehension principle in (6) with three axioms characterizing the existence of maximal sums.
To be clear, this is not the only motivation for pluralism. Indeed, Moltmann (2016) provides various positive linguistic arguments for pluralism, which we intend to address in future work.
A similar idea is sketched in Link (1998, Ch. 6).
In fact, if Link (1998, p. 337) is correct, this is perhaps bound to happen: “We cannot avoid a denotational (and thereby “singularizing”) approach towards pluralities anyway. Linguistic singularization is inevitable”.
Though it is something we intend to return to in future research.
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Snyder, E., Shapiro, S. Mereological Singularism and Paradox. Erkenn 88, 215–234 (2023). https://doi.org/10.1007/s10670-020-00347-9
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DOI: https://doi.org/10.1007/s10670-020-00347-9