The Sortal Theory of Plurals

This paper explores the hypothesis of a semantics for plurals with no atomic partial order defined on the domain of quantification, and thus no ontological distinction between singular and plural individuals. The idea is that the work usually done in the semantics by the atomic partial order is instead done by the syntax, which makes available to the semantics a phonologically covert sortal which provides the suitable granularity. This sortal theory of plurals is compared to the standard atomic theory with the two case studies of partitives and distributivity. 1 The Standard Approach: the Atomic Theory of Plurals In this Section, I introduce the core assumption of the standard approach to count nouns: that the domain of quantification is endowed with an atomic partial order. I discuss the main properties of this approach and illustrate it with the two case studies of plural partitives and distributivity. Finally, I note that the atomic partial order needs to be supplemented with another non-atomic partial order. This observation will be the starting point for an alternative non-atomic framework, introduced in the next Section. Core assumptions. The standard approach to count nouns rests on the following assumption (1) concerning the structure of the domain of quantification D. (1) The domain D is endowed with a partial order ≤ one/many such that (D,≤ one/many ) is isomorphic to1 (℘∗(At),⊆) for some (unique) subset At ⊆ D. The elements of At are called singular or atomic; those of D\At def = Pl are called plural; singular individuals do not have proper ≤ one/many -parts; plural individuals do.2 Let + one/many be the operation on D associated with ≤ one/many , namely such that (D,+ one/many ) is isomorphic to (℘∗(At),∪). The denotations of count nouns are constrained as in (2). ∗I would like to thank Alan Bale, Gennaro Chierchia, Paul Elbourne, Danny Fox and Irene Heim. Let ℘∗(X ) be the collection of all subsets of the set X , but the empty set. Let ≤ be a partial order on a set X ; for every x1, x2 ∈ X , x1 is a proper part of x2 wrt ≤ iff x1 < x2. Grønn, Atle (ed.): Proceedings of SuB12 , Oslo: ILOS 2008 (ISBN 978-82-92800-00-3), 399–413. Giorgio Magri Sortal Theory of Plurals (2) For every singular count noun nSG and corresponding plural noun nPL : a. [[nSG ]] ⊆ At. b. [[nPL ]] def = pl one/many ([[nSG ]]). where the plural operator pl one/many returns the closure of [[nSG ]] under +one/many . Assumption (1) says that ≤ one/many is an atomic partial order. I will thus dub (1)-(2) as the atomic theory of plurals (henceforth: ATP). As it stands, assumption (1) says that standard set theory provides a suitable framework for the semantics of plurals. Yet, at the end of this Section, we’ll see that assumption (1) needs to be supplemented by positing further structure on the domain of quantification. Main properties. After Sharvy (1980), let’s assume the semantics (3) for the definite article: [[the]] takes a property; sums up all its elements wrt + one/many ; checks whether this sum belongs to the given property; if it does, returns that sum; otherwise, is undefined. (3) [[the]] def = ι one/many This semantics for definites yields (4): the singular term ‘the boy’, if defined, denotes a singular individual; the plural term ‘the boys’ denotes a plural individual.3 Thus, the ATP (1)-(2) maps the morphological distinction between singular and plural number into the ontological distinction between singular and plural individuals. (4) First property: morphology/ontology correspondence. a. [[the boy]] ∈ At. b. [[the boys]] ∈ Pl. Consider next the function [[the−1]] defined in (5), which takes an individual and returns the set of its ≤ one/many -parts (namely, the ideal associated with that individual). (5) [[the−1]] def = λxe . λye . y ≤one/many x. The two functions [[the]] and [[the−1]] are related as in (6): the property [[boy(s)]] can be reconstructed from the individual [[the boys]] by means of [[the−1]]. Thus, the ATP (1)-(2) allows the definite article [[the]] to be inverted through [[the−1]]. (6) Second property: invertibility of ‘the’. a. [[the−1]]([[the]]([[boys]])) = [[boys]]. b. [[the−1]]([[the]]([[boys]])) ∩ At = [[boy]]. I now illustrate with the two case studies of plural partitives and distributivity the crucial role played by the invertibility of ‘the’ in (6). This statement is not accurate: if a noun denotes a singleton, its corresponding plural definite denotes a singular individual. I assume that this pathological case is ruled out independently, say by a constraint which forbids vacuous application of the plural operator, as in the case of singleton nouns.