A Vagueness Approach to the Mass/Count Distinction

Outline Although the grammatical mass/count distinction ultimately depends on lexical selection and is subject to cross-linguistic variation, the perceptual mass/count distinction is determined by the intuitive availability of atomic objects. An illuminating definition of atomic object is proposed under the framework built by Chierchia (2010). This analysis facilitates a sensible solution to the problem about counting and it also contributes toward a better understanding of situated interpretation of nouns and the notion of exemplification which are central to situation semantics. Symbolic primitives Part-of : ≤; proper-part-of : d < d′ =def d ≤ d′ ∧ d 6= d′; sum of d′ and d′′: d′ ∪ d′′ =def the minimal object d such that d′, d′′ ≤ d. Mass/count distinction For each noun stem P, [P] is a number-neutral property P . The interpretation Pc of P relative to a context c is threefold: a positive extension P + c consisting of objects being P , a negative extension P− c of those which are not P and a vagueness band P ∗ c being the gap characterizing the inherently vague boundary between P -objects and non-P objects (Chierchia 2010). Generally, an individual d is an atomic P -object (or P -atom, notation: d ∈ PAT c ) in context c iff it is such a P -object that any division will produce non-P -objects, formally, (1). Differently, the atomic objects falling under some certain nouns, e.g., rope and rock, which normally have both mass usage and count usage, are defined by connectedness (Moltmann 1998). Then the perceptual mass/count distinction can be characterized by the lack/existence of atoms defined by (1). Plural objects consist of multiple atoms, while mass objects’ lack of atoms follows from the observation that half of a mass object falls under P ∗ c (if it is too small to count as P) or P+ c (otherwise). Moreover, (1) is compatible with the fact that an atomic object can still be atomic even if some tiny piece is chipped off which is a problem with Bale and Barner (2009) and Chierchia (2010). Counting principle Since a teapot and its residue after losing a tiny fragment both fall under TEAPOT c , it poses a problem about counting: there can be many overlapping teapots within a single teapot (Kratzer 2011). The solution is to divide each P -object into disjoint P -atoms, i.e., for each d ∈ P+ c , pick an Sd ⊆ P+ c consisting of disjoint P -atom(s) whose sum is d, formally, (2). The number |d|P,c =def |Sd|. Furthermore, in each well-defined structure, no matter which Sd is picked out, |Sd| is expected to be the same for the d at issue as long as (2) holds. Situation semantics Without explicit modifiers, people can refer to no P -atom that is a proper part of a P -atom within the same situation by P. Therefore, the interpretation of a count noun P relative to a situation s and a context c (before being associated with determiners) contains the sums of maximal P -atoms within s, formally, (3). In contrast, the situated interpretation of a mass noun P is (4), as the maximality constraint on count nouns doesn’t apply to mass nouns. A situation s exemplifies a proposition p iff (5) is satisfied (Kratzer 2002, 2011). Nonetheless, a situation exemplifying the proposition expressed by (6) remains its exemplification even if the apple loses or gains some water molecules. More generally, the minimality (5) requires should be defined in terms of the discrete part-of relation ≺ relative to the set C of nouns involved in the proposition at issue, e.g., C(6) = {apple} for (6). Specifically, the minimal situations should be such that (a) none of them can still exemplify the proposition if any atom mentioned in the proposition is removed from it, and (b) no irrelevant object is contained. Formally, s′ ≺C,c s iff (7) is satisfied where s \ s′ is the complement of s′ with respect to s, i.e., the maximal part of s that is disjoint with s′. Conclusion The intuitive definition of atomic object sheds new light on the perceptual mass/count distinction and it also grounds the sensible analyses of intimately related issues.