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Early theories of the mass/count distinction in formal semantics and philosophy attempted to analyze the meanings of mass and count nouns in terms of properties like cumulativity, divisivity (due to Quine 1960), atomicity, and homogeneity, among others (see Pelletier 1979 and references therein). Many also agree that such properties are best represented in algebraic or mereological terms. This idea was introduced into formal semantics by Link (1983). His main innovation is to propose that the domain of (concrete) entities has the algebraic structure of a complete join semi-lattice. This allows him to model the differences between mass and count nouns, on the one hand, as well as similarities between mass and plural nouns, on the other hand. There is a sortal semantic distinction between mass and count nouns, which is based on the atomic and non-atomic ontological distinction (see, e.g., Link 1983, 1998), and modeled by means of an atomic and a non-atomic join semi-lattice, respectively. Count nouns are interpreted in the atomic lattice, mass nouns in the non-atomic one. Mass nouns pattern with plurals in having the property of cumulative reference (the term coined by Quine (1960) for the semantics of mass terms, realized as nouns or adjectives).

This chapter analyses how we give linguistic expression to counting as a cognitive process of interpretation in context with a focus on the interaction between aspectual adverbs and phrases with numerical DPs. For example, the use of already or no longer in there are already/no longer three students here conveys the speaker’s knowledge of the recent past of the described situation and gives rise to indexical inferences, for example, that three students were here. In order to capture these indexical inferences, ter Meulen proposes a DRT analysis that models both the information that utterances presuppose in the common ground and the inference triggered by the aspectual adverb in the context. Ter Meulen then considers how this approach could be extended to polar question uses of aspectual adverbs (Still?/Already?), and discusses the interesting and intricate differences between uses of aspectual adverbs and numerical DPs cross-linguistically.

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Lam’s chapter analyses the phenomenon of reduplication, primarily in Cantonese, a language in which reduplication can occur in the nominal domain, usually on the classifier, to express plurality, in the verbal domain to express an iteration of eventualities or a prolongation of an eventuality, and in the adjectival domain to express property attenuation. Lam argues for a uniform treatment of reduplication in terms of summation and a sensitivity to whether the replicated elements are (strictly) quantized or cumulative. For example, entity-denoting nominal classifiers, and verbal predicates denoting quantized sets of eventualities (such as tiu3 ‘jump’) denote pluralities or iterations, respectively, based on the summing of entities/eventualities. For verbal expressions that denote cumulative predicates (such as fan3 ‘sleep’), summation adds up ‘portions’ of unbounded, overlapping eventualities forming one temporally extended eventuality. Lam proposes that Cantonese bare adjectives denote dimensions, e.g. tallness, but not degree or magnitude. This nullifies any semantic effect of summation, and the attenuation effect is derived via competition with other forms using hou2 (‘very’).

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