Degree semantics has been developed to study how the meanings of measurement and comparison are encoded in natural language. Within degree semantics, this paper proposes a difference-based (or subtraction-based) approach to analyze the semantics of comparatives. The motivation is the measurability and comparability of differences involved in comparatives. The main claim is that comparatives encode a subtraction equation among three scalar values: two measurements along an interval scale and the difference between them. We contribute two innovations: (i) using interval arithmetic to implement subtraction, and (ii) analyzing comparative morpheme -er/more as an additive particle, denoting the default, most general, positive difference. Our analysis inherits existing insights in the literature. Moreover, the innovations bring new conceptual and empirical advantages. In particular, we address the interpretation of comparatives containing than-clause-internal quantifiers and various kinds of numerical differentials. We also account for three puzzles with regard to the scope island issue, the monotonicity of than-clauses, and the discourse status of the standard in comparison.

Abstract. NOCH-type additive particles (e.g., German noch, Chinese hái, Hungarian még) have a widespread distribution that roughly covers the uses of English still, also, and even. We propose that with a built-in Gricean Maxim of Quantity Be Informative in its lexical semantics, a NOCH-type particle explicitly requires that the discourse be incremental, and the NOCH-marked sentence add new information and further narrow down the context set, making the whole discourse even more informative. We also show that the cross-linguistic widespread distribution of NOCH-type particles is not arbitrary: there are three ways to build an incremental discourse structure, and these three implementations give rise to the three major uses of NOCH-type particles.

This paper addresses the ambiguity of comparatives that contain a permission-related existential modal in their than-clause. For example, given the context that the interval of permitted speed is between 35 and 50 mph, the sentence Lucinda is driving less fast than allowed is ambiguous between two readings: (i) her speed is below the minimum (i.e., 35 mph); (ii) her speed is below the maximum (i.e., 50 mph). Previously, this ambiguity has been attributed to either the scopal interaction between a negation element and a modal (Heim 2006a) or the optional application of a silent operator (Crnič 2017). Here we show that these two lines of accounts under-or over-generate. Instead, we propose that the source of this ambiguity is located in the ambiguous answerhood for wh-questions corresponding to this kind of than-clauses (e.g., how fast is Lucinda allowed to drive). The current proposal consists of three parts. First, based on Zhang & Ling 2015, 2017a,b, we adopt a generalized interval-arithmetic-based recipe for computing the semantics of comparatives. Second, the semantics of than-clauses is considered equal to that of short answers to corresponding wh-questions. Third, since the use of existential priority modals in wh-questions leads to the 'mention-some/mention-all' ambiguity for answerhood, we propose that this ambiguity projects in further derivation and leads to the two readings for comparatives like the Lucinda sentence.