Language and Linguistics Compass 7/5 (2013): 273–294, 10.1111/lnc3.12018 Bare Numerals and Scalar Implicatures Benjamin Spector* CNRS-ENS-EHESS, Institut Jean Nicod, École Normale Supérieure Abstract Bare numerals present an interesting challenge to formal semantics and pragmatics: they seem to be compatible between various readings (‘at least’, ‘exactly’, and ‘at most’ readings), and the choice of a particular reading seems to depend on complex interactions between contextual factors and linguistic structure. The goal of this article is to present and discuss some of the current approaches to the interpretation of bare numerals in formal semantics and pragmatics. It discusses four approaches to the interpretation of bare numerals, which can be summarized as follows: 1. In the neo-Gricean view, the basic, literal meaning of numerals amounts to an ‘at least interpreta- tion’, and the ‘exactly n’ reading results from a pragmatic enrichment of the literal reading, i.e. it is accounted for in terms of scalar implicatures. 2. In the underspecification view, the interpretation of numerals is ‘underspecified’, with the result that they can freely receive the ‘at least’, ‘exactly’,or‘at most’ reading, depending on which of these three construals is contextually the most relevant. 3. In the ‘exactly’-only view, the numerals’ basic, literal meaning corresponds to the ‘exactly’ reading, and the ‘at least’ and ‘fewer than n’ readings result from the interaction of this literal meaning with background, non-linguistic knowledge. 4. In the ambiguity view, numerals are ambiguous between two readings, the ‘at least’ and ‘exactly’ readings.The article argues that in order to account for all the relevant data, one needs to adopt a certain version of the ambiguity view. But it suggests that numerals should not necessarily be thought of as being lexically ambiguous, but rather as giving rise to ambiguities through their interactions with so-called exhaustivity operators. According to such a view, the ambiguities triggered by numerals are to be explained in terms of a non-Gricean theory of scalar implicatures. 1. Introduction Numerical expressions (‘numerals’ for short) have a multiplicity of uses, some of which are illustrated in (1): (1) a. Determiner use: Three men came in. b. Degree use: (i) Mary is six feet tall. (ii) Fred drove at 55 mph. c. Predicative use: We are three. In this paper, we will be mostly concerned with the interpretation of the determiner use and, to a lesser extent, with the degree use. Furthermore, we will ignore cases where numerals are part of more complex phrases (more than 10, less/fewer than 10, at least 10, at most 10, ‘exactly’ 10, approximately 10, between 5 and 10, etc.), i.e. we will focus on ‘bare’ numerals (even though we will often use such complex phrases in order to paraphrase some of the readings that bare numerals give rise to). ©2013 The Author Language and Linguistics Compass © 2013 John Wiley & Sons Ltd 274 Benjamin Spector Bare numerals present an interesting challenge to formal semantics and pragmatics: they seem to be ambiguous between various readings, and the choice of a particular reading seems to depend on complex interactions between contextual factors and linguistic structure. Thus, consider the following three sentences. (2) Fred has three children. (3) In order to pass, Fred must have solved three problems. (4) If you have three children, you do not qualify for tax exemptions. In (2), the numeral ‘three’ is preferably understood as equivalent to ‘exactly three’, i.e. (2) is perceived as true just in case Fred has ‘exactly’ three children (no more, no less) – we will refer to such readings as ‘exactly’ readings. This is not so in the case of (3): (3) does not normally convey that Fred’s requirement is to solve ‘exactly’ three problems, which would imply that Fred would fail if he solved more than three problems. Rather, (3) is most naturally interpreted as stating that Fred must have solved at least three problems. That is, ‘three’ can be adequately paraphrased as ‘three or more’. We will refer to such readings as ‘at least’ readings. Finally, given reasonable assumptions about how tax exemptions work, one can infer from (4) that you don’t qualify for tax exemptions if you have three or fewer than three children. In this case, it seems that ‘three’ could be paraphrased as ‘three or fewer than three’ or ‘at most three’ (‘at most’ readings). One major goal for a semantic and pragmatic theory of numerals is to account for these various interpretations on the basis of (a) a uniform semantics for numerals and (b) pragmatic principles, which account for the actual interpretation of numerals in various syntactic environments and contexts of use. In this paper, I aim at comparing and assessing various attempts at providing such a theory. I will discuss four approaches to the interpretation of bare numerals, which can be briefly summarized as follows: 1. In the neo-Gricean view (e.g. Horn 1972; van Rooij et al. 2006), the basic, literal meaning of numerals amounts to an ‘at least interpretation’, and the ‘exactly n’-reading results from a pragmatic enrichment of the literal reading, i.e. it is accounted for in terms of conversational implicatures. 2. In the underspecification view (Carston 1988; Carston & Carston 1998), the interpreta- tion of numerals is ‘underspecified’, with the result that they can freely receive the ‘at least’, ‘exactly’,or‘at most’ reading, depending on which of these three construals is contextually the most relevant. 3. In the ‘exactly’-only view (Breheny 2008), numerals’ basic, literal meaning corresponds to the ‘exactly’ reading, and the ‘at least’ and ‘fewer than n’ readings result from the interaction of this literal meaning with background, non-linguistic knowledge. 4. In the ambiguity view (e.g. Geurts 2006), numerals are ambiguous between two readings, the ‘at least’ and ‘exactly’ readings. I will show that in order to account for all the relevant data, one needs to adopt a certain version of the ambiguity view. In particular, I will argue that numerals should not be thought of as being lexically ambiguous, but rather as giving rise to ambiguities through their interactions with so-called exhaustivity operators (Chierchia et al., forthcoming, and references cited therein). Section 2 presents the neo-Gricean approach and discusses its strengths and weaknesses. Section 3 shows that there is no compelling evidence that numerals can ever give rise to a genuinely ‘at most’ reading. Section 4 discusses the ‘exactly’-only view. Section 5 argues that the ambiguity account is necessary and sufficient in order to predict all the relevant data, and Section 6 discusses two possible implementations of the ambiguity ©2013 The Author Language and Linguistics Compass 7/5 (2013): 273–294, 10.1111/lnc3.12018 Language and Linguistics Compass © 2013 John Wiley & Sons Ltd Benjamin Spector 275 account – accounts based on a lexical ambiguity and accounts based on covert exhaustivity operators. Section 7 concludes the paper. 2. The Neo-Gricean Approach: Strengths and Weaknesses 2.1. A SKETCH OF THE NEO-GRICEAN APPROACH The neo-Gricean approach has two components: a semantic component, which determines the contribution of numerals to the literal meaning of the sentences in which they occur, and a pragmatic component, which determines the way in which this literal meaning is enriched as a result of a reasoning about speakers’ communicative intentions. Consider the sentence in (5). (5) Three girls went to the party. According to the neo-Gricean approach, the truth conditions assigned to (5) by the grammatical compositional rules of English make it equivalent to (6). (6) At least three girls went to the party. Before describing the way in which this meaning is enriched so as to yield the ‘exactly’ interpretation, let us be more explicit about how the equivalence between (5) and (6) comes about. Following, e.g. Kadmon (1985) and Kadmon and Kadmon (2001), one can start with the following semantic rule for a numeral such as ‘three’: (7) A sentence of the form three P Q,whereP and Q are one-place predicates (noun- phrases or verb-phrases) is true if and only if there is a collection C of individuals, which contains ‘exactly’ three members, and such that C belongs to the extension of both P and Q. It may seem that such a meaning for ‘three’ gives rise to an ‘exactly’ meaning for a sentence such as (5), but this is not so. Let us see why. In the case of (5), the noun ‘girls’ plays the role of the predicate P in (7), and the verb phrase ‘went to the party’ that of predicate Q. Hence, given (7), (5) is true just in case there is a collection made up of ‘exactly’ three girls and this collection of girls went to the party. Now, this is the case not only if three girls and no more went to the party but also if more than three girls went to the party. For suppose, for instance, that ‘exactly’ seven girls went to the party. Then consider any three girls among these seven girls. They constitute a collection of ‘exactly’ three girls that went to the party. Therefore, (5) is predicted to be true in such a situation and in any situation where more than three girls went to the party. Note that the equivalence between (5) and (6) is due to the fact that ‘girls’ and ‘went to the party’ are distributive predicates. Namely, whenever a certain collection C belongs to the extension of either predicate, any subcollection of C also belongs to their extensions. According to the neo-Gricean approach, the fact that (5) is generally understood to imply that no more than three girls went to the party is due to a reasoning that the hearer performs regarding the speaker’s communicative intentions.