Granularity and Scalar Implicature in Numerical Expressions DRAFT (3/14/2011) Chris Cummins, Uli Sauerland and Stephanie Solt Abstract It has been generally assumed that certain categories of numerical expressions, such as ‘more than n’, ‘at least n’, and ‘fewer than n’, systematically fail to give rise to scalar implicatures. Various proposals have been developed to explain this perceived absence. In this paper, we consider the relevance of scale granularity to scalar implicature, and make two novel predictions: first, that scalar implicatures are in fact available from these numerical expressions at the appropriate granularity level, and second, that these implicatures are attenuated if the numeral has been previously mentioned or is otherwise salient in the context. We present novel experimental data in support of both of these predictions, and discuss the implications of this for granularity and for a recent constraint-based model of numerical quantifier usage. Introduction Modified numerals such as ‘more than nine’ or ‘at least ten’ typically seem to convey the impression that the speaker’s knowledge about the topic under discussion is imprecise. By contrast, unmodified numerals such as ‘ten’ convey precise knowledge. However, on closer inspection, this distinction is not as clear-cut as might be supposed: in particular, modified numerals appear to convey more information than their semantic analyses would suggest. This paper aims to characterise the meaning of these expressions more rigorously, and account for it in terms of semantic and pragmatic theory. The standard account of unmodified and modified numeral meaning claims that unmodified numerals have an upper bound to their interpretation, while the modified numerals do not. So, while ‘ten’ would establish that the cardinality in question is not greater than 10, ‘more than nine’ and ‘at least ten’ would not establish such an upper bound on the cardinality in question. Theories differ on whether the upper bound for the unmodified numerals is part of semantics (Breheny 2008 and others) or derived as a pragmatic implicature (Horn 1972 and others). But the claim that modified numerals have no upper bound is generally agreed on in the literature: Horn (1972, 1984) endorses this implicitly by referring to the interpretation of a numeral without an upper limit as the ‘at least n’ interpretation. More recently Krifka (1999) and Fox and Hackl (2006) have explicitly claimed that modified numerals with ‘more than’ or ‘at least’ have no upper bound and provided theoretical explanations for their empirical claim. Specifically, both note a conflict between the theory of scalar implicatures and the lack of an upper bound for modified numerals. We consider the two proposals in turn in the following section, before we return to the empirical issue of whether modified numerals establish an upper bound. In section 2, we present a proposal for the interpretation of modified numerals based on scale granularity and salience that predicts an upper bound in most cases when a modified numeral is used, and specifically one that depends upon