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Abstract
We investigate the mechanisms proposed in formal semantics to account for the ambiguity generated by simple numerical expressions (e.g., ‘three students’). We explain how these mechanisms, when applied to more complex numerical expressions such as ‘between n and m’ (e.g., ‘between three and five students’), predict a surprising ambiguity between a doubly bounded (e.g., ‘at least three and at most five students’) and a lower-bounded reading (e.g., ‘at least three students’). While the lower-bounded reading is not detectable intuitively, results from three offline experiments and a response time study provide evidence in favour of its existence. Our contribution is twofold. On the experimental side, we present two psycholinguistic methods powerful enough to detect what we call phantom readings, i.e., readings that do not seem to have consequences for actual interpretation, but have detectable effects on processing. On the theoretical side, we show that certain semantic mechanisms that might be thought to overgenerate are in fact vindicated, since they are able to predict certain processing facts that would otherwise remain mysterious. We discuss how these results illustrate the need for a strong integration of formal semantics and psycholinguistic approaches.
Acknowledgements
We would like to thank Lewis Bott, Valentine Hacquard, Ira Noveck, François Récanati, Emmanuel Sander as well as audiences at LSCP in June 2011, at MIT (Gibson Lab) in April 2012 and at SALT in May 2012 for useful comments on earlier versions of this paper. We are grateful to Adrian Staub and the anonymous referee for their careful reading of our manuscript and their thorough reviews. Special thanks go to Isabelle Brunet, Anne-Caroline Fiévet and Amanda Swenson for their invaluable practical help.
Notes
1. For a long time, it was thought that the exact meaning for numerals was derived from the ‘at least’-meaning as a pragmatic, Gricean inference. This view, however, is now believed to be problematic (see e.g., Breheny, Citation2008; Geurts, Citation2006; Spector, Citation2013, among others).
2. Geurts (Citation2006) proposes a slightly different rule, which amounts to the following: [Num N] VP is true if there is a unique group X such that X has the properties represented by Num, N and VP. When Num is a bare numeral and both N and VP are distributive predicates, this does not change anything. However, for More than three Ns VPs, Geurts’ rule gives rise to ‘Four and no more than four Ns have the property represented by VP’, which is not a desirable result. Replacing uniqueness with maximality (as in other accounts) solves this problem. From the point of view of this paper, the most important point is that the Existential Closure rule, which is assumed in one form or another by all accounts, gives rise to the ‘surprising’ at least-reading.
3. Note that we are assuming here that at most does not itself give rise to ‘phantom readings’ through Existential Closure. However, one could entertain the possibility that the basic meaning of ‘at most n’ is ‘being a group consisting in fewer than n + 1 invdividual’. Then, by applying Existential Closure to (18), we would get ‘there is a group of red dots of cardinality less than 6’, which would be true even when there are seven red dots. That ‘at most n’ is not subject to Existential Closure is clear from the fact that even with collective predicates, the maximality component is present (contrary to what we observed with between n and m). That is, a sentence such as ‘At most 100 soldiers surrounded the castle’ is false if there is a group of 150 soldiers that surrounded the castle, even if some other group of 50 soldiers also surrounded the castle.
4. Specifically, the global mean score for the At most sentences in the conditions where they were expected to be true (i.e., when paired with Inferior and Intermediate pictures) was notably lower than the score obtained for the At least sentences in the relevant corresponding conditions (i.e., when paired with Intermediate and Superior pictures): 54% vs. 91%. Plausible explanations exist for this discrepancy. Downward-entailing quantifiers like ‘at most’ have been found to be harder to process than upward-entailing quantifiers like ‘at least’ (Cummins & Katsos, Citation2010; Geurts & van der Slik, Citation2005; Geurts et al., Citation2010). The relatively poor performance on the former in our task is consistent with these previous findings and may reflect the same effect.
5. Under the hypothesis that correct responses might be faster than incorrect responses, one may wonder whether the observed effects of Ambiguity could be partly driven by the current data treatment. For, while it is in principle possible to distinguish correct responses from errors in the unambiguous True/False conditions, no such a distinction can be made in the ambiguous Target conditions, where both response types are assumed to be acceptable. Hence, to rule out this alternative explanation, we carried out an extra analysis that considers RTs for all response types in both ambiguous and unambiguous conditions. Results from this second analysis yielded the same conclusions as the ones currently reported.