Abstract
We consider connections between number sense—the ability to judge number—and the interpretation of natural language quantifiers. In particular, we present empirical evidence concerning the neuroanatomical underpinnings of number sense and quantifier interpretation. We show, further, that impairment of number sense in patients can result in the impairment of the ability to interpret sentences containing quantifiers. This result demonstrates that number sense supports some aspects of the language faculty.
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Notes
This diagram is from Dehaene et al. (2003).
This is similar to the numeron model of Gelman and Gallistel (1978). We will return to this model below.
See, in particular, van Benthem (1986) for an extensive discussion of this point. We will rely heavily on his development of semantic automata for simulating natural language quantifiers throughout.
This is essentially a variant of the Boyer-Moore vote counting algorithm. See Boyer and Moore (1991) for discussion. This algorithm has relatively low complexity but it still requires more in the way of computational resources than is required to test our Aristoteleans and cardinals. Alternatively, we could equip our testing device with a memory that tracks the number of cigar-smoking versus non-cigar-smoking deans. In fact, we could use a push-down automaton (pda) to simulate the quantifier in (7); once again, see van Benthem (1986) for a discussion of this. The behavior of a pda is fully determined by its current input, the state it is in and the content of a memory store. The exact character of this memory need not detain us. The important point, for our purposes, is that natural language quantifiers can be partitioned into two classes that differ as to their complexity. These differences can be seen in expressive power—majority quantifiers can make more finely grained distinctions between models than the cardinals and Aristoteleans. But greater expressive power comes at the price of making greater demands on computational resources than the cardinals and Aristoteleans. Furthermore, these distinctions in expressive power and computational demands will be manifest in any device that implements or simulates these quantifiers. Our firing line simulation of (7) requires us to impose more organization on the model, even if it does not require much in the way of memory.
This section reports work by McMillan et al. (2005) and McMillan et al. (2006).
A more complete discussion of these fMRI results can be found in McMillan et al. (2006).
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The authors wish to acknowledge support from NIH grant, NS44266.
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Clark, R., Grossman, M. Number sense and quantifier interpretation. Topoi 26, 51–62 (2007). https://doi.org/10.1007/s11245-006-9008-2
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DOI: https://doi.org/10.1007/s11245-006-9008-2