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Generalized Quantifiers on Dependent Types: A System for Anaphora

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Modern Perspectives in Type-Theoretical Semantics

Part of the book series: Studies in Linguistics and Philosophy ((SLAP,volume 98))

Abstract

We propose a system for the interpretation of anaphoric relationships between unbound pronouns and quantifiers. The main technical contribution of our proposal consists in combining generalized quantifiers (Mostowski, Fundamenta Mathematicae, 44:12–36, 1957; Lindström, Theoria 32:186–195, 1966; Barwise, Cooper Linguist Philos 4(2):159–219, 1981) with dependent types (Martin-Löf, An intuitionstic theory of types, 1972; Ranta, Type-Theoretical Grammar, 1994, Makkai, First Order Logic with Dependent Sorts, with Applications to Category Theory, 1995). Empirically, our system allows a uniform treatment of the major types of unbound anaphora, with the anaphoric (dynamic) effects falling out naturally as a consequence of having generalized quantification on dependent types.

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Notes

  1. 1.

    For \(a \in \Vert X\Vert , b \in \Vert Y\Vert , c \in \Vert Z\Vert \), a triple \(\langle a, b, c \rangle \) is compatible iff \( \Vert \pi _{Y,x}\Vert (b) = a, \Vert \pi _{Z,y}\Vert (c) = b, \Vert \pi _{Z,x}\Vert (c) = a.\)

  2. 2.

    True in the sense that it is not dummy.

  3. 3.

    By this we mean the (categorical) limit of the described (dependence) diagram in the category Set of sets and functions. The notion of a limit used here is the usual category-theoretic notion. In particular, the notion of a parameter space makes sense in any category with finite limits. However, the definition we give in the text is a standard representation of this limit and does not require any knowledge of Category Theory.

  4. 4.

    This association can be partial.

  5. 5.

    Such sets might not be determined uniquely if one of them is empty.

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Acknowledgements

The work of Justyna Grudzińska was funded by the National Science Center on the basis of decision DEC-2012/07/B/HS1/00301. The authors would like to thank the anonymous reviewers for valuable comments. An early version of the system presented in this chapter appeared in Proceedings of EACL 2014 Type Theory and Natural Language Semantics Workshop (see Grudzinska and Zawadowski 2014).

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Grudzińska, J., Zawadowski, M. (2017). Generalized Quantifiers on Dependent Types: A System for Anaphora. In: Chatzikyriakidis, S., Luo, Z. (eds) Modern Perspectives in Type-Theoretical Semantics. Studies in Linguistics and Philosophy, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-50422-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-50422-3_5

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