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JSAI International Symposium on Artificial Intelligence

JSAI-isAI 2013: New Frontiers in Artificial Intelligence pp 115–124Cite as

Constructive Generalized Quantifiers Revisited

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8417))

Abstract

This paper proposes a proof-theoretic definition for generalized quantifiers (GQs). Sundholm first proposed a proof-theoretic definition of GQs in the framework of constructive type theory. However, that definition is associated with three problems: the proportion problem, absence of strong interpretation and lack of definitional uniformity. This paper presents an alternative definition for “most” based on polymorphic dependent type theory and shows strong potential to serve as an alternative to the traditional model-theoretic approach.

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Notes

  1. 1.

    Linguistics and Philosophy 17 (1994), pp. 109–111.

  2. 2.

    The fact that, in Sundholm [18], A-injections are defined as in the Appendix. Since the type for \(f\) is obviously unsuitable for Sundholm’s purposes, we understand that he intended to define it as we show in this section. What is more, Sundholm mentioned that injections should be simply replaced with A-injections, and did not account for surjections. Therefore, we also define what we call A-surjections, whose domain is of the form \((\varSigma x:B)C\), and address that point. All of these definitions are available in the Appendix.

References

  1. Aczel, P.: Frege structures and the notions of proposition truth and set. In: Barwise, J., Keisler, H.J., Kunen, K. (eds.) The Kleene Symposium, pp. 31–59. North-Holland Publishing Company, Amsterdam (1980)

    Chapter  Google Scholar 

  2. Barendregt, H., et al.: Introduction to generalized type systems. J. Funct. Program. 1(2), 125–154 (1991)

    Article  MathSciNet  Google Scholar 

  3. Bekki, D.: Dependent type semantics: an introduction. In: Christoff, Z., Galeazzi, P., Gierasimczuk, N., Marcoci, A., Smet, S. (eds.) Logic and Interactive RAtionality (LIRa) Yearbook 2012, vol. I, pp. 277–300. University of Amsterdam, The Netherlands (2014)

    Google Scholar 

  4. Chierchia, G.: Anaphora and dynamic binding. Linguist. Philos. 15(2), 111–183 (1992)

    Article  Google Scholar 

  5. Dávila-Pérez, R.: Semantics and Parsing in Intuitionistic Categorial Grammar. Ph.d. thesis, University of Essex (1995)

    Google Scholar 

  6. Fernando, T.: Conservative generalized quantifiers and presupposition. In: Proceedings of the SALT, vol. 11, pp. 172–191 (2001)

    Google Scholar 

  7. Fox, C., Lappin, S.: Foundations of Intensional Semantics. Wiley-Blackwell, Malden (2005)

    Book  Google Scholar 

  8. Geach, P.T.: Reference and Generality: An Examination of Some Medieval and Modern Theories. Cornell University Press Ithaca, Ithaca (1962)

    Google Scholar 

  9. Heim, I.: E-Type pronouns and donkey anaphora. Linguist. Philos. 13(2), 137–177 (1990)

    Article  Google Scholar 

  10. Heim, I., Kratzer, A.: Semantics in Generative Grammar, vol. 13. Blackwell, Oxford (1998)

    Google Scholar 

  11. Jacobs, B.: Studies in Logic and the Foundations of Mathematics, vol. 141. Elsevier, Amsterdam (1999)

    Google Scholar 

  12. Kadmon, N.: Uniqueness. Linguist. Philos. 13(3), 273–324 (1990)

    Article  Google Scholar 

  13. Kanazawa, M.: Weak vs strong readings of donkey sentences and monotonicity inference in a dynamic setting. Linguist. Philos. 17(2), 109–158 (1994)

    Article  Google Scholar 

  14. Kievit, L.: Sets and quantification in a constructive type theoretical semantics for natural language (manuscript) (1995)

    Google Scholar 

  15. Krifka, M.: Pragmatic strengthening in plural predications and donkey sentences. In: Proceedings of the SALT, vol. 6, pp. 136–153 (1996)

    Google Scholar 

  16. Martin-Lof, P., Sambin, G.: Intuitionistic Type Theory. Bibliopolis, Naples (1984)

    Google Scholar 

  17. Ranta, A.: Type-Theoretical Grammar. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  18. Sundholm, G.: Constructive generalized quantifiers. Synthese 79(1), 1–12 (1989)

    Article  MathSciNet  Google Scholar 

  19. Sundholm, G.: Proof theory and meaning. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, pp. 165–198. Springer, New York (1986)

    Google Scholar 

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Acknowledgments

We would like to express our deepest gratitude to the anonymous reviewers of LENLS10, whose comments and suggestions were of tremendous value. We would also like to thank Katsuhiko Yabushita, Matthijs Westera and Yuyu So, who provided invaluable comments and kind encouragement. Daisuke Bekki is partially supported by JST, CREST.

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Authors and Affiliations

  1. Graduate School of Humanities and Sciences, Faculty of Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo, 112–8610, Japan

    Ribeka Tanaka, Yuki Nakano & Daisuke Bekki

  2. National Institute of Informatics, Tokyo, Japan

    Daisuke Bekki

  3. CREST, Japan Science and Technology Agency, Tokyo, Japan

    Daisuke Bekki

Authors
  1. Ribeka Tanaka
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  2. Yuki Nakano
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  3. Daisuke Bekki
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Corresponding author

Correspondence to Ribeka Tanaka .

Editor information

Editors and Affiliations

  1. Seikei University, Musashino-shi, Tokyo, Japan

    Yukiko Nakano

  2. National Institute of Informatics Research Division, Tokyo, Japan

    Ken Satoh

  3. Ochanomizu University, Tokyo, Japan

    Daisuke Bekki

Appendix

Appendix

Definition 1

(Construction of the Natural Numbers). Sundholm [18] presented the construction of natural numbers with the \(M_r\) sequence proposed by Aczel [1]. First put

$$ \begin{array}{l} f(a)=R(a,eq(N,0,0),(x,y)(y+eq(N,0,0))):U.\\ (a:N, U \ \text{ is } \text{ the } \text{ universe } \text{ of } \text{ small } \text{ types }\!, + \ \text{ is } \text{ a } \text{ disjoint } \text{ sum }\!.) \end{array} $$

and

$$ \begin{array}{l} f(0)=eq(N,0,0):U\\ f(s(a))=f(a)+eq(N,0,0):U \ (a:N). \end{array} $$

Finally, put

$$ M(a)=R(a,\bot ,(x,y)f(x)):U \ (a:N) $$

then

$$ \begin{array}{l} M(0)= \bot :U\\ M(1)=eq(N,0,0):U\\ M(s(s(a)))=M(s(a))+eq(N,0,0):U \ (a:N). \end{array} $$

Definition 2

(Primitive Recursive Functions).

$$ \begin{array}{lll} sg(0)&{}=&{}1\\ sg(s(a))&{}=&{}0 \end{array} \qquad \left\{ \begin{array}{lll} rem(0/2)&{}= &{}0\\ rem(s(a)/2)&{}= &{}sg(rem(a/2)) \end{array} \right. \nonumber $$
$$ \begin{array}{lll} \bigl [ 0/2 \bigr ] &{}= &{}0 \\ \bigl [ s(a)/2 \bigr ] &{}=&{} \bigl [ a/2 \bigr ] +rem(a/2) \end{array} $$

Definition 3

( \(| \_ |\) ) .

figure f

Definition 4

( \(\ge \) ) .

figure g

Definition 5

(Surjection).

figure h

Definition 6

(Bijection).

figure i

Definition 7

(Finite).

figure j

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Tanaka, R., Nakano, Y., Bekki, D. (2014). Constructive Generalized Quantifiers Revisited. In: Nakano, Y., Satoh, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2013. Lecture Notes in Computer Science(), vol 8417. Springer, Cham. https://doi.org/10.1007/978-3-319-10061-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-10061-6_8

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  • Online ISBN: 978-3-319-10061-6

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