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Context-Passing and Underspecification in Dependent Type Semantics

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Modern Perspectives in Type-Theoretical Semantics

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

Abstract

Dependent type semantics (DTS) is a framework of discourse semantics based on dependent type theory, following the line of Sundholm (Handbook of Philosophical Logic, 1986) and Ranta (Type-Theoretical Grammar, 1994). DTS attains compositionality as required to serve as a semantic component of modern formal grammars including variations of categorial grammars, which is achieved by adopting mechanisms for local contexts, context-passing, and underspecified terms. In DTS, the calculation of presupposition projection reduces to type checking, and the calculation of anaphora resolution and presupposition binding both reduce to proof search in dependent type theory, inheriting the paradigm of anaphora resolution as proof construction.

Our sincere thanks to Kenichi Asai, Nicholas Asher, Kentaro Inui, Yusuke Kubota, Sadao Kurohashi, Robert Levine, Zhaohui Luo, Ribeka Tanaka and Ayumi Ueyama for many insightful comments. We also thank to Youyou Cong, Yuri Ishishita, Ayako Nakamura, Yuki Nakano, Miho Sato and Maika Utsugi for helpful discussions. This research is partially supported by JST, CREST.

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Notes

  1. 1.

    The representative version of dependent type theory is Martin-Löf Type Theory (MLTT) (Martin-Löf 1984), which is also known as Constructive Type Theory or Intensional Type Theory. In this article, we use the term “dependent type theory” as a term to refer to any type theory with dependent types, including MLTT, \(\lambda P\) (Barendregt 1992), Calculus of Construction (CoC) (Coquand and Huet 1988), and Unified Type Theory (UTT) (Luo 2012b).

  2. 2.

    The subscripts i and j signify that we focus on judgments under a specified reading in which the antecedent of it is a donkey in (1), and the antecedent of He is A man in (2).

  3. 3.

    Francez and Dyckhoff (2010) and Francez et al. (2010) also pursued a proof-theoretic semantics of natural language. The difference in their approach is that the meaning of a word itself is defined via its verification conditions, whereas in our approach the meaning of a word is represented by a term in dependent type theory, as a contribution to the meaning of a sentence it may participate in. Luo (2014) provides a comparison between Francez’s approach and dependent-theoretic approaches, together with an interesting discussion on the proof-theoretic and model-theoretic status of natural language semantics via dependent type theory.

  4. 4.

    DTS also employs a two-dimensional notation for \(\varSigma \)-types as shown in Definition 2, which is reminiscent of the notation for record types in Cooper (2005).

  5. 5.

    We denote the set of free variables in B by \(fv(B)\).

  6. 6.

    DTS employs two sorts: \( \textsf {type}\) and \( \textsf {kind}\), and its terms are stratified into three levels: terms of type A where A is a type, types of sort \( \textsf {type}\), kinds of sort \( \textsf {kind}\). The only axiom is (\( \textsf {type}F\)) in Definition 3. The (\(\varPi F\)) rule allows the four patterns (\( \textsf {type}\), \( \textsf {type}\)), (\( \textsf {type}\), \( \textsf {kind}\)), (\( \textsf {kind}\), \( \textsf {type}\)) and (\( \textsf {kind}\), \( \textsf {kind}\)) as in Definition 1, and the (\(\varSigma F\)) rule allows the three patterns (\( \textsf {type}\), \( \textsf {type}\)), (\( \textsf {type}\), \( \textsf {kind}\)) and (\( \textsf {kind}\), \( \textsf {kind}\)) as in Definition 2. Thus, in this article, DTS employs dependent type theory in which \( \textsf {type}\) is an impredicative universe with respect to \(\varPi \). This setting is stronger than the predicative dependent type theory that Bekki (2014) is founded on, but not too strong to construct a proof of Girard’s paradox (Girard 1972; Coquand 1986; Hook and Howe 1986). We are grateful to Zhaohui Luo (personal communication) for discussions and comments on this issue.

  7. 7.

    Following the notation in logic, we write \( \mathbf{farmer }(x)\) for \(( \mathbf{farmer }\, x)\) and \( \mathbf{own } (x,y)\) for \(( \mathbf{own }\, y)\, x\), and so on. More generally, for an n-place predicate \( \mathbf{f }\), we often write \( \mathbf{f }(x_1, \ldots , x_n)\) for \((\ldots ( \mathbf{f }\, x_n) \ldots x_1)\).

  8. 8.

    Here we use the notation in DTS.

  9. 9.

    Note that the notion of predicate in a dependently typed setting is different from that used in a simply typed setting—the type theory that underlies Montague semantics (Montague 1974) and the standard framework of formal semantics (Heim and Kratzer 1998). In the simply typed setting, we usually use base type e for the type of entities and t for the type of truth-values, that is, we have \(e: \textsf {type}\) and \(t: \textsf {type}\); given these base types, a one-place predicate is assigned type \(e\!\rightarrow \!t\) and a two-place predicate type \(e\!\rightarrow \!e\!\rightarrow \!t\), and so on. In our dependently typed setting, by contrast, we have \( \mathbf{entity }: \textsf {type}\) and assign type \( \mathbf{entity }\rightarrow \textsf {type}\) to one-place predicates and type \( \mathbf{entity }\rightarrow \mathbf{entity }\rightarrow \textsf {type}\) to two-place predicates. In this sense, a predicate in our setting is not a function from entities to truth-values (or, equivalently, a set of entities) but a function from entities to types (that is, propositions); note also that the meanings of types are specified in terms of inference rules, not in terms of their denotation.

  10. 10.

    The problem of negated and conditional forms of predicational sentences is discussed in Tanaka et al. (2015). See also Chatzikyriakidis and Luo (2016) and footnote 1 of that paper for more information.

  11. 11.

    For a recent survey on the interpretation of predicational sentences and predicate nominals, see Mikkelsen (2011).

  12. 12.

    See Sect. 4.2 for more discussion on the formation rule of negation.

  13. 13.

    We will give a more detailed discussion of the notion of presupposition in the context of dependent type theory in Sect. 4.

  14. 14.

    Sundholm (1989) gives an analysis of generalized quantifiers in the framework of dependent type theory in which common nouns are treated as types. Tanaka (2014) points out that Sundholm’s approach faces an “over-counting” problem in the interpretation of the proportional quantifier most, and provides a refined analysis by interpreting common nouns as predicates in the framework of DTS. Also, Tanaka et al. (2014) combines the framework of DTS with the semantics of modals that allows explicit quantification over possible worlds and applies it to the analysis of modal subordination phenomena.

  15. 15.

    The analysis of negation goes back to Chatzikyriakidis and Luo (2014).

  16. 16.

    The definition of the judgment of the form \(\varGamma \vdash M:A\) is that there exists a proof diagram from the assumptions \(\varGamma \) to the consequence M : A. The judgment of the form \(\varGamma \vdash A\ true \) holds if and only if there exists a proof term M such that \(\varGamma \vdash M:A\).

  17. 17.

    In DTS, we assume that the global context \(\mathcal {K}\) at least includes:

    • The basic ontological commitment (e.g. \( \mathbf{entity }: \textsf {type}\))

    • The arities of predicates (e.g. \( \mathbf{whitsle }: \mathbf{entity }\rightarrow \textsf {type}\))

    • Ontological knowledge (e.g. ).

  18. 18.

    The use of the (\( CONV \)) rule in (22) depends on the \(\beta \)-equivalence \( \mathbf{whistle }(\pi _1\pi _1t) =_{\beta } \mathbf{whistle }(\pi _1\pi _1(\pi _1t,\pi _1\pi _2t))\), which is omitted for the sake of space.

  19. 19.

    The dynamic conjunction rule is an extension of the progressive conjunction rule in Ranta (1994) with a context-passing mechanism.

  20. 20.

    The types of the context c and the pair of contexts (c, u) are different. Thus, the two dynamic propositions M and N should be assigned different types. However, this does not require a polymorphic setting at the object-language level since M and N are preterms, and polymorphism is handled at the metalanguage level when type inference takes place.

  21. 21.

    Examples taken from “The Winograd Schema Challenge” (Levesque 2011), slightly adapted.

  22. 22.

    Bekki and Sato (2015) defined a fragment of dependent type theory with underspecified terms which has a decidable type-checking and inference algorithms.

  23. 23.

    See Soames (1989) and Beaver (2001) for useful surveys on the topic.

  24. 24.

    We abbreviate \(\lambda x_1 \ldots \lambda x_n.\, M\) as \(\lambda x_1 \ldots x_n.\, M\).

  25. 25.

    The treatment of gender information of pronoun as presuppositions goes back at least to Cooper (1983). See Sudo (2012) for a recent discussion.

  26. 26.

    See Tanaka et al. (2015) for more details. Earlier work using dependent type theory to analyze factivity includes Fox (1994b), Ranta (1994), and Krahmer and Piwek (1999).

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  1. Ochanomizu University/CREST, Japan Science and Technology Agency (JST), Tokyo, Japan

    Daisuke Bekki & Koji Mineshima

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  1. Daisuke Bekki
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  1. University of Gothenburg, Sweden & Open University Cyprus, Gothenburg, Sweden

    Stergios Chatzikyriakidis

  2. Dept. of Computer Science, University of London Royal Holloway, London, United Kingdom

    Zhaohui Luo

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Bekki, D., Mineshima, K. (2017). Context-Passing and Underspecification in Dependent Type Semantics. 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_2

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