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International Workshop on Logic, Language, Information, and Computation

WoLLIC 2019: Logic, Language, Information, and Computation pp 415–432Cite as

Modal Auxiliaries and Negation: A Type-Logical Account

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11541))

Abstract

This paper proposes an analysis of modal auxiliaries in English in Type-Logical Grammar. The proposed analysis captures the scopal interactions between different types of modal auxiliaries and negation by incorporating the key analytic idea of Iatridou and Zeijlstra [6], who classify English modal auxiliaries into PPI and NPI types. In order to technically implement this analysis, we build on Kubota and Levine’s [8, 10] treatment of modal auxiliaries as higher-order operators that take scope at the clausal level. The proposed extension of the Kubota/Levine analysis is shown to have several interesting consequences, including a formal derivability relation from the higher-order entry for auxiliaries to a lower-order VP/VP entry traditionally recognized in categorial grammar (CG) research. The systematic analysis of the scopal properties of auxiliaries and the somewhat more abstract meta-comparison between ‘transformational’ and ‘non-transformational’ analytic ideas that become possible in a type-logical setup highlight the value of taking a logical perspective on the syntax of natural language embodied in Type-Logical Grammar research.

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Notes

  1. 1.

    As a reviewer notes, transportability of the analysis depends significantly what is common between the two frameworks. Since the Displacement Calculus [14] is largely similar to Hybrid TLG, translation of the present analysis to the Displacement Calculus should for the most part be straightforward (see Morrill and Valentín [15] in this connection). Lowering to VP/VP is of course not available in Linear Categorial Grammar [12] and Abstract Categorial Grammar [4], but lowering to (NP \(\multimap \) S) \(\multimap \) (NP \(\multimap \) S) should be possible.

  2. 2.

    One might wonder about the classification of must and should as PPIs, given that they can appear unproblematically in the scope of negation in sentences such as I don’t think that John should be even one little bit nice to anyone in that room, where the NPIs even, anyone and one little bit appear with no hint of ill-formedness. But here it is crucial to bear in mind that polarity items as a broad class are known to be sensitive to not only semantic scope effects but syntactic contexts as well; see Richter and Soehn [19] for a survey of syntactic conditions on a range of NPIs in German. Iatridou and Zeijlstra argue that the same syntactic sensitivity holds for PPIs, and note that

    figure c
  3. 3.

    We remain agnostic about the exact formal implementation of syntactic features in the present paper. This could be done, for example, via some mechanism of unification as in HPSG. Another approach would involve the use of dependent types, along lines suggested by Morrill [13] and worked out in some detail by Pompigne [17]. So far as we can tell, the results of the current paper does not hinge on the specific choice on this matter.

  4. 4.

    Extending the present analysis to cases involving negative quantifiers (e.g. Nothing need be said about this) is a task that we leave for future work.

  5. 5.

    Though we have chosen to posit two distinct lexical entries for the ‘neutral’ modals (can, could and may) for high and low scoping possibilities with respect to negation, corresponding respectively to the scoping properties of the unambiguous modals, it is easy to collapse these two entries for these modals by making the polarity features for the two S’s and two VPs in the complex higher-order category for the modal totally underspecified and unconstrained (except for one constraint , to exclude the possibility of double negation marking *can not not), along the following lines:

    By (partially) resolving underspecification, we can derive both the ‘PPI’ and ‘NPI’ variants of the modal lexical entry in (10) from (i), thus capturing scope ambiguity via a single lexical entry. (i) allows for other instantiations of feature specification, but these are either redundant (yielding either high or low scope that are already derivable with the PPI and NPI instantiations in (10)), or useless (i.e. cannot be used in any well-formed syntactic derivation), and hence harmless. Thus, if desired, the lexical ambiguity we have tentatively assumed in the main text can be eliminated by adopting the more general lexical entry along the lines of (i) without the danger of overgeneration.

  6. 6.

    Here and below, \(\mathsf {{\epsilon }}\) denotes the null string.

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Acknowledgments

We would like to thank two anonymous reviewers for helpful comments. This work was supported by JSPS KAKENHI JP15K16732 and the NINJAL collaborative research project ‘Cross-linguistic Studies of Japanese Prosody and Grammar’.

Author information

Authors and Affiliations

  1. National Institute for Japanese Language and Linguistics, Tachikawa, Japan

    Yusuke Kubota

  2. Ohio State University, Columbus, USA

    Robert Levine

Authors
  1. Yusuke Kubota
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Correspondence to Yusuke Kubota .

Editor information

Editors and Affiliations

  1. Department of Philosophy and Religious Studies, Utrecht University, Utrecht, The Netherlands

    Rosalie Iemhoff

  2. Utrecht Institute of Linguistics OTS, Utrecht University, Utrecht, The Netherlands

    Michael Moortgat

  3. Centro de Informática, Universidade Federal de Pernambuco (UFPE), Recife, Brazil

    Ruy de Queiroz

A Hybrid Type-Logical Grammar

A Hybrid Type-Logical Grammar

1.1 A.1 Syntactic Types

figure bj

Note: The algebra of syntactic types is not a free algebra generated over the set of atomic types with the three binary connectives /, \(\backslash \), and . Specifically, given the definitions in (30), in Hybrid TLG, a vertical slash cannot occur ‘under’ a directional slash. Thus, is not a well-formed syntactic type. This is a deliberate design, and Hybrid TLCG differs from closely related variants of TLG (such as the Displacement Calculus Morrill [14] and NL\(_\lambda \) Barker and Shan [1]) in this respect.

1.2 A.2 Mapping from Syntactic Types to Semantic Types

1.3 A.3 Mapping from Syntactic Types to Prosodic Types

1.4 A.4 Deductive Rules

Notes: Corresponding to the asymmetry in the status of the directional slashes (/, \(\backslash \)) and the vertical slash ( ) in the definitions of syntactic types, there is an asymmetry in the definitions of the deductive rules for the two types of slashes.

Note in particular that in the Introduction rules for / (\(\backslash \)), instead of lambda binding, the prosodic variable of the hypothesis that is withdrawn is removed from the prosodic term on the condition that it appears on the right (left) edge of the prosody of the expression that feeds into the rule. (One way to make sense of this is to take the /,\(\backslash \) Introduction rules as abbreviations of theorems in which the variable is first bound by left and right lambda abstraction as usual [23], immediately followed by a step of feeding an empty string to the prosodic function thus obtained.)

So far as we can tell, fixing the prosodic type to be st for directional (i.e. Lambek) syntactic types is crucial for ensuring the particular way in which the directional and vertical slashes interact with one another in the various Slanting lemma and related results (which play important roles in the linguistic analyses we have presented above).

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Kubota, Y., Levine, R. (2019). Modal Auxiliaries and Negation: A Type-Logical Account. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_25

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  • DOI: https://doi.org/10.1007/978-3-662-59533-6_25

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