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JSAI International Symposium on Artificial Intelligence

JSAI-isAI 2019: New Frontiers in Artificial Intelligence pp 164–181Cite as

Variables vs. Parameters in the Interpretation of Natural Language

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12331))

Abstract

This paper compares two systems of functional type logic that have been applied to the analysis of meaning composition in natural language: Montague’s Intensional Logic IL and its extensional substratum Ty2 of two-sorted type theory. The two systems differ in their treatment of reference and quantification over indices (like possible worlds or times): whereas the denotations of IL-formulae (inter alia) depend on indices as parameters, their Ty2-counterparts contain explicit free and bound variables for them. Building on earlier results, it is argued that, appearances to the contrary, the two systems are largely equivalent; that any differences in expressivity are irrelevant to said applications; and that the equivalence also extends to variations of the systems that make use of multiple indices (as in mixed systems of modal and temporal interpretation) or additional dimensions (as in standard accounts of context dependence).

The first two sections are based on [32] and cover the material the second author presented at his invited LENLS lecture; the results in the third section had also been hinted at in that presentation, but were only obtained by a joint effort of both authors well after the LENLS conference. We would like to thank the LENLS audience for an interesting discussion following the presentation and the editors for providing an opportunity to publish our work.

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Notes

  1. 1.

    With the variability of the underlying Logical Space of possibilities, the intensions are generalizations of Kripke’s [17] possible worlds, as propagated in [19].

  2. 2.

    The \(\lambda \)-operator indicates functional abstraction: \([\lambda x. \dots \ x \ \dots ]\) is that function f that assigns \(\dots \ u \ \dots \) to any u in its domain D (which is left implicit); given the set-theoretic account of functions we thus have:

    $$ [\lambda x. \dots \ x \ \dots ] \ = \ \{(u,v) \vert u \in D \ \& \ v = \dots \ u \dots \}.$$
  3. 3.

    Following [20], the sets \(\textit{Con}_{(s,b)}\) are usually referred to as \(\textit{Con}_b\), which we feel is confusing (given their interpretation) and moreover leads to complications when it comes to comparing IL and Ty2.

  4. 4.

    See [20, p. 387] and [9, p. 15] for other, less transparent ways of defining conjunction.

  5. 5.

    It ought to be mentioned that there is every reason to believe that Montague had been aware of Theorem (6), given the very design of IL.

  6. 6.

    In fact, it is almost that fragment, except for a little twist concerning constants occurring without index argument; cf. [30, p. 75].

  7. 7.

    Tautologies are among the obvious exceptions.

  8. 8.

    Hint: (7-a) and (8-a) boil down to a cardinality condition on \(D_s\); and the \(^*\)-counterpart of (8-b) is:

    • \((\lambda F^{((s,t),t)}. \ (\lambda \upsilon ^s_0. \ ( \lambda p^{(s,t)}. \ (p(\upsilon ^s_0) \ = F)))) \ (\lambda p. p(\upsilon ^s_0)).\)

  9. 9.

    The classic elimination algorithm from [4, p. 189] cannot be applied directly because it outputs terms of types outside \(\textit{IT}^+\). Thanks are due to Oleg Kiselyov for bringing up the question during the discussion following the LENLS presentation.

  10. 10.

    The syntactic variable t indicates the surface position from which the underlying noun plush toys has been moved and corresponds to the \(\lambda \)-bound Ty2-variable P in (13-b). See [10, p. 199f.] for an early formulation (and a different motivation) of such an analysis.

  11. 11.

    \(\chi '\), too, would have to be subjected to a meaning postulate in order to make sure it denotes a choice function \(\chi \), i.e. that \(w\in \chi (p)\) whenever \(p \ne \emptyset \).

  12. 12.

    The exact nature of the representation relation between contexts and indices varies across different versions of two-dimensional semantics and need not concern us here.

  13. 13.

    Cf. [15, p. 510], where the ban is presented as a descriptive observation on English and related languages. The more natural interpretation, following [18], takes it to be a defining criterion for context-dependence: unlike indices, contexts comprise those denotation-determining factors that cannot be shifted.

  14. 14.

    Here we slightly simplify matters: contexts are usually taken to contain more components than indices (cf. [14]); alternatively, they may be taken to be fully specified (utterance) situations (cf. [18]). In any case, these complications are orthogonal to our concerns.

  15. 15.

    The reasoning runs parallel to the proof Corollary (50b) in [16, Sec. 2.6]; see the next sub-section for this connection.

  16. 16.

    The (contextual) truth conditions of (23) are captured by equating index time \(t_0\) and context time \(t^*\) in (24), which still need to be kept apart to account for embedded occurrences of (23).

  17. 17.

    Apart from that a further result was proved to the effect that denotational equivalence holds across all types at diagonal points; obviously, this result is also covered by (27).

  18. 18.

    The ‘Y’ stands for ‘Yanovich’ on whose account [29] of backwards-looking operators (25) and (26) are based.

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  1. Goethe-Universität, Frankfurt am Main, Germany

    Jan Köpping & Thomas Ede Zimmermann

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  1. Jan Köpping
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  1. The University of Electro-Communications, Tokyo, Japan

    Maki Sakamoto

  2. Tokyo Institute of Technology, Tokyo, Japan

    Naoaki Okazaki

  3. Keio University, Tokyo, Japan

    Koji Mineshima

  4. National Institute of Informatics, Tokyo, Japan

    Ken Satoh

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Köpping, J., Zimmermann, T.E. (2020). Variables vs. Parameters in the Interpretation of Natural Language. In: Sakamoto, M., Okazaki, N., Mineshima, K., Satoh, K. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2019. Lecture Notes in Computer Science(), vol 12331. Springer, Cham. https://doi.org/10.1007/978-3-030-58790-1_11

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