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A Model-Theoretic Reconstruction of Type-Theoretic Semantics for Anaphora

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Formal Grammar (FG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10686))

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Abstract

I present an analysis of the interpretation of anaphora that takes concepts from type-theoretic semantics, in particular the use of the \(\varSigma \) and \(\varPi \) dependent type constructors, and incorporates them into a model-theoretic framework. The analysis makes use of (parametrically) polymorphic lexical entries. The key ideas are that, in the simplest case, eventualities can play the role that proof objects do in type-theoretic semantics; that more complex, compositionally-defined, structures can play that role in other cases; and that pronouns can be modelled by context-dependent functions from proof objects of the preceding discourse (in this sense) to entities.

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Notes

  1. 1.

    \(\varSigma \) can also be used to give the meaning of conjunction, and \(\varPi \) implication; this is reflected in the lexical entries given in Fig. 1. Limitations of space prevent any further consideration of these connections here.

  2. 2.

    Here and throughout the paper, a dot following a variable binder will often be used instead of parentheses to indicate unbounded scope to the right.

  3. 3.

    One option, therefore, is to change the model theory so that (5) would be interpreted in the desired way. This, explicitly, is the approach taken in Dynamic Predicate Logic (DPL) [8]. Discourse Representation Theory (DRT) [13] and File Change Semantics (FCS) [10] take a similar approach.

  4. 4.

    By this, I mean that meanings will be given as expressions of a logical language, which are taken to be dispensable in favour of their interpretations in a model (as in [16]), which is where the ‘real’ semantics is. Expressions of the language of type theory are not understood this way in TTS; see [14] and [18], Sect. 2.27.

  5. 5.

    In the type annotations, here and throughout the rest of the paper, brackets are omitted where possible, on the understanding that both \(\times \) and associate to the right and that \(\times \) binds more tightly than .

  6. 6.

    The same issue prompted [2] to switch from treating common nouns as type-denoting to predicate-denoting.

  7. 7.

    In the rest of the paper this will be referred to as ‘the null context’, and will generally be assumed.

  8. 8.

    This corresponds closely to the truth definition for DRT proposed in [12], p. 149.

  9. 9.

    Some speakers may allow an interpretation of (24) on which a donkey takes wider scope than negation. In that case, the pronoun could anaphorically refer back to the donkey.

  10. 10.

    Figure 1 defines VP negation, which is derived from sentential negation in the obvious way. The VP formulation is more transparent in terms of compositional semantics, and also makes Giles available for anaphoric reference.

  11. 11.

    As stated, \(\lambda b.[[b]_1]_0\) is also a possible resolution function, which would have it varying with farmers rather than donkeys, which is obviously not a possible reading of (25). This reading could be ruled out by tweaking the lexical entry for every, but only at the cost of ruling out interpretations that we do want when we have an embedded clause. The mechanism for ruling out violations of ‘Principle B’ must come from somewhere else.

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Authors and Affiliations

  1. University of Oslo, Oslo, Norway

    Matthew Gotham

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  1. Matthew Gotham
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Correspondence to Matthew Gotham .

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  1. IRISA, University of Rennes 1, Rennes, France

    Annie Foret

  2. Department of Philosophy, Tilburg University, Tilburg, The Netherlands

    Reinhard Muskens

  3. LORIA/Inria Nancy, Villers-les-Nancy, France

    Sylvain Pogodalla

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Gotham, M. (2018). A Model-Theoretic Reconstruction of Type-Theoretic Semantics for Anaphora. In: Foret, A., Muskens, R., Pogodalla, S. (eds) Formal Grammar . FG 2017. Lecture Notes in Computer Science(), vol 10686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56343-4_3

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