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International Conference on Formal Grammar

FG 2019: Formal Grammar pp 18–34Cite as

A Topos-Based Approach to Building Language Ontologies

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11668)

Abstract

A common tendency in lexical semantics is to assume the existence of a hierarchy of types for fine-grained analyses of semantic phenomena. This paper provides a formal account of the existence of such a structure. A type system based on the categorical notion of topos is introduced, and is shown to be possibly adaptable to several existing formal approaches where such hierarchies are used. A refinement of the type hierarchy based on Fred Sommers’ ontological theory is also proposed.

Keywords

  • Formal semantics
  • Lexical semantics
  • Type theory
  • Type ontology
  • Category theory

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Notes

  1. 1.

    Which correspond respectively to \(\iota \) and o in Church’s notation. Although it is worth studying, the additional s type, denoting intension, will not be discussed in this paper, for it is assumed to be a necessary feature for the set-theoretic models of the logic used by Montague. The conception of types as denotations of sets will be overlooked here as it is too specific.

  2. 2.

    See the website https://wordnet.princeton.edu/. The online application provides the opportunity to explore the hierarchy by browsing a word, selecting its synset and accessing the list of hyponyms and hypernyms.

  3. 3.

    As a convention, I will distinguish between the adjective categorial when talking about linguistics categories such as Chomsky’s or Sommers’, and the adjective categorical when talking about things from category theory.

  4. 4.

    For linguistically motivated reasons, classes of subobjects of an object and classes of morphisms between two objects will be considered as sets in the rest of this paper.

  5. 5.

    I am grateful to an anonymous reviewer for bringing this work to my attention.

  6. 6.

    Actually, a monoidal closed category would suffice if we wanted a linear \(\lambda \)-calculus, but such a restriction is not justified here.

  7. 7.

    This morphism is actually the character of A as a subobject of itself.

  8. 8.

    Whether or not the common span of ‘cat’ and ‘dog’ is really animate entities could be debated, in particular with some examples as in (i) where ‘rock’ seems also to belong to the span of ‘dog’:

    1. (i)

            This is not a dog but just a rock.

    However, the main idea to keep is that the spans of the two predicates are probably the same, as it does not seem absurd to say that anything that is not a dog could be a cat or not, and conversely.

  9. 9.

    This transformation corresponds to a function \(\mathcal T(\textsc {{e}},\textsc {{t}})\rightarrow \mathcal T(A,\textsc {{t}})\) in Set, which is a specific case of application of the contravariant hom-functor \(\mathcal T(-,\textsc {{t}})\).

  10. 10.

    This is actually a well-known property for Hilbert algebras, but it applies here as a Heyting algebra is a particular case of Hilbert algebra. The existence of a monomorphism between two subobjects A and B corresponds to the natural order of those algebras: if we note \(A\le B\) when such a monomorphism exists, then \(A\le B\) iff \((E\cap A)\le B\) iff \(E\le (A\Mapsto \!B)\), which is equivalent to as expected for a natural order.

  11. 11.

    In the remainder of this paper both types will be assumed to be ontological.

  12. 12.

    There is actually more subtleties in his construction, but they will not be detailled here due to lack of space. The whole reasoning can be found in [1].

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Authors and Affiliations

  1. LORIA, Université de Lorraine, CNRS, Inria Nancy Grand Est, Nancy, France

    William Babonnaud

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  1. William Babonnaud
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Correspondence to William Babonnaud .

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Editors and Affiliations

  1. University of Trento, Povo, Italy

    Raffaella Bernardi

  2. Universität Leipzig, Leipzig, Germany

    Greg Kobele

  3. LORIA, Inria Nancy, Villers-lès-Nancy, France

    Sylvain Pogodalla

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Babonnaud, W. (2019). A Topos-Based Approach to Building Language Ontologies. In: Bernardi, R., Kobele, G., Pogodalla, S. (eds) Formal Grammar. FG 2019. Lecture Notes in Computer Science(), vol 11668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59648-7_2

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  • DOI: https://doi.org/10.1007/978-3-662-59648-7_2

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