Skip to main content

A Topos-Based Approach to Building Language Ontologies

  • 242 Accesses

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


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.


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

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  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 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].


  1. Asher, N.: Lexical Meaning in Context: A Web of Words. Cambridge University Press, Cambridge (2011)

    CrossRef  Google Scholar 

  2. Asher, N., Pustejovsky, J.: A type composition logic for generative lexicon. J. Cogn. Sci. 7(1), 1–38 (2006)

    Google Scholar 

  3. Berry, G.: Some syntactic and categorical constructions of lambda-calculus models, RR-0080. Inria (1981)

    Google Scholar 

  4. Brown, R.: A First Language: The Early Stages. Harvard University Press, Cambridge (1973)

    CrossRef  Google Scholar 

  5. Chatzikyriakidis, S., Luo, Z.: On the interpretation of common nouns: types versus predicates. In: Chatzikyriakidis, S., Luo, Z. (eds.) Modern Perspectives in Type-Theoretical Semantics. SLP, vol. 98, pp. 43–70. Springer, Cham (2017).

    CrossRef  Google Scholar 

  6. Chomsky, N.: Some methodological remarks on generative grammar. Word 17(2), 219–239 (1961)

    CrossRef  Google Scholar 

  7. Church, A.: A formulation of the simple theory of types. J. Symb. Logic 5(2), 56–68 (1940)

    CrossRef  MathSciNet  Google Scholar 

  8. Geeraerts, D.: Prototype theory. Linguistics 27(4), 587–612 (1989)

    CrossRef  Google Scholar 

  9. Goldblatt, R.: Topoi: The Categorial Analysis of Logic. Studies in Logic and the Foundations of Mathematics, vol. 98. North-Holland Publishings, Amsterdam (1979)

    MATH  Google Scholar 

  10. Kiefer, F.: Some semantic relations in natural language. In: Josselson, H.H. (ed.) Proceedings of the Conference on Computer-related Semantic Analysis, pp. VII/1–23. Wayne State University, Detroit (1966)

    Google Scholar 

  11. La Palme Reyes, M., Macnamara, J., Reyes, G.E.: Reference, kinds and predicates. In: Macnamara, J., Reyes, G.E. (eds.) The Logical Foundations of Cognition, Vancouver Studies in Cognitive Science, vol. 4, pp. 91–143. Oxford University Press, Oxford (1994)

    Google Scholar 

  12. Luo, Z.: Type-theoretical semantics with coercive subtyping. In: Li, N., Lutz, D. (eds.) Proceedings of SALT 20, vol. 20, pp. 38–56 (2010)

    CrossRef  Google Scholar 

  13. Miller, G.A.: Nouns in WordNet. In: Fellbaum, C. (ed.) WordNet: An Electronic Lexical Database, pp. 23–46. The MIT Press, Cambridge (1998)

    Google Scholar 

  14. Montague, R.: The proper treatment of quantification in ordinary english. In: Suppes, P., Moravcsik, J., Hintikka, J. (eds.) Approaches to Natural Language, pp. 221–242. Springer, Dordrecht (1973).

    CrossRef  Google Scholar 

  15. Pustejovsky, J.: The semantics of lexical underspecification. Folia Linguistica 32(3–4), 323–348 (1998)

    Google Scholar 

  16. Retoré, C.: The montagovian generative lexicon \(\Lambda Ty_n\): a type theoretical framework for natural language semantics. In: Matthes, R., Schubert, A. (eds.) Proceedings of the 19th International Conference on Types for Proofs and Programs. LIPICS, vol. 26, pp. 202–229 (2014)

    Google Scholar 

  17. Rosch, E.H.: Natural categories. Cogn. Psychol. 4, 328–350 (1973)

    CrossRef  Google Scholar 

  18. Saba, W.S.: Logical semantics and commonsense knowledge: where did we go wrong, and how to go forward, again (2018). arXiv preprint

    Google Scholar 

  19. Seely, R.A.G.: Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Logic 52(4), 969–989 (1987)

    CrossRef  MathSciNet  Google Scholar 

  20. Sommers, F.: The ordinary language tree. Mind 68(2), 160–185 (1959)

    CrossRef  Google Scholar 

  21. Sommers, F.: Type and ontology. Philos. Rev. 72(3), 327–363 (1963)

    CrossRef  Google Scholar 

  22. Sommers, F.: Structural ontology. Philosophia 1(1–2), 21–42 (1971)

    CrossRef  Google Scholar 

  23. Suzman, J.: The ordinary language lattice. Mind 81(3), 434–436 (1972)

    CrossRef  Google Scholar 

  24. Westerhoff, J.: The construction of ontological categories. Australas. J. Philos. 82(4), 595–620 (2004)

    CrossRef  Google Scholar 

  25. Wittgenstein, L.: Philosophical Investigations. Macmillan, New York (1953)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to William Babonnaud .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer-Verlag GmbH Germany, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-59647-0

  • Online ISBN: 978-3-662-59648-7

  • eBook Packages: Computer ScienceComputer Science (R0)