Skip to main content
SpringerLink
Log in

A Constructive Type-Theoretical Formalism for the Interpretation of Subatomically Sensitive Natural Language Constructions

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Löf) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Boldini, P., Vagueness and type theory, in C. Retoréd.), Logical Aspects of Computational Linguistics, LNCS 1328, Springer, 1997, pp. 134–148.

  2. Fernando, T., Conservative generalized quantifiers and presupposition, in Proceedings of Semantics and Linguistic Theory XI,Cornell University, 2001, pp. 172–191.

  3. Fernando T.: Situations as strings. Electronic Notes in Theoretical Computer Science 165, 23–36 (2006)

    Article  Google Scholar 

  4. Francez, N., and R. Dyckhoff, Proof-theoretic semantics for a natural language fragment, in C. Ebert, G. Jäger, and J. Michaelis (eds.), Mathematics of Language 10/11,LNAI 6149, Springer, 2010, pp. 56–71.

  5. Francez N., Dyckhoff R.: Proof-theoretic semantics for a natural language fragment. Linguistics and Philosophy 33, 447–477 (2010)

    Article  Google Scholar 

  6. Francez N., Dyckhoff R., Ben-Avi G.: Proof-theoretic semantics for subsentential phrases. Studia Logica 94, 381–401 (2010)

    Article  Google Scholar 

  7. Krahmer, E., and P. Piwek, Presupposition projection as proof construction, in H. Bunt, and R. Muskens (eds.), Computing Meaning,Vol. 1, Kluwer Academic Publishers, 1999, pp. 281–300.

  8. Martin-Löf, P., Intuitionistic Type Theory, Bibliopolis, 1984

  9. Nordström, B., K. Petersson, and J. M. Smith, Programming in Martin-Löf’s Type Theory, Clarendon Press, 1990.

  10. Prawitz, D., Ideas and results in proof theory, in J.E. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium (Oslo 1970), North-Holland, 1971, pp. 235–309.

  11. Prawitz, D., Towards a foundation of a general proof theory, in P. Suppes et al. (eds.), Logic, Methodology, and Philosophy of Science IV, North-Holland, 1973, pp. 225–250.

  12. Prawitz, D., Meaning approached via proofs, Synthese 148:1–12. Special issue on Proof-Theoretic Semantics edited by R. Kahle, and P. Schroeder-Heister, 2006.

  13. Primiero, G., and B. Jespersen, Two kinds of procedural semantics for privative modification, in K. Nakakoji, Y. Murakami, and E.McCready (eds.), New Frontiers in Artificial Intelligence, LNAI 6284, Springer, 2010, pp. 252–271.

  14. Ranta A.: Intuitionistic categorial grammar. Linguistics and Philosophy 14, 203–239 (1991)

    Article  Google Scholar 

  15. Ranta, A., Type-Theoretical Grammar, Clarendon Press, 1994.

  16. Rathjen M.: The constructive Hilbert program and the limits of Martin-Löf Type Theory. Synthese 147, 81–120 (2005)

    Article  Google Scholar 

  17. Sundholm, G., Proof-theory and meaning, in D. M. Gabbay, and F.Guenthner (eds.), Handbook of Philosophical Logic, 2nd Edition, Vol. 9, Kluwer Academic Publishers, 2002, pp. 165–198. (First published in 1984.)

  18. Sundholm G.: Constructive generalized quantifiers. Synthese 79, 1–12 (1989)

    Article  Google Scholar 

  19. Więckowski B. (2010) Associative substitutional semantics and quantified modal logic. Studia Logica 94:105–138

    Google Scholar 

  20. Więckowski B. (2011) Rules for subatomic derivation, The Review of Symbolic Logic 4:219–236

    Google Scholar 

  21. Zimmermann T. E.: On the proper treatment of opacity in certain verbs. Natural Language Semantics 1, 149–179 (1993)

    Article  Google Scholar 

  22. Zimmermann T. E.: Monotonicity in opaque verbs. Linguistics and Philosophy 29, 715–761 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Philosophie, Universität Greifswald, Baderstraße 6–7, 17489, Greifswald, Germany

    Bartosz Więckowski

Authors
  1. Bartosz Więckowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bartosz Więckowski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Więckowski, B. A Constructive Type-Theoretical Formalism for the Interpretation of Subatomically Sensitive Natural Language Constructions. Stud Logica 100, 815–853 (2012). https://doi.org/10.1007/s11225-012-9431-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-012-9431-x

Keywords

Search

Navigation