Linguistics and Philosophy

, Volume 33, Issue 6, pp 447–477 | Cite as

Proof-theoretic semantics for a natural language fragment

Research Article

Abstract

The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a “dedicated” natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the “meaning as use” paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument does not obtain, and assertions are always warranted, having grounds of assertion. The proof system is shown to satisfy Dummett’s harmony property, justifying the ND rules as meaning conferring. The semantics is suitable for incorporation into computational linguistics grammars, formulated in type-logical grammar.

Keywords

Proof-theoretic semantics Natural language Harmony Natural deduction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barwise J., Cooper R. (1981) Generalized quantifiers and natural language. Linguistics and Philosophy 4(2): 159–219CrossRefGoogle Scholar
  2. Belnap N. (1962) Tonk, Plonk and Plink. Analysis 22: 130–134CrossRefGoogle Scholar
  3. Ben-Avi, G., & Francez, N. (2004). Categorial grammar with ontology-refined types. In Categorial Grammars 2004: An efficient tool for Natural Language Processing, Montpellier, France, June 2004.Google Scholar
  4. Ben-Avi, G., & Francez, N. (2005). A proof-theoretic semantics for the syllogistic fragment. In Proceedings of the Amsterdam Colloquium, Amsterdam, The Netherlands, December.Google Scholar
  5. Ben-Avi, G., & Francez, N. (2011). A proof-theoretic reconstruction of generalized quantifiers. In preparation.Google Scholar
  6. Boulter S. (2001) Whose challenge. Which semantics. Synthese 126: 325–337Google Scholar
  7. Brandom R.B. (2000) Articulating reasons. MA: Harvard University Press, CambridgeGoogle Scholar
  8. de Groote, P., & Retoré, C. (1996). On the semantic readings of proof-nets. In G.-J. Kruijff & D. Oehrle (Eds.), Formal grammar (pp. 57–70). Prague: FOLLI.Google Scholar
  9. Dummett M. (1991) The logical basis of metaphysics. MA: Harvard University Press, CambridgeGoogle Scholar
  10. Fernando, T. (2001). Conservative generalized quantifiers and presupposition. In Semantics and linguistic Theory XI (pp. 172–191). Cornell: CLC Publications.Google Scholar
  11. Fitch F.B. (1973) Natural deduction rules for English. Philosophical Studies 24: 89–104CrossRefGoogle Scholar
  12. Forbes G. (2000) Attitude problems. Oxford University Press, OxfordGoogle Scholar
  13. Francez, N., & Ben-Avi, G. (2011). Proof-theoretic semantic values for logical operators. Review of Symbolic Logic. Under refereeing.Google Scholar
  14. Francez N., & Dyckhoff, R. (2010). A note on harmony. Journal of Philosophical Logic, accepted.Google Scholar
  15. Francez N., Dyckhoff R., Ben-Avi G. (2010) Proof-theoretic semantics for subsentential phrases. Studia Logica 94: 381–401CrossRefGoogle Scholar
  16. Gentzen, G. (1935). Investigations into logical deduction. In M. E. Szabo (Ed.), The collected papers of Gerhard Gentzen (pp. 68–131). Amsterdam: North-Holland. (English translation of the 1935 paper in German.)Google Scholar
  17. Hinzen W. (2000) Anti-realist semantics. Erkenntnis 52: 281–311CrossRefGoogle Scholar
  18. Lappin, S. (eds) (1997) The handbook of contemporary semantic theory. Blackwell, OxfordGoogle Scholar
  19. Larson R.K. (2001) The grammar of intensionality. In: Preyer G., Peter G. (eds) Logical form and natural language.. Oxford University Press, Oxford, pp 228–262Google Scholar
  20. Moltmann F. (1997) Intensional verbs and quantifiers. Natural Language Semantics 5(1): 1–52CrossRefGoogle Scholar
  21. Moltmann F. (2008) Intensional verbs and their intentional objects. Natural Language Semantics 16(3): 239–270CrossRefGoogle Scholar
  22. Moltmann, F. (to appear). Abstract objects and the semantics of natural language. Oxford: Oxford University Press.Google Scholar
  23. Moortgat M. (1997) Categorial type logics. In: Benthem J., ter Meulen A. (eds) Handbook of logic and language.. North Holland, Amsterdam, pp 93–178CrossRefGoogle Scholar
  24. Montague, R. (1973). The proper treatment of quantification in ordinary English. In J. Hintikka, J. Moravcsik, & P. Suppes (Eds.), Approaches to natural language. Proceedings of the 1970 Stanford workshop on grammar and semantics. Dordrecht: Reidel.Google Scholar
  25. Moss L. (2010) Syllogistic logics with verbs. Journal of Logic and Computation 20(4): 947–967CrossRefGoogle Scholar
  26. Nishihara N., Morita K., Iwata S. (1990) An extended syllogistic system with verbs and proper nouns, and its completeness proof. Systems and Computers in Japan 21(1): 96–111CrossRefGoogle Scholar
  27. Peregrin J. (1997) Language and models: Is model theory a theory of semantics?. Nordic Journal of Philosophical Logic 2(1): 1–23Google Scholar
  28. Pfenning F., Davies R. (2001) A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science 11: 511–540CrossRefGoogle Scholar
  29. Prawitz, D. (1965). Natural deduction: A proof-theoretical study. Stockholm: Almqvist and Wicksell. (Soft cover edition by Dover, 2006.)Google Scholar
  30. Prawitz, D. (1978). Proofs and the meaning and completeness of logical constants. In J. Hintikka, I. Niiniluoto, & E. Saarinen (Eds.), Essays in mathematical and philosophical logic (pp. 25–40). Dordrecht: Reidel.Google Scholar
  31. Prior A.N. (1960) The runabout inference-ticket. Analysis 21: 38–39CrossRefGoogle Scholar
  32. Quine W.v.O. (1969) Ontological relativity and other essays. Columbia University Press, New YorkGoogle Scholar
  33. Ranta A. (1994) Type-theoretical grammar. Oxford University Press, OxfordGoogle Scholar
  34. Ranta A. (2004a) Computational semantics in type theory. Mathematics and Social Sciences 165: 31–57Google Scholar
  35. Ranta A. (2004) Grammatical frameworks: A type-theoretical grammar formalism. The Journal of Functional Programming 14(2): 145–189CrossRefGoogle Scholar
  36. Rayo, A., Uzquiano, G. (eds) (1997) Absolute generality. Oxford University Press, OxfordGoogle Scholar
  37. Read S. (2000) Harmony and autonomy in classical logic. Journal of Philosophical Logic 29: 123–154CrossRefGoogle Scholar
  38. Read, S. (2004). Identity and harmony. Analysis, 64(2), 113–119. See correction in Kremer, M., 2007, Read on identity and harmony—A friendly correction and simplification. Analysis, 67(2), 157–159.Google Scholar
  39. Read, S. (2008). Harmony and modality. In C. Dégremont, L. Keiff, & H. Rückert (Eds.), On dialogues, logics and other strange things: Essays in honour of Shahid Rahman (pp. 285–303). London: College Publication.Google Scholar
  40. Restall, G. (2010). Proof theory and meaning: on the context of deducibility. In Delon, F., Kohlenbach, U., Maddy, P., & Sephan, F. (Eds.), Proceedings of Logica 07, Hejnice, Czech Republic (pp. 204–219). Cambridge: Cambridge University Press.Google Scholar
  41. Richard, R. (2001). Seeking a centaur, Adoring Adonis: Intensional transitives and empty terms. In P. French & H. Wettstein (Eds.), Midwest studies in philosophy, Vol. 25: Figurative language (pp. 103–127). Oxford: Blackwell.Google Scholar
  42. Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. In R. Kahle & P. Schroeder- Heister (Eds.), Proof-theoretic semantics, Special issue of Synthese (Vol. 148, pp. 525–571).Google Scholar
  43. Tennant N. (1987) Anti-realism and logic Clarendon Library of Logic and Philosophy. Oxford University Press, OxfordGoogle Scholar
  44. van Dalen, D. (1986). Intuitionistic logic. In D. Gabbay & F. Günthner (Eds.), Handbook of philosophical logic (Vol. III, pp. 225–339). Dordrecht: Reidel.Google Scholar
  45. von Plato J. (2001) Natural deduction with general elimination rules. Archive for Mathematical Logic 40: 541–567CrossRefGoogle Scholar
  46. Wieckowski, B. (2011). Rules for subatomic derivation. Review of Symbolic Logic, to appear.Google Scholar
  47. Zimmermann T.E. (1993) On the proper treatment of opacity in certain verbs. Natural Language Semantics 1: 149–179CrossRefGoogle Scholar
  48. Zimmermann, T. E. (2006). Monotonicity in opaque verbs. Linguistics and Philosophy, 29, 715–761. doi:10.1007/s10988-006-9009-z.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Computer Science Departmentthe Technion-IITHaifaIsrael
  2. 2.School of Computer ScienceUniversity of St AndrewsSt AndrewsScotland, UK

Personalised recommendations