Skip to main content
Log in

Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We present a general framework for proof systems for relational theories. We discuss principles of the construction of deduction rules and correspondences reflecting relationships between semantics of relational logics and the rules of the respective proof systems. We illustrate the methods developed in the paper with examples relevant for the Lambek calculus and some of its extensions.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andréka, H., and S. Mikulaś, ‘Lambek calculus and its relational semantics. Completeness and incompleteness’, Journal of Logic, Language and Information 3 (1994), 1-37.

    Google Scholar 

  2. Avron, A., F. Honsell, M. Miculan, and C. Paravano, ‘Encoding modal logics in logical frameworks’, Studia Logica 60,1, (1998), 161-208.

    Google Scholar 

  3. Bachmair, L., and H. Ganzinger, ‘Ordered chaining calculi for first order theories of transitive relations’, Journal of ACM 45,6 (1998), 1007-1049.

    Google Scholar 

  4. Basin, D., S. Matthews, and L. Vigano, ‘A modular presentation of modal logics in a logical framework’, Studia Logica 60,1 (1998), 119-160.

    Google Scholar 

  5. Buszkowski, W., ‘Mathematical linguistics and proof theory’, in: J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, The MIT Press, Cambridge Mass., 1997, pp. 683-735.

    Google Scholar 

  6. Buszkowski, W., and M. Kołowska-Gawiejnowicz, ‘Representation of residuated semigroups in some algebras of relations. The method of canonical models’, Fundamenta Informaticae 31 (1997), 1-12.

    Google Scholar 

  7. Buszkowski, W., and E. Orłowska, ‘Indiscernibility-based formalization of dependencies in information systems’, in: E. Orłowska (ed.), Incomplete Information: Rough Set Analysis, Studies in Fuzziness and Soft Computing 13, Physica Verlag, 1997, pp. 347-380.

  8. Coquand, T., and G. Huet, ‘The calculus of constructions’, Information and Control 76 (1988), 95-120.

    Google Scholar 

  9. Demri, S., ‘Sequent calculi for nominal tense logics: a step towards mechanization?’, Lecture Notes in Computer Science 1289, Springer, 1999, 140-154.

  10. Demri, S., and E. Orłowska, ‘Logical analysis of demonic nondeterministic programs’, Theoretical Computer Science 166 (1966), 173-202.

    Google Scholar 

  11. Došen, K., ‘A brief survey of frames for the Lambek calculus’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 38 (1992), 179-187.

    Google Scholar 

  12. Došen, K., and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 1993.

    Google Scholar 

  13. Düntsch, I., W. MacCaull, and E. Orłowska, ‘Structures with many-valued information and their relational proof theory’, Proc. of 30th IEEE International Symposium on Multiple-Valued Logic, Portland, Oregon, 2000, pp. 293-301.

  14. Düntsch, I., E. Orłowska, and H. Wang, ‘An algebraic and logical approach to the approximation of regions’, Proc. of RelMiCS'2000, Quebec, Canada, 2000, 65-74.

  15. Düntsch, I., and E. Orłowska, ‘A proof system for contact relation algebras’, Journal of Philosophical Logic 29 (2000), 241-262.

    Google Scholar 

  16. Düntsch, I., and E. Orłowska, ‘Beyond modalities: sufficiency and mixed algebras’, in: E. Orłowska and A. Szałas (eds.), Relational Methods for Computer Science Applications, Physica Verlag, Heidelberg, 2001, pp. 263-285.

    Google Scholar 

  17. Frias, M., and E. Orłowska, ‘A proof system for fork algebras and its applications in logics based on intuitionism’, Logique et Analyse 150, 151, 152 (1995), 239-284.

    Google Scholar 

  18. Ganziger, H., and V. Sofronie-Stokkermans, ‘Chaining techniques for automated theorem proving in many-valued logics’, Proc. of 30th IEEE International Symposium on Multiple-Valued Logics, Portland, Oregon, 2000, pp. 337-344.

  19. Harper, R., F. Honsell, and G. Plotkin, ‘A framework for defining logics’, Journal of ACM 40,1 (1993), 143-184.

    Google Scholar 

  20. Konikowska, B., C. Morgan, and E. Orłowska, ‘Relational semantics for arbitrary finite-valued logics’, in: E. Orłowska, and A. Szałas (eds.), Relational Methods in Logic, Algebra and Computer Science, Proc. of RelMiCS'98, Warsaw, 1998, pp. 138-143.

  21. Konikowska, B., C. Morgan, and E. Orłowska, ‘A relational formalization of arbitrary finite-valued logics’, Logic Journal of IGPL 6,5 (1998), 755-774.

    Google Scholar 

  22. Konikowska, B., and E. Orłowska, ‘A relational formalization of a generic many-valued modal logic’, in: E. Orłowska and A. Szałas (eds.), Relational Methods for Computer Science Applications, Physica Verlag, Heidelberg, 2001, pp. 183-202.

    Google Scholar 

  23. Lambek, J., ‘The mathematics of sentence structure’, The American Mathematical Monthly 65 (1958), 154-170.

    Google Scholar 

  24. Lambek, J., ‘On the calculus of syntactic types’, in: R. Jakobson (ed.), Structure of Language and Its Mathematical Aspects, AMS, Providence, 1961.

    Google Scholar 

  25. Lambek, J., ‘Relations old and new’, in: E. Orłowska and A. Szałas (eds.), Relational Methods for Computer Science Applications, Physica Verlag, Heidelberg, 2001, pp. 135-147.

    Google Scholar 

  26. MacCaull, W., ‘Relational proof theory for linear and other substructural logics’, Logic Journal of IGPL 5 (1997), 673-697.

    Google Scholar 

  27. MacCaull, W., ‘Relational tableaux for tree models, language models and information networks’, in: E. Orłowska (ed.), Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, Physica Verlag, Heidelberg, 1998, pp. 354-382.

    Google Scholar 

  28. MacCaull, W., ‘Relational semantics and a relational proof system for full Lambek calculus’, Journal of Symbolic Logic 63,2 (1998), 623-637.

    Google Scholar 

  29. Moortgat, M., ‘Categorial type logics’, in: J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, The MIT Press, Cambridge Mass., 1997, pp. 93-177.

    Google Scholar 

  30. Ono, H., and Y. Komori, ‘Logics without the contraction rules’, Journal of Symbolic Logic 50 (1985), 169-201.

    Google Scholar 

  31. Orłowska, E., ‘Proof system for weakest prespecification’, Information Processing Letters 27 (1988), 309-313.

    Google Scholar 

  32. Orłowska, E., ‘Relational interpretation of modal logics’, in: H. Andréka, D. Monk and I. Nemeti (eds.), Algebraic Logic., Col. Math. Soc. J. Bolyai 54, North Holland, Amsterdam, 1988, pp. 443-471.

    Google Scholar 

  33. Orłowska, E., ‘Relational proof system for relevant logics’, Journal of Symbolic Logic 57 (1992), 1425-1440.

    Google Scholar 

  34. Orłowska, E., ‘Relational proof system for modal logics’, in: H. Wansing (ed.), Proof Theory of Modal Logics, Kluwer, Dordrecht, 1996, pp. 55-77.

    Google Scholar 

  35. Orłowska, E., and A. Szałas (eds.), Relational Methods in Logic, Algebra and Computer Science, Proc. of RelMiCS'98, Warsaw, 1998.

  36. Pentus, M., ‘Lambek calculus is L-complete’, ILLC Prepublication Series LP-93-14, University of Amsterdam, 1993.

  37. Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, Polish Science Publishers, Warsaw, 1963.

    Google Scholar 

  38. Spencer, B., and W. MacCaull, ReVAT — Relational Validator via Analytic Tableaux, Technical Report, University of New Brunswick, 2001.

  39. Szczerba, M., ‘Relational models for the nonassociative Lambek calculus’, in: E. Orłowska and A. Szałas (eds.), Relational Methods for Computer Science Applications, Physica Verlag, Heidelberg, 2001, pp. 149-159.

    Google Scholar 

  40. Tarski, A., ‘On the calculus of relations’, Journal of Symbolic Logic 6 (1941), 73-89.

    Google Scholar 

  41. Tarski, A., A. Mostowski and A. Robinson, Undecidable Theories, Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, 1968.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

MacCaull, W., Orłlowska, E. Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus. Studia Logica 71, 389–414 (2002). https://doi.org/10.1023/A:1020572931854

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020572931854

Navigation