Skip to main content
SpringerLink
Log in

Quantized Linear Logic, Involutive Quantales and Strong Negation

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's (commutative) involutive quantales. Some cut-free sequent calculi with a new property “quantization principle” and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramsky, S., and S. Vickers, 'Quantales, observational logic and process semantics', Mathematical Structures in Computer Science 3:161–227, 1993.

    Google Scholar 

  2. Allwein, G., and W. MacCaull, 'A Kripke semantics for the logic of Gelfand quantales', Studia Logica 68:173–228, 2001.

    Google Scholar 

  3. Elbl, G., 'A declarative semantics for depth-first logic programs', The Journal of Logic Programming 41:27–66, 1999.

    Google Scholar 

  4. Engberg, U., and G. Winskel, 'Completeness results for linear logic on Petri nets', Annals of Pure and Applied Logic 86:101–135, 1997.

    Google Scholar 

  5. Girard, J-Y., 'Linear logic', Theoretical Computer Science 50:1–102, 1987.

    Google Scholar 

  6. Hoare, C. A. R., and He Jifeng, 'A weakest pre-specification', Information Processing Letter 24:127–132, 1987.

    Google Scholar 

  7. Ishihara, K., and K. Hiraishi, 'The completeness of linear logic for Petri net models', Logic Journal of the IGPL 9, No. 4: 549–567, 2001.

    Google Scholar 

  8. Kamide, N., 'Sequent calculi for intuitionistic linear logic with strong negation', Logic Journal of the IGPL 10, No. 6:653–678, 2002.

    Google Scholar 

  9. Kamide, N., 'A canonical model construction for substructural logics with strong negation', Reports on Mathematical Logic 36:95–116, 2002.

    Google Scholar 

  10. Kamide, N., 'Normal modal substructual logics with strong negation', Journal of Philosophical Logic 32, No 6:589–612, 2003.

    Google Scholar 

  11. Larchey-Wendling, D., and D. Galmiche, 'Provability in intuitionistic linear logic from a new interpretation on Petri nets —extended abstract —', Electronic Notes in Theoretical Computer Science 17, 18 pages, 1998.

  12. Larchey-Wendling, D., and D. Galmiche, 'Quantales as completions of ordered monoids: revised semantics for intuitionistic linear logic', Electronic Notes in Theoretical Computer Science 35, 15 pages, 2000.

  13. Lilius, J., 'High-level nets and linear logic', Lecture Notes in Computer Science 616:310–327, Springer-Verlag, 1992.

  14. MacCaull, W., 'Relational proof system for linear and other substructural logics', Logic Journal of the IGPL 5:673–697, 1997.

    Google Scholar 

  15. Mulvey, C. J., and J. W. Pelletier, 'A quantisation of the calculus of relations', Category Theory 1991, CMS Conference Proceedings, 13:345–360, Amer. Math. Soc., Providence, RI, 1992.

    Google Scholar 

  16. Mulvey, C. J., and J. W. Pelletier, 'On the quantization of points', Journal of Pure and Applied Algebra 159:231–295, 2001.

    Google Scholar 

  17. Nelson, D., 'Constructible falsity', Journal of Symbolic Logic 14:16–26, 1949.

    Google Scholar 

  18. Ono, H., 'Semantics for substructural logics', Substructural Logics (edited by K. Došen and P. Schroeder-Heister), Oxford University Press, 1993, pp. 259–291

  19. Ono, H., and Y. Komori, 'Logics without the contraction rule', Journal of Symbolic Logic 50:169–201, 1985.

    Google Scholar 

  20. Pelletier, J. W., and J. Rosický, 'Simple involutive quantales', Journal of Algebra 195:367–386, 1997.

    Google Scholar 

  21. Resende, P., 'Quantales, finite observations and strong bisimulation', Theoretical Computer Science 254:95–149, 2001.

    Google Scholar 

  22. Troelstra, A. S., Lectures on linear logic, CSLI Lecture Notes 29, 1992.

  23. Wagner, G., 'Logic programming with strong negation and inexact predicates', Journal of Logic and Computation 1, No. 6:835–859, 1991.

    Google Scholar 

  24. Wansing, H., 'The logic of information structures', Lecture Notes in Artificial Intelligence 681:1–163, Springer-Verlag, 1993.

  25. Wansing, H., 'Informational interpretation of substructural propositional logics', Journal of Logic, Language and Information 2:285–308, 1993.

    Google Scholar 

  26. Wansing, H., 'Diamonds are a philosopher's best friends —the knowability paradox and modal epistemic relevant logic', Journal of Philosophical Logic 31:591–612, 2002.

    Google Scholar 

  27. Yetter, D. N., 'Quantales and (noncommutative) linear logic', Journal of Symbolic Logic 55:41–64, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Keio University, JAPAN

    Norihiro Kamide

Authors
  1. Norihiro Kamide
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamide, N. Quantized Linear Logic, Involutive Quantales and Strong Negation. Studia Logica 77, 355–384 (2004). https://doi.org/10.1023/B:STUD.0000039030.03885.7c

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:STUD.0000039030.03885.7c

Search

Navigation