Skip to main content
SpringerLink
Log in

Disentangling Structural Connectives or Life Without Display Property

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The work is concerned with the so called display property of display logic. The motivation behind it is discussed and challenged. It is shown using one display calculus for intuitionistic logic as an example that the display property can be abandoned without losing subformula, cut elimination and completeness properties in such a way that results in additional expressive power of the system. This is done by disentangling structural connectives so that they are no longer context-sensitive. A recipe for characterizing structural extensions of display calculi is provided.

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. Avron, A. (1996). The method of hypersequents in the proof theory of propositional non-classical logics. In Hodges, W., Hyland, M., Steinhorn, C., Truss, J. (Eds.) Logic: from Foundation to Applications, European Logic Colloquium (pp. 1–32). Oxford: Oxford University Press.

  2. Belnap, N.D. (1982). Display logic. Journal of Philosophical Logic, 11(4), 375–417. https://doi.org/10.1007/bf00284976.

    Article  Google Scholar 

  3. Belnap, N.D. (1989). Linear logic displayed. Notre Dame Journal of Formal Logic, 31(1), 14–25. https://doi.org/10.1305/ndjfl/1093635329.

    Article  Google Scholar 

  4. Belnap, N.D. (1996). The display problem. In Wansing, H. (Ed.) Proof Theory of Modal Logic, Applied Logic Series, (Vol. 2 pp. 79–92). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-2798-3_6.

  5. van Benthem, J. (2001). Correspondence theory. In Gabbay, D.M., & Guenther, F. (Eds.) Handbook of Philosophical Logic, (Vol. 3 pp. 325–408): Springer. https://doi.org/10.1007/978-94-017-0454-0_4.

  6. Blamey, S., & Humberstone, L. (1991). A perspective on modal sequent logic. Publications of the Research Institute for Mathematical Sciences, 27(5), 763–782. https://doi.org/10.2977/prims/1195169271.

    Article  Google Scholar 

  7. Brünnler, K. (2009). Deep sequent systems for modal logic. Archive for Mathematical Logic, 48(6), 551–577. https://doi.org/10.1007/s00153-009-0137-3.

    Article  Google Scholar 

  8. Chagrov, A., & Zakharyaschev, M. (1997). Modal logic oxford logic guides Vol. 35. Oxford: Clarendon Press.

    Google Scholar 

  9. Ciabattoni, A., & Ramanayake, R. (2016). Power and limits of structural display rules. ACM Transactions on Computational Logic, 17(3), 1–39. https://doi.org/10.1145/2874775.

    Article  Google Scholar 

  10. Ciabattoni, A., Ramanayake, R., Wansing, H. (2014). Hypersequent and display calculi – a unified perspective. Studia Logica, 102(6), 1245–1294. https://doi.org/10.1007/s11225-014-9566-z.

    Article  Google Scholar 

  11. Cintula, P., & Paoli, F. (2016). Is multiset consequence trivial? Synthese, Special issue: Substructural Approaches to Paradox pp. 1–25. https://doi.org/10.1007/s11229-016-1209-7.

  12. Conradie, W., & Palmigiano, A. (2012). Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic, 163 (3), 338–376. https://doi.org/10.1016/j.apal.2011.10.004.

    Article  Google Scholar 

  13. Došen, K. (1988). Sequent-systems and groupoid models I. Studia Logica, 47 (4), 353–385. https://doi.org/10.1007/bf00671566.

    Article  Google Scholar 

  14. Drobyshevich, S. (2017). On displaying negative modalities. Logic and Logical Philosophy. https://doi.org/10.12775/llp.2017.023. Online-first.

  15. Dunn, J.M. (1991). Gaggle theory: an abstraction of galois connections and residuation, with applications to negation, implication, and various logical operators. In van Eijck, J. (Ed.) Logics in AI: European workshop on logics in artificial intelligence JELIA 1990, lecture notes in computer science, (Vol. 478 pp. 31–51). Berlin: Springer. https://doi.org/10.1007/bfb0018431.

  16. Dunn, J.M., Gehrke, M., Palmigiano, A. (2005). Canonical extensions and relational completeness of some substructural logics. The Journal of Symbolic Logic, 70(3), 713–740. https://doi.org/10.2178/jsl/1122038911.

    Article  Google Scholar 

  17. Dyckhoff, R., & Negri, S. (2011). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51(1-2), 71–92. https://doi.org/10.1007/s00153-011-0254-7.

    Article  Google Scholar 

  18. Frittella, S., Greco, G., Kurz, A., Palmigiano, A. (2016). Multi-type display calculus for propositional dynamic logic. Journal of Logic and Computation, 26(6), 2067–2104. https://doi.org/10.1093/logcom/exu064.

    Article  Google Scholar 

  19. Frittella, S., Greco, G., Kurz, A., Palmigiano, A., Sikimić, V. (2016). Multi-type display calculus for dynamic epistemic logic. Journal of Logic and Computation, 26(6), 2017–2065. https://doi.org/10.1093/logcom/exu068.

    Article  Google Scholar 

  20. Gehrke, M., & Harding, J. (2001). Bounded lattice expansions. Journal of Algebra, 238(1), 345–371. https://doi.org/10.1006/jabr.2000.8622.

    Article  Google Scholar 

  21. Gehrke, M., & Jónsson, B. (2004). Bounded distributive lattice expansions. Mathematica Scandinavica, 94(1), 13–45. https://doi.org/10.7146/math.scand.a-14428.

    Article  Google Scholar 

  22. Goré, R. (1995). Solving the display problem via residuation. Tech. rep., Automated Reasoning Project TR-ARP-12-95 Australian National University.

  23. Goré, R. (1995). A uniform display system for intuitionistic and dual intuitionistic logic. Tech. rep., Automated Reasoning Project TR-ARP-6-95 Australian National University.

  24. Goré, R. (1998). Substructural logics on display. Logic Journal of IGPL, 6 (3), 451–504. https://doi.org/10.1093/jigpal/6.3.451.

    Article  Google Scholar 

  25. Goré, R. (2000). Dual intuitionistic logic revisited. In Dyckhoff, R. (Ed.) TABLEAUX 2000: Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Computer Science, (Vol. 1847 pp. 252–267): Springer. https://doi.org/10.1007/10722086_21.

  26. Goré, R., Postniece, L., Tiu, A. (2011). On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. Logical Methods in Computer Science 7(2). https://doi.org/10.2168/lmcs-7(2:8)2011.

  27. Goré, R., & Ramanayake, R. (2012). Labelled tree sequents, tree hypersequents and nested (deep) sequents. In Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (Eds.) Advances in modal logic, (Vol. 9 pp. 279–299). London: College Publications.

  28. Greco, G., Ma, M., Palmigiano, A., Tzimoulis, A., Zhao, Z. (2016). Unified correspondence as a proof-theoretic tool. Journal of Logic and Computation. https://doi.org/10.1093/logcom/exw022. Open access.

  29. Kracht, M. (1996). Power and weakness of the modal display calculus. In Wansing, H. (Ed.) Proof Theory of Modal Logic, Applied Logic Series. https://doi.org/10.1007/978-94-017-2798-3_7, (Vol. 2 pp. 93–121). Dordrecht: Kluwer Academic Publishers.

  30. Kushida, H., & Okada, M. (2003). A proof-theoretic study of the correspondence of classical logic and modal logic. The Journal of Symbolic Logic, 68(04), 1403–1414. https://doi.org/10.2178/jsl/1067620195.

    Article  Google Scholar 

  31. Lahav, O., & Avron, A. (2013). A unified semantic framework for fully structural propositional sequent systems. ACM Transactions on Computational Logic, 14(4), 1–33. https://doi.org/10.1145/2528930.

    Article  Google Scholar 

  32. Lambek, J. (1961). On the calculus of syntactic types. In Jackobson, R. (Ed.) Structure of Language and its Mathematical Aspects (pp. 166–178). Providence.

  33. Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5-6), 507–544. https://doi.org/10.1007/s10992-005-2267-3.

    Article  Google Scholar 

  34. Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  35. Onishi, T. (2015). Substructural negations. The Australasian Journal of Logic 12(4).

  36. Onishi, T. (2016). Understanding negation implicationally in the relevant logic R. Studia Logica, 104(6), 1267–1285. https://doi.org/10.1007/s11225-016-9676-x.

    Article  Google Scholar 

  37. Poggiolesi, F. (2009). The method of tree-hypersequents for modal propositional logic. In Makinson, D., Malinowski, J., Wansing, H. (Eds.) Towards Mathematical Philosophy, Trends in Logic, (Vol. 28 pp. 31–51). Netherlands: Springer. https://doi.org/10.1007/978-1-4020-9084-4_3.

  38. Poggiolesi, F. (2010). Display calculi and other modal calculi: a comparison. Synthese, 173(3), 259–279. https://doi.org/10.1007/s11229-008-9425-4.

    Article  Google Scholar 

  39. Pottinger, G. (1983). Uniform, cut-free formulations of t, s4 and s5 (abstract). The Journal of Symbolic Logic, 48, 900–901.

    Google Scholar 

  40. Ramanayake, R. (2014). Embedding the hypersequent calculus in the display calculus. Journal of Logic and Computation, 25(3), 921–942. https://doi.org/10.1093/logcom/exu061.

    Article  Google Scholar 

  41. Restall, G. (1995). Display logic and gaggle theory. Tech. rep., Reports on Mathematical Logic.

  42. Restall, G. (1998). Displaying and deciding substructural logics 1: Logics with contraposition. Journal of Philosophical Logic, 27(2), 179–216. https://doi.org/10.1023/a:1017998605966.

    Article  Google Scholar 

  43. Sahlqvist, H. (1975). Completeness and correspondence in the first and second order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium (pp. 110–143): Elsevier. https://doi.org/10.1016/s0049-237x(08)70728-6.

  44. Wansing, H. (1994). Sequent calculi for normal modal propositional logics. Journal of Logic and Computation, 4(2), 125–142. https://doi.org/10.1093/logcom/4.2.125.

    Article  Google Scholar 

  45. Wansing, H. (1998). Displaying modal logic, trends in logic Vol. 3. Netherlands: Springer. https://doi.org/10.1007/978-94-017-1280-4.

    Book  Google Scholar 

  46. Wansing, H. (2008). Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics, 18(2-3), 341–364. https://doi.org/10.3166/jancl.18.341-364.

    Article  Google Scholar 

  47. Wansing, H. (2010). Proofs, disproofs and thier duals. In Beklemishev, L., Goranko, V., Shekhtman, V. (Eds.) Advances in modal logic, (Vol. 8 pp. 483–505). London: College Publications.

Download references

Acknowledgements

I would like to thank Prof. Heinrich Wansing specifically and Bochum logic group in general for valuable feedback on early presentations of the result.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russian Federation

    Sergey Drobyshevich

  2. Novosibirsk State University, Novosibirsk, Russian Federation

    Sergey Drobyshevich

Authors
  1. Sergey Drobyshevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Drobyshevich.

Additional information

This work was supported by the Alexander von Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drobyshevich, S. Disentangling Structural Connectives or Life Without Display Property. J Philos Logic 48, 279–303 (2019). https://doi.org/10.1007/s10992-018-9466-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-018-9466-1

Keywords

Search

Navigation