Skip to main content
Log in

A Cut-Elimination Proof in Positive Relevant Logic with Necessity

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This paper presents a sequent calculus for the positive relevant logic with necessity and a proof that it admits the elimination of cut.

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.

Similar content being viewed by others

References

  1. Anderson, A., and N. D. Belnap Jr., Entailment: the logic of relevance and necessity, vol. 1, Princeton University Press, Princeton, New Jersey, 1975.

    Google Scholar 

  2. Anderson, A., N. D. Belnap Jr., and J. M. Dunn, Entailment: the logic of relevance and necessity, vol. 2, Princeton University Press, Princeton, New Jersey, 1992.

    Google Scholar 

  3. Belnap Jr. N. D. A. Gupta, and J. M. Dunn, A consecutive calculus for positive relevant implication with necessity, Journal of Philosophical Logic 9:343–362, 1980.

  4. Belnap Jr., N. D., Display logic, Journal of Philosophical Logic 11:375–417, 1982.

    Article  Google Scholar 

  5. Bimbó, K., Relevance logics, in D. Jacquette, (ed.), Philosophy of Logic, (Handbook of the Philosophy of Science, vol. 5, D. Gabbay, P. Thagard, and J. Woods, eds.), Elsevier, 2007, pp. 723–789.

  6. Bimbó, K., Proof theory: Sequent Calculi and Related Formalisms, CRC Press, Boca Raton, FL, 2015.

    Google Scholar 

  7. Bimbó, K., and J. M. Dunn, Two extensions of the structurally free logic \(LC\), Logic Journal of the IGPL 6:403–424, 1998.

    Article  Google Scholar 

  8. Bimbó, K., and J. M. Dunn, New consecution calculi for \(R_{\rightarrow }^t\), Notre Dame Journal of Formal Logic 53:491–501, 2012.

    Article  Google Scholar 

  9. Brady, R. T., The Gentzenization and decidability of RW, Journal of Philosophical Logic 19:35–73, 1990.

    Article  Google Scholar 

  10. Curry, H. B., A Theory of Formal Deducibility, Notre Dame Mathematical Lectures, No. 6, University of Notre Dame, 1950.

  11. Curry, H. B., Foundations of Mathematical Logic, McGraw-Hill Book Company, New York, NY, 1963.

    Google Scholar 

  12. Dunn, J. M., The Algebra of Intensional Logics, PhD Thesis, University of Pittsburgh, 1966.

  13. Dunn, J. M., A ’Gentzen system’ for positive relevant implication. The Journal of Symbolic Logic 38:356-357, 1973.

    Google Scholar 

  14. Dunn, J. M., Relevance logic and entailment, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 3, D. Reidel Publishing Company, 1986, pp. 117–224.

  15. Dunn, J. M., and G. Restall, Relevance logic, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 6, Kluwer Academic Publlishers, 2002, pp. 1–128.

  16. Gentzen, G., Investigations into logical deduction, in M. E. Szabo, (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, 1969, pp. 68–131.

  17. Giambrone, S., \(TW_+\) and \(RW_+\) are decidable, Journal of Philosophical Logic 14:235–254, 1985.

    Article  Google Scholar 

  18. Ilić, M., An alternative Gentzenization of \(RW_+^\circ \), Mathematical Logic Quarterly 62(6):465–480, 2016.

    Article  Google Scholar 

  19. Ilić, M., and B. Boričić, A cut-free sequent calculus for relevant logic \(RW\), Logic Journal of IGPL 22(4): 673–695, 2014.

    Article  Google Scholar 

  20. Meyer, R. K., and M. A. McRobbie, Multisets and relevant implication, Australasian Journal of Philosophy 60:107–139, 1982.

    Article  Google Scholar 

  21. Minc, G., Cut elimination theorem for relevant logics, Journal of Soviet Mathematics 6:422–428, 1976.

    Article  Google Scholar 

Download references

Acknowledgements

I am grateful to the anonymous referee of SL for very valuable suggestions and helpful comments regarding the earlier version of this paper.

Funding

This work is supported by the Ministry of Science and Technology of Serbia [Grant Number ON174026].

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mirjana Ilić.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by Heinrich Wansing

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ilić, M. A Cut-Elimination Proof in Positive Relevant Logic with Necessity. Stud Logica 109, 607–638 (2021). https://doi.org/10.1007/s11225-020-09920-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-020-09920-3

Keywords

Navigation