Notre Dame Journal of Formal Logic

An Abelian Rule for BCI—and Variations

Tomasz Kowalski and Lloyd Humberstone

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show the admissibility for BCI of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from (αβ)β to α. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of BCI.

Article information

Notre Dame J. Formal Logic, Volume 57, Number 4 (2016), 551-568.

Received: 6 November 2012
Accepted: 9 November 2013
First available in Project Euclid: 12 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F03: Proof theory, general 03F52: Linear logic and other substructural logics [See also 03B47]
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

admissible rules structural completeness BCI BCK abelian logic


Kowalski, Tomasz; Humberstone, Lloyd. An Abelian Rule for BCI—and Variations. Notre Dame J. Formal Logic 57 (2016), no. 4, 551--568. doi:10.1215/00294527-3679398.

Export citation


  • [1] Blok, W. J., and D. Pigozzi, Algebraizable Logics, vol. 77 of Memoirs of the American Mathematical Society, American Mathematical Society, Providence, 1989.
  • [2] Blok, W. J., and J. G. Raftery, “Assertionally equivalent quasivarieties,” International Journal of Algebra and Computation, vol. 18 (2008), pp. 589–681.
  • [3] Bunder, M. W., “Combinators, proofs and implicational logics,” pp. 229–86 in Handbook of Philosophical Logic, Second Edition, edited by D. M. Gabbay and F. Guenthner, vol. 6 of Handbook of Philosophical Logic, Kluwer, Dordrecht, 2002.
  • [4] Butchart, S., and S. Rogerson, “On the algebraizability of the implicational fragment of abelian logic,” Studia Logica, vol. 102 (2014), pp. 981–1001.
  • [5] Cintula, P., and G. Metcalfe, “Structural completeness in fuzzy logics,” Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 153–82.
  • [6] Došen, K., “A historical introduction to substructural logics,” pp. 1–30 in Substructural Logics (Tübingen, 1990), edited by P. Schroeder-Heister and K. Došen, vol. 2 of Studies in Logic and Computation, Oxford University Press, New York, 1993.
  • [7] Humberstone, L., “Extensions of intuitionistic logic without the deduction theorem: Some simple examples,” Reports on Mathematical Logic, vol. 40 (2006), pp. 45–82.
  • [8] Humberstone, L., “Variations on a theme of Curry,” Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 101–31.
  • [9] Humberstone, L., The Connectives, MIT Press, Cambridge, Mass., 2011.
  • [10] Kabziński, J. K., “BCI-algebras from the point of view of logic,” Bulletin of the Section of Logic, vol. 12 (1983), pp. 126–29.
  • [11] Kowalski, T., “Self-implications in BCI,” Notre Dame Journal of Formal Logic, vol. 49 (2008), pp. 295–305.
  • [12] Kowalski, T., “BCK is not structurally complete,” Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 197–204.
  • [13] Kowalski, T., and S. Butchart, “A note on monothetic BCI,” Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 541–44.
  • [14] Meyer, R. K., and J. K. Slaney, “Abelian logic (from A to Z),” pp. 245–88 in Paraconsistent Logic: Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Philosophia, Munich, 1989.
  • [15] Meyer, R. K., and J. K. Slaney, “A, still adorable,” pp. 241–60 in Paraconsistency (São Sebastião, 2000), edited by W. A. Carnielli, M. E. Coniglio, and I. M. L. D’Ottaviano, vol. 228 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 2002.
  • [16] Olson, J. S., J. G. Raftery, and C. J. van Alten, “Structural completeness in substructural logics,” Logic Journal of the IGPL, vol. 16 (2008), pp. 455–95.
  • [17] Paoli, F., “Logic and groups,” pp. 109–28 in Paraconsistency, Part III (Toruń, 1998), edited by J. Perzanowski and A. Pietruszczak, vol. 9 of Logic and Logical Philosophy, 2001.
  • [18] Paoli, F., M. Spinks, and R. Veroff, “Abelian logic and the logics of pointed lattice-ordered varieties,” Logica Universalis, vol. 2 (2008), pp. 209–33.
  • [19] Pogorzelski, W. A., “Structural completeness of the propositional calculus,” Bulletin of the Polish Academy of Sciences, vol. 19 (1971), pp. 349–51.
  • [20] Raftery, J. G., and C. J. van Alten, “Residuation in commutative ordered monoids with minimal zero,” Reports on Mathematical Logic, vol. 34 (2000), pp. 23–57.
  • [21] Raftery, J. G., and C. J. van Alten, “Corrigendum: Residuation in commutative ordered monoids with minimal zero,” Reports on Mathematical Logic, vol. 39 (2005), pp. 133–35.
  • [22] Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations, vol. 199 of Synthese Library, Kluwer, Dordrecht, 1988.