Notre Dame Journal of Formal Logic

Self-implications in BCI

Tomasz Kowalski


Humberstone asks whether every theorem of BCI provably implies φ φ for some formula φ . Meyer conjectures that the axiom B does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.

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Notre Dame J. Formal Logic, Volume 49, Number 3 (2008), 295-305.

First available in Project Euclid: 15 July 2008

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Zentralblatt MATH identifier

Primary: 03F07: Structure of proofs 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

BCI logic sequent system self-implication


Kowalski, Tomasz. Self-implications in BCI. Notre Dame J. Formal Logic 49 (2008), no. 3, 295--305. doi:10.1215/00294527-2008-013.

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  • [1] Blok, W. J., and J. G. Raftery, "Fragments of $R$"-mingle, Studia Logica, vol. 78 (2004), pp. 59--106.
  • [2] Buszkowski, W., "Type logics in grammar", pp. 337--82 in Trends in Logic, vol. 21 of Trends in Logic-Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2003.
  • [3] Curry, H. B., R. Hindley, and J. Seldin, Combinatory Logic II, North Holland, Amsterdam, 1972.
  • [4] Humberstone, L., "Variations on a theme of Curry", Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 101--31.
  • [5] Kowalski, T., and S. Butchart, "A note on monothetic BCI", Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 541--44.
  • [6] Nagayama, M., "On a property of BCK"-identities, Studia Logica, vol. 53 (1994), pp. 227--34.