Notre Dame Journal of Formal Logic

An Algebraic Approach to the Disjunction Property of Substructural Logics

Daisuke Souma


Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the substructural logics for which syntactic methods don't work well.

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Notre Dame J. Formal Logic, Volume 48, Number 4 (2007), 489-495.

First available in Project Euclid: 29 October 2007

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Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} 03G25: Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35]

substructural logic residuated lattice disjunction property well-connectedness


Souma, Daisuke. An Algebraic Approach to the Disjunction Property of Substructural Logics. Notre Dame J. Formal Logic 48 (2007), no. 4, 489--495. doi:10.1305/ndjfl/1193667706.

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  • [1] Bayu Surarso, and H. Ono, "Cut elimination in noncommutative substructural logics", Reports on Mathematical Logic, no. 30 (1996), pp. 13--29.
  • [2] Galatos, N., and H. Ono, "Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL", Studia Logica, vol. 83 (2006),pp. 279--308.
  • [3] Hori, R., H. Ono, and H. Schellinx, "Extending intuitionistic linear logic with knotted structural rules", Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 219--42.
  • [4] Maksimova, L. L., "On maximal intermediate logics with the disjunction property", Studia Logica, vol. 45 (1986), pp. 69--75.
  • [5] Meyer, R. K., "Metacompleteness", Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501--16.
  • [6] Ono, H., "Proof-theoretic methods in nonclassical logic---An introduction", pp. 207--54 in Theories of Types and Proofs (Tokyo, 1997), vol. 2 of MSJ Memoirs, Mathmatical Society of Japan, Tokyo, 1998.
  • [7] Ono, H., ``Substructural logics and residuated lattices---An introduction,'' pp. 193--228in Trends in Logic. 50 Years of Studia Logica, vol. 21 of Trends in Logic Studia Logica Library, Kluwer Academic Publications, Dordrecht, 2003.
  • [8] Slaney, J. K., "A metacompleteness theorem for contraction-free relevant logics", Studia Logica, vol. 43 (1984), pp. 159--68.
  • [9] Souma, D., "Algebraic approach to disjunction property of substructural logics", pp. 26--28 in Proceedings of 38th MLG Meeting at Gamagori, Japan 2004, 2004.
  • [10] Souma, D., "An algebraic approach to the disjunction property of substructural logics", Master's thesis, Japan Advanced Institute of Science and Technology, Ishikawa, 2005.