Notre Dame Journal of Formal Logic

A Syntactic Approach to Maksimova's Principle of Variable Separation for Some Substructural Logics

H. Naruse, H. Ono, and Bayu Surarso


Maksimova's principle of variable separation says that if propositional formulas $A_1 \supset A_2$ and $B_1 \supset B_2$ have no propositional variables in common and if a formula $A_1\wedge B_1 \supset A_2\vee B_2$ is provable, then either $A_1 \supset A_2$ or $B_1 \supset B_2$ is provable. Results on Maksimova's principle until now are obtained mostly by using semantical arguments. In the present paper, a proof-theoretic approach to this principle in some substructural logics is given, which analyzes a given cut-free proof of the formula $A_1\wedge B_1 \supset A_2\vee B_2$ and examines how the formula is derived. This analysis will make clear why Maksimova's principle holds for these logics.

Article information

Notre Dame J. Formal Logic, Volume 39, Number 1 (1998), 94-113.

First available in Project Euclid: 7 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03F03: Proof theory, general


Naruse, H.; Surarso, Bayu; Ono, H. A Syntactic Approach to Maksimova's Principle of Variable Separation for Some Substructural Logics. Notre Dame J. Formal Logic 39 (1998), no. 1, 94--113. doi:10.1305/ndjfl/1039293022.

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  • Anderson, A. R., and N. D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity, vol. 1, Princeton University Press, Princeton, 1975. Zbl 0323.02030 MR 53:10542
  • Bayu Surarso, “Interpolation theorem for some distributive logics,” forthcoming in Mathematica Japonica. Zbl 0957.03032 MR 2001d:03057
  • Bayu Surarso, and H. Ono, “Cut elimination in noncommutative substructural logics,” Reports on Mathematical Logic, vol. 30 (1996), pp. 13–29. Zbl 0896.03048 MR 2000a:03096
  • Chagrov, A., and M. Zakharyaschev, “The undecidability of the disjunction property of propositional logics and other related problems,” The Journal of Symbolic Logic, vol. 58 (1993), pp. 967–1002. Zbl 0799.03009 MR 94i:03048
  • Dunn, J. M., “Consecution formulation of positive $R$ with co-tenability and $t$,” pp. 381–91 in Entailment: The Logic of Relevance and Necessity, vol. 1, edited by A. R. Anderson and N. D. Belnap, Princeton University Press, Princeton,1975.
  • Dunn, J. M., “Relevance logic and entailment,” pp. 117–224 in Handbook of Philosophical Logic, vol. 3, edited by D. Gabbay and F. Guenthner, D. Reidel, Dordrecht, 1986. Zbl 0875.03051
  • Giambrone, S., “$TW_{+}$ and $RW_{+}$ are decidable,” The Journal of Philosophical Logic, vol. 14 (1985), pp. 235–54. Zbl 0587.03014 MR 87i:03021a
  • Maksimova, L., “The principle of separation of variables in propositional logics,” Algebra i Logika, vol. 15 (1976), pp. 168–84. Zbl 0363.02024 MR 58:21417
  • Maksimova, L., “Interpolation properties of superintuitionistic logics,” Studia Logica, vol. 38 (1979), pp. 419–28. Zbl 0435.03021 MR 81f:03035
  • Maksimova, L., “Relevance principles and formal deducibility,” pp. 95–97 in Directions in Relevant Logic, edited by J. Norman and R. Sylvan, Kluwer Academic Publishers, Boston, 1989.
  • Maksimova, L., “On variable separation in modal and superintuitionistic logics,” Studia Logica, vol. 55 (1995), pp. 99–112. Zbl 0840.03017 MR 96j:03034
  • Mints, G. E., “Cut elimination theorem for relevant logics,” Journal of Soviet Mathematics, vol. 6 (1976), pp. 422–28. Zbl 0379.02011 MR 49:8823
  • Ono, H., “Structural rules and a logical hierarchy,” pp. 95–104 in Mathematical Logic, edited by P. P. Petkov, Plenum Press, New York, 1990. Zbl 0790.03007 MR 91j:03073
  • Ono, H., “Semantics for substructural logics,” pp. 259–91 in Substructural Logics, edited by K. Došen and P. Schröeder-Heister, Oxford University Press, Oxford, 1993. Zbl 0941.03522 MR 95f:03013
  • Ono, H., and Y. Komori, “Logics without the contraction rule,” The Journal of Symbolic Logic, vol. 50 (1985), pp. 169–201. Zbl 0583.03018 MR 87a:03053
  • Slaney, J., “Solution to a problem of Ono and Komori,” The Journal of Philosophical Logic, vol. 18 (1989), pp. 103–11. Zbl 0671.03036 MR 90c:03050