Geodesic
Predicate and set-theoretic calculi based on logic without contractions
Izvestiya. Mathematics , Tome 18 (1982) no. 1, pp. 41-59.

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In this paper the author studies the effect of eliminating the contraction rules from the Gentzen predicate calculus, and also from set-theoretic systems based on this calculus. It turns out that the predicate calculus without contraction rules is complete relative to finite models, and eliminating contraction rules from set-theoretic systems does not change the number of derivable formulas as long as the systems contain certain natural axioms. Bibliography: 4 titles. 
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V. N. Grishin. Predicate and set-theoretic calculi based on logic without contractions. Izvestiya. Mathematics , Tome 18 (1982) no. 1, pp. 41-59. http://geodesic.mathdoc.fr/item/IM2_1982_18_1_a2/

[1] Gentsen G., “Issledovaniya logicheskikh vyvodov”, Matematicheskaya teoriya logicheskogo vyvoda, Nauka, M., 1967, 9–76 | MR

[2] Grishin V. N., “Ob odnoi nestandartnoi logike i ee primenenii k teorii mnozhestv”, Issledovaniya po formalizovannym yazykam i neklassicheskim logikam, Nauka, M., 1974, 135–171 | MR

[3] Grishin V. N., “Ob algebraicheskoi semantike logiki bez sokraschenii”, Issledovaniya po teorii mnozhestv i neklassicheskim logikam, Nauka, M., 1976, 247–264 | MR

[4] Mints G. E., “Gentsena formalnaya sistema”, Matematicheskaya Entsiklopediya, t. I, Sovetskaya Entsiklopediya, M., 1977, 919–921