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 Mat. Sb. (N.S.), 1977, Volume 102(144), Number 2, Pages 314–323 (Mi msb2686)

On admissible rules of intuitionistic propositional logic

A. I. Citkin

Abstract: This paper studies modus rules of deduction admissible in intuitionistic propositional logic (a rule is called a modus rule if it corresponds to some sequence and allows passage from the results of any substitution in the formulas in its antecedent to the result of the same substitution in its succedent). Examples of such rules are considered, as well as the derivability of certain rules from others by means of the intuitionistic propositional calculus. An infinite independent system of admissible modus rules is constructed. It is proved that a finite Gödel pseudo-Boolean algebra in which all modus rules are valid (i.e. the quasi-identities corresponding to them are valid) is isomorphic to a sequential union of Boolean algebras of power not greater than 4.
Figures: 3.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 31:2, 279–288

Bibliographic databases:

UDC: 517.12
MSC: Primary 02B05, 02C15, 02J05, 02D99, 02E05, 06A35; Secondary 02B99, 02E99, 06A25, 02H10, 06A40, 08A15

Citation: A. I. Citkin, “On admissible rules of intuitionistic propositional logic”, Mat. Sb. (N.S.), 102(144):2 (1977), 314–323; Math. USSR-Sb., 31:2 (1977), 279–288

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Rybakov, “Bases of admissible rules of the modal system Grz and of intuitionistic logic”, Math. USSR-Sb., 56:2 (1987), 311–331
2. V. V. Rybakov, “Decidability of admissibility in the modal system $\mathrm{Grz}$ and in intuitionistic logic”, Math. USSR-Izv., 28:3 (1987), 589–608
3. V. V. Rybakov, “Criteria for admissibility of rules of inference with parameters in the intuituonistc propositional calculus”, Math. USSR-Izv., 37:3 (1991), 693–703
4. V. V. Rybakov, “Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability”, Math. USSR-Izv., 36:2 (1991), 369–390
5. Vladimir V. Rybakov, “Intermediate logics preserving admissible inference rules of heyting calculus”, MLQ-Math Log Quart, 39:1 (1993), 403
6. V. V. Rybakov, “Hereditarily structurally complete modal logics”, J. symb. log, 60:01 (1995), 266
7. V. V. Rybakov, “Logical consecutions in discrete linear temporal logic”, J. symb. log, 70:04 (2005), 1137
8. B. R. Fedorishin, “An explicit basis for the admissible inference rules in the Gödel–Löb logic $GL$”, Siberian Math. J., 48:2 (2007), 339–345
9. Vitalii V. Rimatskii, “Yavnyi bazis dopustimykh pravil vyvoda logik konechnoi shiriny”, Zhurn. SFU. Ser. Matem. i fiz., 1:1 (2008), 83–91
10. V. V. Rimatskii, “An explicit basis for admissible inference rules in table modal logics of width 2”, Algebra and Logic, 48:1 (2009), 72–86
11. V. V. Rimatskii, “An explicit basis for the admissible inference rules of the modal logics extending $S4.1$ and $Grz$”, Siberian Math. J., 50:4 (2009), 692–699
12. Rutskii A.N., “Finitnaya approksimiruemost kak dostatochnoe uslovie razreshimosti po dopustimosti dlya tranzitivnykh modalnykh i superintuitsionistskikh logik”, Vestn. Krasnoyarskogo gos. ped. un-ta im. V. P. Astafeva, 2009, no. 3, 86–93
13. Alexander Citkin, “A note on admissible rules and the disjunction property in intermediate logics”, Arch. Math. Logic, 2011
14. V. V. Rimatskii, “On construction of an explicit basis for admissible inference rules of modal logics extending $S4.1$”, Discrete Math. Appl., 21:5-6 (2011), 741–760
15. G. V. Bokov, “Criterion for propositional calculi to be finitely generated”, Discrete Math. Appl., 23:5-6 (2013), 399–427
16. J.P.. Goudsmit, “Admissibility and refutation: some characterisations of intermediate logics”, Arch. Math. Logic, 2014
17. Alex Citkin, “Characteristic Inference Rules”, Log. Univers, 2015
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