Geodesic
Sequential Reflexive Logics with Noncontingency Operator
Matematičeskie zametki, Tome 72 (2002) no. 6, pp. 853-868.

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Hilbert systems $L^\vartriangleright$ and sequential calculi $[L^\vartriangleright]$ for the versions of logics $L=\mathbf T,\mathbf {S4},\mathbf B,\mathbf {S5}$, and $\mathbf {Grz}$ stated in a language with the single modal noncontingency operator $\vartriangleright A=\square A\vee \square \neg A$ are constructed. It is proved that cut is not eliminable in the calculi $[L^\vartriangleright]$, but we can restrict ourselves to analytic cut preserving the subformula property. Thus the calculi $[\mathbf T^\vartriangleright]$, $[\mathbf {S4}^\vartriangleright]$, $[\mathbf {S5}^\vartriangleright ]$ ($[\mathbf {Grz}^\vartriangleright]$, respectively) satisfy the (weak, respectively) subformula property; for $[\mathbf B_2^\vartriangleright]$, this question remains open. For the noncontingency logics in question, the Craig interpolation property is established. 
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E. E. Zolin. Sequential Reflexive Logics with Noncontingency Operator. Matematičeskie zametki, Tome 72 (2002) no. 6, pp. 853-868. http://geodesic.mathdoc.fr/item/MZM_2002_72_6_a6/

[1] Brogan A. P., “Aristotle's logic of statements about contingency”, Mind, 76 (1967), 49–81 | DOI

[2] Montgomery H., Routley R., “Contingency and non-contingency bases for normal modal logics”, Logique et Analyse, 9 (1966), 318–328 | MR | Zbl

[3] Montgomery H., Routley R., “Noncontingency axioms for $S4$ and $S5$”, Logique et Analyse, 11 (1968), 422–424 | MR | Zbl

[4] Montgomery H., Routley R., “Modalities in a sequence of normal non-contingency modal systems”, Logique et Analyse, 12 (1969), 225–227 | MR | Zbl

[5] Mortensen C., “A sequence of normal modal systems with non-contingency bases”, Logique et Analyse, 19 (1976), 341–344 | MR | Zbl

[6] Cresswell M. J., “Necessity and contingency”, Studia Logica, 47 (1988), 145–149 | DOI | MR | Zbl

[7] Humberstone I. L., “The logic of non-contingency”, Notre Dame J. Formal Logic, 36:2 (1995), 214–229 | DOI | MR | Zbl

[8] Kuhn S. T., “Minimal non-contingency logic”, Notre Dame J. Formal Logic, 36:2 (1995), 230–234 | DOI | MR | Zbl

[9] Zolin E., “Epistemic non-contingency logic”, Notre Dame J. Formal Logic, 40:4 (1999), 533–547 | DOI | MR | Zbl

[10] Zolin E., “Sekventsialnaya logika arifmeticheskoi razreshimosti”, Vestn. MGU. Ser. 1. Matem., mekh., 2001, no. 6, 43–48 | MR | Zbl

[11] Mints G. E., “Sistemy Lyuisa i sistema $T$ (1965–1973)”, R. Feis, Modalnaya logika, Nauka, M., 1974, 423–509

[12] Ohnishi M., Matsumoto K., “Gentzen method in modal calculi”, Osaka Math. J., 9:2 (1957), 113–130 ; Correction, Osaka Math. J., 10:1 (1958), 147 | MR | Zbl | MR

[13] Ohnishi M., Matsumoto K., “Gentzen method in modal calculi, II”, Osaka Math. J., 11:2 (1959), 115–120 | MR | Zbl

[14] Takano M., “Subformula property as a substitute for cut elimination in modal propositional logics”, Math. Japonica, 37:6 (1992), 1129–1145 | MR | Zbl

[15] Smullyan R. M., “Analytic cut”, J. Symbolic Logic, 33 (1968), 560–564 | DOI | MR | Zbl

[16] Boolos G., The Logic of Provability, Cambridge Univ. Press, Cambridge, 1993 | Zbl

[17] Chagrov A., Zakharyaschev M., Modal Logic, Oxford Sci. Publ., 1997 | Zbl