Geodesic
Graded modal operators and fixed points
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 1, pp. 70-76
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There is the well-known Fixed Point Theorem in the theory of modal logics. In the article this theorem is generalized from monomodal case to graded modalities. The following theorem is proved Theorem. For any graded modalized operator $F_\varphi$, there is unique fixed point of the operator $F_\varphi$ in every strictly partially ordered model with the ascending chain condition and there is a graded formula $\omega$, which defines the fixed point in every such model. The formula $\omega$ contains only those graded modalities, which are contained in $\varphi$. 
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S. I. Mardaev. Graded modal operators and fixed points. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 1, pp. 70-76. http://geodesic.mathdoc.fr/item/VNGU_2006_6_1_a4/

[1] P. Aczel, “An introduction to inductive definitions”, Handbook in Mathematical Logic, Studies in Logic and the Foundations of Mathematics, 90, ed. J. Barwise, North-Holland Publishing Company, 1977, 739–782 ; P. Atsel, “Vvedenie v teoriyu induktivnykh opredelenii”, Spravochnaya kniga po matematicheskoi logike, v. 3, ed. Yu. L. Ershov, Nauka, M., 1982, 224–268 | DOI | MR

[2] C. Bernardi, “The fixed-point theorem for diagonalizable algebras”, Studia Logica, 34:3 (1975), 239–251 | DOI | MR | Zbl

[3] A. Chagrov, M. Zakharyaschev, Modal Logic, Oxford Logic Guides, 35, Clarendon Press, 1997 | MR | Zbl

[4] S. I. Mardaev, “Least Fixed Points in Modal Logic”, Proceedings of International Conferences on Mathematical logic, eds. S. S. Goncharov a.o., Novosibirsk State University, Novosibirsk, 2002, 92–103

[5] L. Reidhaar-Olson, “A new proof of the fixed-point theorem of provability logic”, Notre Dame Journal of Formal Logic, 31:1 (1990), 37–43 | DOI | MR | Zbl

[6] V. V. Rybakov, Admissibility of Logical Inference Rules, Studies in Logic and the Foundations of Mathematics, 136, Elsevier Science Publishers B. V., 1997 | MR | Zbl

[7] G. Sambin, “An effective fixed-point theorem in intuitionistic diagonalizable algebras”, Studia Logica, 35:4 (1976), 345–361 | DOI | MR | Zbl

[8] G. Sambin, S. Valentini, “The modal logic of provability: The sequential approach”, J. of Philosophical Logic, 11:3 (1982), 311–342 | DOI | MR | Zbl

[9] K. Segerberg, An Essay in Classical Modal Logic, Filosofiska Studier, 13, Uppsala Universitet, Uppsala, 1971 | MR | Zbl

[10] C. Smorýnski, Consistency and related metamathematical properties, Technical Report 75-02, Univ. of Amsterdam, Mathematics Institute, 1975