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The fixed point property in modal logic. (English) Zbl 1031.03039
Summary: This paper deals with the modal logics associated with (possibly nonstandard) provability predicates of Peano Arithmetic. One of our goals is to present some modal systems having the fixed point property and not extending the Gödel-Löb system GL. We prove that, for every \(n\geq 2\), \(K+\square (\square^{n-1} p\to p)\to\square p\) has the explicit fixed point property. Our main result states that every complete modal logic \(L\) having Craig’s interpolation property and such that \(L\vdash\Delta (\nabla(p)\to p)\to \Delta(p)\), where \(\nabla(p)\) and \(\Delta(p)\) are suitable modal formulas, has the explicit fixed point property.

03B45 Modal logic (including the logic of norms)
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
03F40 Gödel numberings and issues of incompleteness
03F30 First-order arithmetic and fragments
Full Text: DOI
[1] Boolos, G., The Unprovability of Consistency , Cambridge University Press, Cambridge, 1979. · Zbl 0409.03009
[2] Boolos, G., The Logic of Provability , Cambridge University Press, Cambridge, 1993. · Zbl 0891.03004
[3] Feferman, S., ”The arithmetization of metamathematics in a general setting”, Fundamentum Mathematicae , vol. 49 (1960), pp. 35–92. · Zbl 0095.24301 · eudml:213578
[4] Gabbay, D. M., ”Craig’s interpolation theorem for modal logics”, pp. 111–27 in Conference in Mathematical Logic—London ’70 , edited by W. Hodges, Springer, Berlin, 1972. · Zbl 0233.02009
[5] Hughes, G. E., and M. J. Cresswell, A New Introduction to Modal Logic , Routledge, London, 1996. · Zbl 0855.03002
[6] Maksimova, L., ”Amalgamation and interpolation in normal modal logic”, Studia Logica , vol. 50 (1991), pp. 457–71. · Zbl 0754.03013 · doi:10.1007/BF00370682
[7] Rautenberg, W., ”Modal tableau calculi and interpolation”, Journal of Philosophical Logic , vol. 12 (1983), pp. 403–23. · Zbl 0547.03015 · doi:10.1007/BF00249258
[8] Sacchetti, L., ”Modal logics with the fixed-point property”, Bollettino della Unione Matematica Italiana , vol. 2 (1999), pp. 279–90. · Zbl 0929.03026 · eudml:195361
[9] Sambin, G., and S. Valentini, ”The modal logic of provability. The sequential approach”, Journal of Philosophical Logic , vol. 11 (1982), pp. 311–42. · Zbl 0523.03014 · doi:10.1007/BF00293433
[10] Smoryński, C., ”Beth’s theorem and self-referential sentences”, pp. 253–61 in Logic Colloquium ’77 , edited by A. Macintyre, L. Pacholski, and J. Paris, North-Holland, Amsterdam, 1978. · Zbl 0453.03018
[11] Smoryński, C., Self-reference and Modal Logic , Springer-Verlag, New York, 1985. · Zbl 0596.03001
[12] van Benthem, J., Modal Logic and Classical Logic , Bibliopolis, Naples, 1985. · Zbl 0639.03014
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