Interpolation and implicit definability in extensions of the provability logic
DOI:
https://doi.org/10.12775/LLP.2008.008Słowa kluczowe
provability logic, interpolation, Beth propertyAbstrakt
The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog—the intuitionistic provability logic—and investigated these two logics and their extensions. In the present paper, different versions of interpolation and of the Beth property in normal extensions of the provability logic GL are considered. It is proved that in a large class of extensions of GL (including all finite slice logics over GL) almost all versions of interpolation and of the Beth property are equivalent. It follows that in finite slice logics over GL the three versions CIP, IPD and IPR of the interpolation property are equivalent. Also they are equivalent to the Beth properties B1, PB1 and PB2.Bibliografia
Beth, E.W., “On Padoa’s method in the theory of definitions”, Indagationes Math. 15, 4 (1953), 330–339.
Blok, W.J., “Pretabular varieties of modal algebras”, Studia Logica 39 (1980), 101–124.
Craig, W., “Three uses of Herbrand-Gentzen theorem in relating model theory”, J. Symbolic Logic 22 (1957), 269–285.
Gabbay, D.M., and L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Clarendon Press, Oxford, 2005.
Kuznetsov, A.V., “Proof intuitionistic logic”, pages 75–79 in Modal and Intensional Logics (abstracts of conference), Moscow, 1978.
Kuznetsov, A.V., “On proof-intuitionistic propositional calculus”, Doklady Academii Nauk SSSR 283, 1 (1985), 27–29.
Kuznetsov, A.V., and A.Y. Muravitsky, “Provability as modality”, pages in 193–230 in Actual Problems on Logics and Methodology of Science, Naukova Dumka, Kiev, 1980.
Maksimova, L., “On interpolation in normal modal logics”, pages 40–56 in Non-classical Logics, Stiinca, Kishinev, 1987.
Maksimova, L.L., “A continuum of normal extensions of the modal provability logics with interpolation property”, Siberian Math. J. 30, 6 (1989), 122–131.
Maksimova, L.L., “An analog of Beth’s theorem in normal extensions of the modal K4 logic”, Siberian Math. J. 33, 6 (1992), 118–130.
Maksimova, L.L., “Modal logics and varieties of modal algebras: the Beth property, interpolation and amalgamation”, Algebra and Logic 31, 2 (1992), 145–166.
Maksimova, L., “Explicit and implicit definability in modal and related logics”, Bulletin of the Section of Logic 27, 1/2 (1998), 36–39.
Maksimova, L.L., “Intuitionistic logic and implicit definability”, Annals of Pure and Applied Logic 105, 1–3 (2000), 83–102.
Maksimova, L., “Projective Beth’s properties in infinite slice extensions of K4”, pages 349–363 in F. Wolter, H. Wansing, M. de Rijke, M. Zakharyaschev (eds.), Advances in Modal Logics, vol. 3, World Scientific, Singapore, 2002.
Maksimova, L.L., “Implicit definability in positive logics”, Algebra and Logic 42, 1 (2003), 65–93.
Maksimova, L., “Restricted interpolation in modal logics”, pages 297–311 in Advances in Modal Logic, vol. 4, King’s College Publications, London, 2003.
Maksimova, L., “Definability in normal extensions of S4”, Algebra and Logic 43, 4 (2004), 387–410.
Maksimova, L., “Interpolation and Joint Consistency”, pages 293–305 in S. Artemov, H. Barringer, A. d’Avila Garcez, L. Lamb and J. Woods (eds.), We Will Show Them! Essays in Honour of Dov Gabbay, vol. 2, King’s College Publications, London, 2005.
Maksimova, L.L., “Projective Beth property and interpolation in positive and related logics”, Algebra and Logic 44 (2005), 85–113.
Maksimova, L., “Definability and interpolation in non-classical logics”, Studia Logica 82, 2 (2006), 271–291.
Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, PWN, Warszawa, 1963.
Segerberg, K., An Essay in Classical Modal Logic, Uppsala University, 1971.
Solovay, R.M., “Provability interpretations of modal logics”, Israel Journal of Mathematics 25 (1976), 287–304.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Statystyki
Liczba wyświetleń i pobrań: 594
Liczba cytowań: 0
