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Failure of the Blok-Esakia Theorem in the monadic setting
Published 15 May 2024 in math.LO | (2405.09401v2)
Abstract: The Blok-Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok-Esakia isomorphism $\sigma$ does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that $\sigma$ is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.
- Monadic intuitionistic and modal logics admitting provability interpretations. J. Symb. Log., 88(1):427–467, 2023.
- G. Bezhanishvili and L. Carai. Failure of Esakia’s Theorem in the monadic setting. In preparation, 2024.
- G. Bezhanishvili. Varieties of monadic Heyting algebras. I. Studia Logica, 61(3):367–402, 1998.
- G. Bezhanishvili. Varieties of monadic Heyting algebras. II. Duality theory. Studia Logica, 62(1):21–48, 1999.
- G. Bezhanishvili and C. Meadors. Local finiteness in varieties of MS4-algebras. Submitted. Available at arXiv:2312.16754, 2024.
- S. Burris and H. P. Sankappanavar. A course in universal algebra, volume 78 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1981.
- R. A. Bull. MIPC as the formalisation of an intuitionist concept of modality. J. Symb. Log., 31(4):609–616, 1966.
- A. Chagrov and M. Zakharyaschev. Modal logic, volume 35 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1997.
- Introduction to lattices and order. Cambridge University Press, New York, second edition, 2002.
- L. Esakia. Topological Kripke models. Soviet Math. Dokl., 15:147–151, 1974.
- L. Esakia. On the variety of Grzegorczyk algebras. In Studies in nonclassical logics and set theory (Russian), pages 257–287. “Nauka”, Moscow, 1979.
- L. Esakia. Provability logic with quantifier modalities. In Intensional logics and the logical structure of theories (Russian), pages 4–9. “Metsniereba”, Tbilisi, 1988.
- L. Esakia. Heyting algebras. Duality theory, volume 50 of Trends in Logic. Translated from the Russian by A. Evseev. Edited by G. Bezhanishvili and W. Holliday. Springer, 2019.
- G. Fischer Servi. On modal logic with an intuitionistic base. Studia Logica, 36(3):141–149, 1977.
- G. Fischer Servi. The finite model property for MIPQ and some consequences. Notre Dame J. Formal Logic, 19(4):687–692, 1978.
- Many-dimensional modal logics: theory and applications. Amsterdam; Boston: Elsevier North Holland, 2003.
- Quantification in nonclassical logic. Vol. 1, volume 153 of Studies in Logic and the Foundations of Mathematics. Elsevier B. V., Amsterdam, 2009.
- M. Kracht. Tools and techniques in modal logic, volume 142 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1999.
- The lattice of normal modal logics. Algebra and Logic, 13:105–122, 1974.
- J. C. C. McKinsey and A. Tarski. Some theorems about the sentential calculi of Lewis and Heyting. J. Symb. Log., 13:1–15, 1948.
- A. Monteiro and O. Varsavsky. Álgebras de Heyting monádicas. Actas de las X Jornadas de la Unión Matemática Argentina, Bahía Blanca, pages 52–62, 1957.
- P. G. Naumov. Modal logics that are conservative over intuitionistic predicate calculus. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Russian), (6):86–90, 1991.
- A. N. Prior. Time and modality. Greenwood Press, 1957.
- H. Rasiowa and R. Sikorski. The mathematics of metamathematics. Polish Scientific Publishers, 1963.
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