Hostname: page-component-5b777bbd6c-sbgtn Total loading time: 0 Render date: 2025-06-19T22:22:04.778Z Has data issue: false hasContentIssue false
  • English
  • Français

NON–WELL-FOUNDED DERIVATIONS IN THE GÖDEL-LÖB PROVABILITY LOGIC

Published online by Cambridge University Press:  26 November 2019

DANIYAR SHAMKANOV
DANIYAR SHAMKANOV*
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences and National Research University Higher School of Economics
*
*STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES GUBKINA STR. 8, 119991, MOSCOW, RUSSIA and NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS FACULTY OF MATHEMATICS USACHEVA STR. 6, 119048, MOSCOW, RUSSIA E-mail: daniyar.shamkanov@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider Hilbert-style non–well-founded derivations in the Gödel-Löb provability logic GL and establish that GL with the obtained derivability relation is globally complete for algebraic and neighbourhood semantics.

Keywords

03B65provability logicalgebraic semanticsneighbourhood semanticsscattered topological spacesglobal consequence relationsnon–well-founded derivations
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 776 - 796
Check for updates
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

BIBLIOGRAPHY

[1]Abashidze, M. (1985). Ordinal completeness of the Gödel-Löb modal system. Intensional Logics and the Logical Structure of Theories. Tbilisi: Metsniereba, pp. 49–73. In Russian.Google Scholar
[2]Aguilera, J. & Fernández-Duque, D. (2017). Strong completeness of provability logic for ordinal spaces. The Journal of Symbolic Logic, 82 (2), 608–628.CrossRefGoogle Scholar
[3]Beklemishev, L. & Gabelaia, D. (2014). Topological interpretations of provability logic. In Bezhanishvili, G., editor. Outstanding Contributions to Logic. Leo Esakia on Duality in Modal and Intuitionistic Logics, Vol. 4. Dordrecht: Springer, pp. 257–290.Google Scholar
[4]Blass, A. (1990). Infinitary combinatorics and modal logic. The Journal of Symbolic Logic, 55(2), 761–778.CrossRefGoogle Scholar
[5]Esakia, L. (1981). Diagonal constructions, Löb’s formula and Cantor’s scattered space. Studies in Logic and Semantics, 132(3), 128–143. In Russian.Google Scholar
[6]Hakli, R. & Negri, S. (2011). Does the deduction theorem fail for modal logic? Synthese, 187(3), 849–867.CrossRefGoogle Scholar
[7]Iemhoff, R. (2016). Reasoning in circles. In van Eijck, J., Iemhoff, R., and Joosten, J. J., editors. Liber Amicorum Alberti. A Tribute to Albert Visser. London: College Publications, pp. 165–178.Google Scholar
[8]Kracht, M. (1999). Tools and Techniques in Modal Logic. Studies in Logic, Vol. 142. Amsterdam: Elsevier.CrossRefGoogle Scholar
[9]Litak, T. (2005). An Algebraic Approach to Incompleteness in Modal Logic. Ph.D. Thesis, Japan Advanced Institute of Science and Technology.Google Scholar
[10]Magari, R. (1975). The diagonalizable algebras (the algebraization of the theories which express Theor. II). Bollettino dell’Unione Matematica Italiana, 4(12), 117–125.Google Scholar
[11]Mazurkiewicz, S. & Sierpiński, W. (1920). Contribution à la topologie des ensembles dénombrables. Fundamenta Mathematicae, 1, 17–27.CrossRefGoogle Scholar
[12]Montague, R. (1970). Universal grammar. Theoria, 36, 373–398.CrossRefGoogle Scholar
[13]Pakhomov, F. & Walsh, J. (2018). Reflection ranks and ordinal analysis. arXiv:1805.02095.Google Scholar
[14]Scott, D. (1970). Advice on modal logic. In Lambert, K., editor. Philosophical Problems in Logic. Doredrecht: Reidel, pp. 143–173.CrossRefGoogle Scholar
[15]Shamkanov, D. (2014). Circular proofs for the Gödel-Löb provability logic. Mathematical Notes, 96(3), 575–585.CrossRefGoogle Scholar
[16]Shamkanov, D. (2017). Global neighbourhood completeness of the Gödel-Löb provability logic. In Kennedy, J. and de Queiroz, R., editors. Logic, Language, Information, and Computation. 24th International Workshop, WoLLIC 2017 (London, UK, July 18–21, 2017). Lecture Notes in Computer Science, Vol. 103888. Berlin: Springer, pp. 358–371.Google Scholar
[17]Shehtman, V. (2005). On neighbourhood semantics thirty years later. In Artemov, S., Barringer, H., d’Avila Garces, A., Lamb, L. C., and Woods, J., editors. We Will Show Them ! Essays in Honour of Dov Gabbay, Vol. 2. London: College Publications, pp. 663–692.Google Scholar
[18]Simmons, H. (1975). Topological aspects of suitable theories. Proceedings of the Edinburgh Mathematical Society, 19(4), 383–391.CrossRefGoogle Scholar
[19]Smoryński, C. (1985). Self-Reference and Modal Logic. New York: Springer.CrossRefGoogle Scholar
[20]Solovay, R. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25, 287–304.CrossRefGoogle Scholar
[21]Wen, X. (2019). Modal logic via global consequence. arXiv:1910.02446v1.Google Scholar