Skip to main content

Modal Logic and Self-Reference

  • Chapter
  • 473 Accesses

Part of the book series: Handbook of Philosophical Logic ((HALO,volume 11))

Abstract

Ever since Epimenides made his startling confession, philosophers and mathematicians have been fascinated by self-reference. Of course, mathematicians are not free to admit this.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. S. N. Artyomov. Arithmetically complete modal theories. Semiotics and Information Science,14:115–133, 1980. In Russian.

    Google Scholar 

  2. C. Bernardi. The fixed-point theorem for diagonalizable algebras. Studia Logica, 34: 239–251, 1975.

    Article  Google Scholar 

  3. G. Boolos. The Unprovability of Consistency. Cambridge University Press, 1979. 2nd edition: The Logic of Provability, Cambridge University Press, 1995.

    Google Scholar 

  4. G. Boolos. Extremely undecidable sentences. Journal of Symbolic Logic, 47: 191–196, 1982.

    Article  Google Scholar 

  5. H. Friedman. The disjunction property implies the numerical existence property. Proc Nat Acad Sci, 62: 2877–2878, 1975.

    Article  Google Scholar 

  6. D. Guaspari and R. M. Solovay. Rosser sentences. Ann. Math. Logic, 16: 81–99, 1979.

    Article  Google Scholar 

  7. D. Guaspari. Partially conservative extensions of arithmetic. Trans. AMS, 254: 47–68, 1979.

    Article  Google Scholar 

  8. P. Hajek. On interpretability in theories containing arithmetic II Com . Math. Univ. Carol., 22: 667–688, 1982.

    Google Scholar 

  9. D. Jensen and A. Ehrenfeucht. Some problem in elementary arithmetics. Fund. Math, 92: 223–245, 1976.

    Google Scholar 

  10. G. Kreisel and G. Takeuti. Formally self-referential propo sitions in cut-free classical analysis and related systems. Diss. Math, 118: 1–50, 1974.

    Google Scholar 

  11. A. Macintyre and H. Simmons. Gödel’s diagonalization technique and related properties of theories. Colloq. Math, 28: 165–180, 1973.

    Google Scholar 

  12. F. Montagna. On the diagonalizable algebra of Peano arithmetic. Bolletino UMI (5), 16-B: 795–812, 1979.

    Google Scholar 

  13. G. Sambin. Fixed points through the finite model property. Studia Logica, 37: 287–289, 1976.

    Article  Google Scholar 

  14. C. Smorynski. Beth’s theorem and self-referential sentences. In A. Macintyre, L. Pacholski, and J. Paris, editors, Logic Colloquium 77. North-Holland, Amsterdam, 1978.

    Google Scholar 

  15. C. Smorynski. Calculating self-referential statementsl: explicit calculations. Studia Logica, 38: 17–36, 1979.

    Article  Google Scholar 

  16. C. Smorynski. Calculating self-referential statements. Fund Math, 109: 189–210, 1980.

    Google Scholar 

  17. C. Smorynski. Fifty years of self-reference in arithmetic. Notre Dame Journal of Formal Logic, 22: 357–374, 1981.

    Article  Google Scholar 

  18. C. Smorynski. Commutativity and self-reference. Notre Dame Journal of Formal Logic, 213: 443–452, 1982.

    Article  Google Scholar 

  19. C. Smorynski. Self-Reference and Modal Logic. Springer-Verlag, 1985.

    Google Scholar 

  20. C. Smorynski. An ubiquitous fixed point calculation. In L. Szczerba and K. Prazmowski, editors, Interpretability. Proceedings of the First Joint Conference on the Foundations of Mathematics Organised by the Bialystok Branch of the Warsaw University and Humboldt University (Berlin) in Doktorce 1980. Filia Uniwersytetu Warszawskiego w Bialmstoku, 1985.

    Google Scholar 

  21. R. M. Solovay. Provability interpretations of modal logic. Israel J. Math, 25: 287–304, 1976.

    Article  Google Scholar 

  22. R. M. Solovay. On interpretability in set theory, To appear. Editor’s note: Results contained in a letter to P. Hajek, paper still not published.

    Google Scholar 

  23. V. Švejdar. On interpretability in set theory, To appear Editor’s note: Published as: A sentence that is difficult to interpret. Comm. Math. Univ. Carolinae. See also: Modal analysis of generalised Rosser sentences. Journal of Symbolic Logic, 48: 986–999, 1983.

    Google Scholar 

  24. A. Visser. Aspects of diagonalisation and provability. PhD thesis, University of Utrecht, 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Smoryński, C. (2002). Modal Logic and Self-Reference. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0466-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0466-3_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6554-4

  • Online ISBN: 978-94-017-0466-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics