Skip to main content
SpringerLink
Log in

The Power of a Propositional Constant

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akishev, G., & Goldblatt, R. (2010). Monadic bounded algebras. Studia Logica, 96(1), 1–40.

    Article  Google Scholar 

  2. Anderson, A.R. (1958). A reduction of deontic logic to alethic modal logic. Mind (New Series), 67, 100–103.

    Article  Google Scholar 

  3. Bezhanishvili, G., Ghilardi, S., Jibladze, M. (2011). An algebraic approach to subframe logics. Modal case. Notre Dame Journal of Formal Logic, 52(2), 187–202.

    Article  Google Scholar 

  4. Bull, R.A. (1966). That all normal extensions of S4.3 have the finite model property. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 12, 341–344.

    Article  Google Scholar 

  5. Chagrov, A., & Zakharyaschev, M. (1997). Modal logic. Oxford University Press.

  6. Esakia, L. (2004). Intuitionistic logic and modality via topology. Annals of Pure and Applied Logic, 127, 155–170.

    Article  Google Scholar 

  7. Fine, K. (1971). The logics containing S4.3. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 17, 371–376.

    Article  Google Scholar 

  8. Fine, K. (1974). Logics containing K4. Part I. The Journal of Symbolic Logic, 39(1), 31–42.

    Article  Google Scholar 

  9. French, R. (2012). Denumerably many post-complete normal modal logics with propositional constants. Notre Dame Journal of Formal Logic, 53(4), 549–556.

    Article  Google Scholar 

  10. Goldblatt, R. (1985). An algebraic study of well-foundedness. Studia Logica, 44(4), 423–437.

    Article  Google Scholar 

  11. Halmos, P.R. (1962). Algebraic logic. New York: Chelsea.

  12. Harel, D. (1984). Dynamic logic. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic, volume II: Extensions of classical logic (pp. 497–604). D. Reidel.

  13. Jónsson, B., & Tarski, A. (1951). Boolean algebras with operators, part I. American Journal of Mathematics, 73, 891–939.

    Article  Google Scholar 

  14. Kowalski, T. (1998). Varieties of tense algebras. Reports on Mathematical Logic, 32, 53–95.

    Google Scholar 

  15. Kowalski, T., & Kracht, M. (2006). Semisimple varieties of modal algebras. Studia Logica, 83, 351–363.

    Article  Google Scholar 

  16. Kripke, S.A. (1965). Semantical analysis of modal logic II. Non-normal modal propositional calculi. In J. W. Addison, L. Henkin, A. Tarski (Eds.), The Theory of Models (pp. 206–220). North-Holland.

  17. Litak, T. (2005). On notions of completeness weaker than Kripke completeness. In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, H. Wansing (Eds.), Advances in Modal Logic (Vol. 5, pp. 149–169). London: King’s College Publications. www.aiml.net.

  18. Makinson, D.C. (1971). Some embedding theorems for modal logic. Notre Dame Journal of Formal Logic, 12, 252–254.

    Article  Google Scholar 

  19. Nowell-Smith, P.H., & Lemmon, E.J. (1960). Escapism: the logical basis of ethics. Mind (New Series), 69, 289–300.

    Article  Google Scholar 

  20. Post, E.L. (1921). Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43, 163–185. Reprinted in [27], 264–283.

    Article  Google Scholar 

  21. Prior, A.N. (1985). Escapism: The logical basis of ethics. In A.I. Melden (Ed.), Essays in moral philosophy (pp. 135–146). University of Washington Press.

  22. Segerberg, K. (1971). An Essay in Classical Modal Logic, volume 13 of Filosof iska Studier. Uppsala Universitet.

  23. Segerberg, K. (1972). Post completeness in modal logic. The Journal of Symbolic Logic, 37(4), 711–715.

    Article  Google Scholar 

  24. Segerberg, K. (1975). That every extension of S4.3 is normal. In S. Kanger (Ed.), Proceedings of the third scandinavian logic symposium (pp. 194–196). North-Holland.

  25. Streett, R.S. (1982). Propositional dynamic logic of looping and converse is elementarily decidable. Information and Control, 54, 121–141.

    Article  Google Scholar 

  26. Thomason, S.K. (1972). Semantic analysis of tense logic. The Journal of Symbolic Logic, 37, 150–158.

    Article  Google Scholar 

  27. van Heijenoort, J. (Ed.). (1967). From Frege to Gödel: A source book in mathematical logic (pp. 1879–1931). Harvard University Press.

  28. Williamson, T. (1998). Continuum many maximal consistent normal bimodal logics with inverses. Notre Dame Journal of Formal Logic, 39(1), 128–134.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, P.O. Box 600, Wellington, 6140, New Zealand

    Robert Goldblatt

  2. Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria, 3086, Australia

    Tomasz Kowalski

Authors
  1. Robert Goldblatt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Kowalski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Goldblatt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldblatt, R., Kowalski, T. The Power of a Propositional Constant. J Philos Logic 43, 133–152 (2014). https://doi.org/10.1007/s10992-012-9256-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-012-9256-0

Keywords

Search

Navigation