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Logic Journal of IGPL Advance Access originally published online on October 12, 2007
Logic Journal of IGPL 2007 15(5-6):741-757; doi:10.1093/jigpal/jzm046
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Restricted Classical Modal Logics

Cezar A. Mortari

Departamento de Filosofia, Universidade Federal de Santa Catarina, Trindade, 88040-900 Florianópolis, SC, Brazil. E-mail: cmortari{at}cfh.ufsc.br

We consider a family of noncongruential modal logics obtained by restricting the smallest classical modal logic E and some of its extensions. We show that these logics are also properly contained in Lemmon's S0.5; semantics for them are adapted from Cresswell's semantics for S0.5 and from neighborhood semantics for classical modal logics. Some extensions of these logics by means of usual modal logical axioms are also considered, and determination results proved. As a further example, we also show how to obtain restricted versions of some of Chellas and Segerberg's prenormal modal logics.

Key Words: modal logic • non-normal modal logics • noncongruential modal logics.

Received for publication 15 September 2006.

References

  1. Chellas BF. Modal Logic: an introduction. (1980) Cambridge: Cambridge University Press.

  2. Chellas BF, Segerberg K. "Modal Logics in the Vicinity of S1." Notre Dame Journal of Formal Logic (1996) 37(1):1–25.[CrossRef]

  3. Hughes GE, Cresswell MJ. A New Introduction to Modal Logic. (1996) London, New York: Routledge.

  4. Hintikka J. "Impossible possible worlds vindicated". Journal of Philosophical Logic (1975) 4:475–84.[CrossRef][ISI]

  5. Addison J, Henkind L, Tarski A. "Semantical analysis of modal logic II. Non-normal modal propositional calculi". (1965) The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley: Amsterdam, North-Holland. 206–20.

  6. Lemmon EJ. "New Foundations for Lewis Modal Systems". Journal of Symbolic Logic (1957) 22(6):176–86.[CrossRef]

  7. Lemmon EJ. "Algebraic Semantics for Modal Logics, I". Journal of Symbolic Logic (1966) 31(1):46–65.[CrossRef][ISI]

  8. Lemmon EJ. "Algebraic Semantics for Modal Logics, II". Journal of Symbolic Logic (1966) 31(2):191–218.[CrossRef][ISI]

  9. Routley R. Decision procedures and semantics for C1, E1 and S0.5*. Logique et Analyse (1968) 11:468–71.

  10. Surendonk TJ. "Canonicity for intensional logics without iterative axioms". Journal of Philosophical Logic (1997) 26:391–409.[CrossRef][ISI]

  11. Sylvan R. Relational semantics for all Lewis, Lemmon and Feys’ modal logics, most notably for the systems between S0.3° and S1. The Journal of Non-Classical Logic (1989) 6(2):19–40.


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This Article
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