## Notre Dame Journal of Formal Logic

### Modal Logics in the Vicinity of S1

#### Abstract

We define prenormal modal logics and show that S1, S1, S0.9, and S0.9 are Lewis versions of certain prenormal logics, determination and decidability for which are immediate. At the end we characterize Cresswell logics and ponder C. I. Lewis's idea of strict implication in S1.

#### Article information

Source
Notre Dame J. Formal Logic Volume 37, Number 1 (1996), 1-24.

Dates
First available in Project Euclid: 16 December 2002

http://projecteuclid.org/euclid.ndjfl/1040067312

Digital Object Identifier
doi:10.1305/ndjfl/1040067312

Mathematical Reviews number (MathSciNet)
MR1379545

Zentralblatt MATH identifier
0859.03011

#### Citation

Chellas, Brian F.; Segerberg, Krister. Modal Logics in the Vicinity of S1 . Notre Dame J. Formal Logic 37 (1996), no. 1, 1--24. doi:10.1305/ndjfl/1040067312. http://projecteuclid.org/euclid.ndjfl/1040067312.

#### References

• [1] Benton, Roy A., Strong modal completeness with respect to neighborhood semantics,'' unpublished manuscript, Department of Philosophy, The University of Michigan, 1975.
• [2] Chellas, Brian F., Modal Logic: An Introduction, Cambridge University Press, Cambridge and New York, 1980; reprinted with corrections 1988.
• [3] Cresswell, M. J., The interpretation of some Lewis systems of modal logic,'' Australasian Journal of Philosophy, vol. 45 (1967), pp. 198--206.
• [4] Cresswell, M. J., The completeness of S1 and some related problems,'' Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 485--496.
• [5] Cresswell, M. J., S1 is not so simple,'' pp. 29--40 in Modality, Morality and Belief: Essays in Honor of Ruth Barcan Marcus, edited by Walter Sinnott-Armstrong, Diana Raffman, and Nicholas Asher, Cambridge University Press, Cambridge, 1995.
• [6] Feys, Robert, Modal Logics, edited with some complements by Joseph Dopp, E. Nauwelaerts, Louvain, and Gauthiers-Villars, Paris, 1965.
• [7] Girle, Roderick A., S1$\neq$ S0.9,'' Notre Dame Journal of Formal Logic, vol. 16 (1975), pp. 339--344.
• [8] Hughes, G. E., and M. J. Cresswell, An Introduction to Modal Logic, Methuen and Co. Ltd., London, 1968; reprinted with corrections 1972.
• [9] Hughes, G. E., and M. J. Cresswell, A Companion to Modal Logic, Methuen and Co. Ltd., London and New York, 1984.
• [10] Kripke, Saul A., The semantical analysis of modal logic ii. Nonnormal modal propositional calculi,'' pp. 206--220 in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, edited by J. Addison, L. Henkin, and A. Tarski, North-Holland Publishing Co., Amsterdam, 1965.
• [11] Lemmon, E. J., New foundations for Lewis modal systems,'' The Journal of Symbolic Logic, vol. 22 (1957), pp. 176--186.
• [12] Lemmon, E. J., in collaboration with Dana Scott, The Lemmon Notes'': An Introduction to Modal Logic, edited by Krister Segerberg, no. 11 in the American Philosophical Quarterly Monograph Series, edited by Nicholas Rescher, Basil Blackwell, Oxford, 1977.
• [13] Lewis, C. I., A Survey of Symbolic Logic, University of California Press, Berkeley, 1918.
• [14] Lewis, C. I., and C. H. Langford, Symbolic Logic, New York, 1932; second edition, Dover Publications, 1959.
• [15] Lewis, David K., Intensional logics without iterative axioms,'' Journal of Philosophical Logic, vol. 3 (1974), pp. 457--466.
• [16] McKinsey, J. C. C., A reduction in number of the postulates for C. I. Lewis' system of strict implication,'' Bulletin of the American Mathematical Society, vol. 40 (1934), pp. 425--427.
• [17] Schotch, Peter K., Remarks on the semantics of non-normal logics,'' Topoi, vol. 3 (1984), pp. 85--90.
• [18] Segerberg, Krister, An Essay in Classical Modal Logic, The Philosophical Society, Uppsala, 1971.
• [19] Shukla, Anjan, Decision procedures for Lewis system S1 and related modal systems,'' Notre Dame Journal of Formal Logic, vol. 11 (1970), pp. 141--180.
• [20] Surendonk, Timothy J., Canonicity for intensional logics without iterative axioms,'' Technical Report TR-ARP-4-95, Automated Reasoning Project (Research School of Information Sciences and Engineering, and Centre for Information Science Research), Australian National University, November 27, 1995.
• [21] Sylvan, Richard, Relational semantics for all Lewis, Lemmon and Feys modal logics, most notably for systems between S0.3$^\circ$ and S1,'' The Journal of Non-classical Logic, vol. 6 (1989), pp. 19--40.