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The decidability of normal K5 logics1

Published online by Cambridge University Press:  12 March 2014

Michael C. Nagle
Michael C. Nagle*
Affiliation:
Department of Philosophy, University of Calgary, Calgary T2N 1N4, Canada
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Extract

The literature on modal logic includes a number of general completeness and decidability results. The work of Schiller Joe Scroggs [5], R.A. Bull [1], Kit Fine [2], and Krister Segerberg [6] provide examples.

Scroggs showed that the proper extensions of S5 have the finite model property and are axiomatizable. (Harrop [3] then argued that logics having these properties are decidable.) Bull extended Scroggs' result by showing that the normal extensions of S4.3 have the finite model property. Fine subsequently provided a model-theoretic proof of Bull's result and also proved the axiomatizability of these logics. In a different direction Segerberg proved that every normal logic containing the characteristic axioms of Lewis' systems S4 and S5 is decidable.

The present paper is in this tradition. We extend the results of Scroggs and Segerberg by showing that every normal modal logic containing the S5 axiom has the finite model property, is axiomatizable, and thus is decidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 319 - 328
Copyright
Copyright © Association for Symbolic Logic 1981

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Footnotes

1

At the meeting of the Society for Exact Philosophy at the University of Pittsburgh, June 5, 1978, Brian Chellas conjectured that every normal modal logic extending K5 has the finite model property. His conjecture led to our interest in the subject. I would like to thank Professors Brian Chellas, Verena H. Dyson, John Heintz, Craig and the referee, for their comments on earlier versions of this paper. Special thanks are due Professor Chellas, whose encouragement and criticism have been invaluable.

References

REFERENCES

[1]Bull, R.A., That all normal extensions of S4.3 have the finite model property, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 341–344.CrossRefGoogle Scholar
[2]Fine, K., Logics containing S4.3, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 371–376.CrossRefGoogle Scholar
[3]Harrop, R., On the existence of finite models and decision procedures for propositional calculi, Proceedings of the Cambridge Philosophical Society, vol. 54 (1958), pp. 1–13.CrossRefGoogle Scholar
[4]Lemmon, E.J. and Scott, D., An introduction to modal logic (the “Lemmon notes”) (Segerberg, K., Editor), Basil Blackwell, Oxford, 1977.Google Scholar
[5]Scroggs, S.J., Extensions of the Lewis system S5, this Journal, vol. 16 (1951), pp. 112–120.Google Scholar
[6]Segerberg, K., An essay in classical modal logic, Uppsala University, Uppsala, 1971.Google Scholar
[7]Segerberg, K., Franzén's proof of Bull's theorem, Ajatus, vol. 35 (1973), pp. 216–221.Google Scholar