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On McKinsey's syntactical characterizations of systems of modal logic

Published online by Cambridge University Press:  12 March 2014

F. R. Drake
F. R. Drake*
Affiliation:
University of Cambridge
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Extract

In [1] McKinsey gives a method of defining possibility in terms of “substitutions”, or transformations of one sentence into another. Roughly, the motivation is that a sentence be possible if a sentence of “like form”, i.e. a transformation of it, is true.

The purpose of this paper is to show that the two systems given in [1], shown by McKinsey to be at least as strong as Lewis' systems S4 and S5, do exactly characterize S4 and S5, and that the system given by omitting rule A4 from the first system exactly characterizes von Wright's system M.1

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 4 , December 1962 , pp. 400 - 406
Copyright
Copyright © Association for Symbolic Logic 1962

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References

[1] McKinsey, J. C. C., On the syntactical construction of systems of modal logic. Journal of Symbolic Logic, Vol. 10 (1945), pp. 83–94.Google Scholar
[2] Joe Scroggs, Schiller, Extensions of Lewis' system S5. Journal of Symbolic Logic, Vol. 16 (1951), pp. 112–120.Google Scholar
[3] McKinsey, J. C. C., A solution of the decision problem for S2 and S4, with an application to topology. Journal of Symbolic Logic, Vol. 6 (1941), pp. 117–134.CrossRefGoogle Scholar
[4] Bergmann, G., The finite representations of S5. Methodos, Vol. 1 (1949), pp. 217–219.Google Scholar
[5] Smiley, T. J., Natural systems of logic. Ph. D. Thesis, University of Cambridge (1956).Google Scholar
[6] Anderson, A. R., Improved decision procedures for Lewis' S4 and von Wright's M. Journal of Symbolic Logic, Vol. 19 (1954), pp. 201–214, and Correction, ibid. Vol. 20 (1955), p. 150.Google Scholar