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New foundations for Lewis modal systems1

Published online by Cambridge University Press:  12 March 2014

E. J. Lemmon
E. J. Lemmon*
Affiliation:
Magdalen College, Oxford, England
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Extract

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:

We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 22 , Issue 2 , June 1957 , pp. 176 - 186
Copyright
Copyright © Association for Symbolic Logic 1957

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Footnotes

1

A version of this paper was read to a colloquium of logicians held in Oxford, England, in July, 1956. It appeared there that much work on the same lines had been done independently and at an earlier date by Dr T. J. Smiley, of Clare College, Cambridge.

References

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