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Alternative postulate sets for Lewis's S5

Published online by Cambridge University Press:  12 March 2014

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

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.

The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:

Df. M: M = NLN;

and two special rules:

L1: ⊦Cαβ → ⊦CLαβ;

L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).

Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 4 , December 1956 , pp. 347 - 349
Copyright
Copyright © Association for Symbolic Logic 1956

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References

1 I use ‘⊦α’ in what follows as an abbreviation for ‘α is the form of a thesis of the system in question.’

2 Prior's formulation of L2 (p. 306) is slightly wrong, as he has pointed out to me. It allows CpLp to be proved as a thesis. The above is his own correction.

3 Prior actually shows that P contains von Wright's system M″; von Wright showed M″ to be equivalent to S5 in An essay in modal logic (1951), Appendix II.

4 Cf. Lewis and Langford, , Symbolic logic (1932), p. 140Google Scholar.

5 C'CpqCLpq is provable in S1 from Lewis and Langford's (op. cit.) 16.15, substituting q/Mp, r/q, detaching C'ApqAMpq, and, after substituting p/Np in this, working the result into the required thesis by substitution of equivalents.

6 Cf. Becker, O., Zur Logik der Modalitäten, I para. 2, Jahrbuch für Philosophie und phänomenologische Forschung, Vol. 11, 1930Google Scholar. S5 has six modalities including γ and Nγ, but only four proper modalities.

7 Cf. Lewis and Langford, op. cit., p. 501.

8 Of these six theses, (i), (ii), (iv) and (v) are Lewis and Langford's 12.11, 19.81, C 10 (p. 497) and 18.42 respectively; (iii) is a simple consequence of the definition of ‘L’, and (vi) follows from 18.53 by 14.26 and 18.7.

9 This is merely a special case of the rule ha ⊦α → ⊦Lα, shown to be a derived rule of S4 and so of S5 by McKinsey, and Tarski, , Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13, 1948 (cf. Theorem 2.1, p. 5)Google Scholar.