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Results concerning the decision problem of Lewis's calculi S3 and S6

Published online by Cambridge University Press:  12 March 2014

Sören Halldén
Sören Halldén*
Affiliation:
Stockholm University
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Extract

The calculus generated by the addition of the postulate

to S2 will, after Miss Alban, be called “S6”, the calculus generated by the addition of the same postulate to S3, “S7”, and the calculus generated by the addition of the postulate

to S3, “S8”. No interesting interpretation of the calculi S6–8 is known, but they are of some indirect interest because of the connections between them and the five calculi S1–S5. Certain questions concerning them will be treated in the following. In §1 it will be shown that every method of decision for S7 can be turned into a method of decision for S3, and in §2 that the number of complete extensions of S3 is equal to the number of complete extensions of S7 plus one. In §3 it will be shown that McKinsey's method of decision for S2 and S4 can be modified so as to cover S6.

The letters “P”, “Q”, “R”, and “S” will be employed as syntactic variables denoting formulas. Logical expressions will sometimes be used as selfdenotative. “C” and “C” stand for arbitrary calculi. “C + P” is the calculus which is the result of adding to C as new postulates all formulas which can be derived from P by substitution. With “substitution” I mean the operation performed on propositional variables.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 14 , Issue 4 , January 1950 , pp. 230 - 236
Copyright
Copyright © Association for Symbolic Logic 1950

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References

1 It might be mentioned that the three calculi S6–8 are consistent. This can be shown by group I of Lewis, and Langford's, Symbolic logic, pp. 493, 498Google Scholar. For definition of the calculi S1–S5, see the same work, pp. 500–501.

2 Parry, W. T., Zum Lewisschen Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, no. 4 (1931–1932), pp. 15–16Google Scholar.

3 McKinsey, J. C. C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134Google Scholar.

4 Methods have been discovered by Wajsberg, Parry and Carnap. See Wajsberg, M., Ein erweiterter Klassenkalkül, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 113–126CrossRefGoogle Scholar, Parry, op.cit., Carnap, R., Modalities and quantification, this Journal, vol. 11 (1946), pp. 41, 43–46Google Scholar.

It may be remarked that every formula of S5 is reducible to a modal function of the first degree. Therefore, every method of decision for S2–S5, or any intermediate calculus, is a method of decision for S5.

5 Parry, W. T., Modalities in the “Survey” system of strict implication, this Journal, vol. 4 (1939), p. 141Google Scholar.

6 McKinsey, J. C. C. and Tarski, A., Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), pp. 2–3Google Scholar.

My original proof contained a certain deficiency. I am indebted to the referee for pointing out to me that by utilizing the circumstance mentioned above this deficiency could be remedied. I am also indebted to the referee for several suggestions concerning the formulation of Theorems 1 and 2 and concerning the formulation of the proofs of these.

7 Symbolic logic, p. 493.

8 This circumstance, together with the proof for it, was pointed out to me by the referee.

9 McKinsey, J. C. C., On the number of complete extensions of the Lewis systems of sentential calculus, this Journal, vol. 9 (1944), pp. 42–46Google Scholar.

10 See McKinsey, , On the number of complete extensions …, p. 42Google Scholar.

11 Cf. MK, pp. 121, 126.