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On the number of complete extensions of the Lewis systems of sentential calculus

Published online by Cambridge University Press:  12 March 2014

J. C. C. McKinsey
J. C. C. McKinsey*
Affiliation:
Montana State College
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Extract

We say that a system S of sentential calculus is an extension of a system R, if R and S have the same class of (meaningful) sentences, and every provable sentence of R is also provable in S. If the two classes of provable sentences do not coincide, we call S a proper extension of R.

By a complete system of sentential calculus is meant one which is itself consistent, but has no consistent proper extensions. Thus one cannot add a new independent primitive sentence to a complete system without obtaining an inconsistent system. The usual two-valued sentential calculus is complete in the sense defined.

It has been shown by Lindenbaum that every incomplete system of sentential calculus possesses at least one complete extension. In this paper we shall examine how many complete extensions there are of some of the Lewis systems of sentential calculus. We shall show that there is only one complete extension of S4 (and hence also of S5, which is an extension of S4), and that there are infinitely many complete extensions of S2 (and hence also of S1, since S2 is an extension of S1). We leave open the question how many complete extensions there are of S3.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 9 , Issue 2 , June 1944 , pp. 42 - 45
Copyright
Copyright © Association for Symbolic Logic 1944

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References

1 See Tarski, A., Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I, Monatshefte für Mathematik und Physik , vol. 37 (1930), p. 394 f.Google Scholar, Satz I. 56.

2 For an account of these systems, see Lewis and Langford, Symbolic logic, pp. 122–178 and pp. 492–502.

3 It will be noticed that this result contradicts Ward's, Morgan paper, A determination of all possible systems of strict implication, American journal of mathematics, vol. 57 (1935), pp. 261–266Google Scholar, in which it is erroneously stated that there are just four complete extensions of S2.

4 See Tarski, A., Über die Erweiterungen den unvollständigen Systeme des Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums, Heft 7, pp. 53–54Google Scholar, Satz 1.

5 We use the sign “▀” to indicate the operation of strict equivalence, so as to avoid confusion with the sign “▄”, which is used for identity.

6 That (1) through (6) are provable in S4 follows directly from Lewis' work (loc. cit.), since they are already provable in S2. The fact that (6) and (7) are provable in S4 can be concluded from the author's paper, A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.