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Postulational Methods. II

Published online by Cambridge University Press:  14 March 2022

Louis Osgood Kattsoff
Louis Osgood Kattsoff*
Affiliation:
University of North Carolina, Chapel Hill, N. C.
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Extract

In this study we shall analyze the postulate set into its various elements. In order to carry out our analysis, it is advisable to have before us a typical set.

We take the set of five postulates for Boolean Algebras given by H. M. Sheffer in Trans. Am. Math Soc. 14, p. 482 ff.

Type
Research Article
Information
Philosophy of Science , Volume 3 , Issue 1 , January 1936 , pp. 67 - 89
Copyright
Copyright © Philosophy of Science Association 1936

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References

1 For this reason Carnap's attempt to make a “Logische Aufbau der Welt” is misleading and doomed to failure. Carnap bases his work on Principia Mathematica.

2 Cf. Fraenkel, p. 347. And also W. Dubislav, “Die Philosophie der Mathematik in der Gegenwart,” Berlin, 1932.

3 E.g. E. L. Post has offered a method for the solution of this problem in the propositional calculus by means of truth-value matrices. American Journal of Math., 1921. Cf. also Behmann in Math. Ann. 1922, Lowenheim in Math. Ann. 1915.

4 “Den Axiomen kommt also keine logische Notwendigkeit zu” says Holder in Die Mathematische Methode, Berlin, 1924, p. 358.

5 Einleitung in die Mengenlehre, p. 247.

6 Les Definitions Mathématiques, Bibliotheque due Congrès International de Philosophie 3, 1901.

7 Erkenntnis 1.

8 Carnap defines a tautology as a proposition which is entailed by every proposition (Philosophic Problems in Philosophy of Science, January, 1934, p. 11) but this is equivalent to the definition given since every proposition implied by any proposition is a true proposition.

9 Proc. London Math. Soc., 1926, p. 17.

10 Nature of Systems, Open Court Publishing Co., Chicago, p. 43.

11 Carnap, “Bericht über Unterzuchungen zur Allgemeinen Axiomatik,” Erkenntnis 1, p. 303.

12 Journal of Philos. Psych. and Sci. Methods 15, 1918. “Doctrinal Functions.”

13 Cf. also H. Behmann, in Math. Ann. 1922, and Bernays and Schönfinkel, Math. Ann. 1928. Post's article is in American Journal of Math. 1921.

14 This idea is due directly to Prof. H. B. Smith who suggested it during the course of a discussion with the author on these problems. This explains the way in which such a remark as Prof. Lewis makes that no two propositions can be consistent and independent at the same time, can be shown to argue against material implication.

15 Due to its technicality, the remainder of this analysis will be published as a Supplement to this Journal.