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Published online by Cambridge University Press: 14 March 2022
We now proceed to define certain terms which we shall apply to sets of axioms and derive a few properties. A great deal of what follows in this section is still based on a two-valued logic, while our criticism of the usual independence and consistency proofs is based on an n-valued logic. The necessary alterations are now being worked out by the author.
1 This term was first applied by H. M. Sheffer to two postulates, neither of which implies any part of the other. A set of postulates mutually prime by pairs he calls a set of mutually prime postulates. Cf. Abstract in Bull. Am. Math. Soc., v. 22, 1916, p. 287. We here extend the idea to two sets of postulates. All our results were reached independently, since the abstracts of Prof. Sheffer's paper merely gave the definitions.
2 Sheffer defines an independent set as a set of postulates such that if m is the number of postulates no m-l of them imply the mth. Then he says that a mutually prime set of postulates imply independence. Bull. Am. Math. Soc., 22, 1916, p. 287. We do not develop the properties of primeness. Sheffer's definition is identical with ours.
3 For other properties of postulate sets and the definition of further concepts cf. “Axiomensysteme für beliebige Satzsysteme,” Teil 1, “Sätze erster grade,” Teil II “Sätze höheren grades.” Math. Ann. v. 87, 1922, p. 246 ff. and v. 89, 1923, p. 76 ff. This is so far as I know the only other work on this topic.
4 Cf. Weyl “Handbuch der Phil. der Math,” p. 18.
5 We mean by concept here the subjectival concept as opposed to the adjectival and relational concept.
6 Cf. Lewis and Langford Symbolic Logic, p. 337.
7 Cf. A. Church, On Irredundant Sets of Postulates. Trans. Am. Math. Soc., 1925, p. 318 ff.
8 In an article in the Monist (v. 29, p. 152 ff.) Lenzen shows that the only set of postulates which are independent is one of only one postulate. The proof is based on the assumption that q ·→· p→q (2.02 of Principia). He reduces Huntington's sets of independent postulates to non-independence as follows for A2 and M5
Proof
This can obviously be done for all postulates in any set.
9 Cf. E. V. Huntington's discussion in Postulates for Betweenness, Trans. Am. Math. Soc., 26, 1924, p. 277.
10 V. 27, 1925, p. 318 ff.
11 Cf. H. B. Smith Abstract Logic or the Science of Modality, Jan. 1934 (privately distributed manuscript).
12 Cf. Verhandlungen des Intern. Math. Kongresses Zurich, 1932, p. 342–343.
13 Another good illustration is in Veblen and Young's “Projective Geometry,” p. 2–3. We do not reproduce it here.
14 Carnap in the Abriss der Logistik sets up two columns, one for concepts and one for propositions (p. 70). We show this to be unnecessary.
15 Cf. Einleit in die Mengenlehre, p. 365.
16 A great deal of what follows is due to Prof. H. B. Smith. It is advisable to consult his works before reading what follows. Cf. especially “Abstract Logic.”
