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Published online by Cambridge University Press: 14 March 2022
Despite the skepticism of many mathematicians and logicians as to the possibility of any test which will show conclusively the consistency or independence of the members of a postulate set, several methods have nevertheless been devised and employed, e.g., the empirical methods of Russell and Huntington, the internal method of Hilbert, and the reflective method of Royce. However, with the possible exception of Hilbert's method (which is still available only in the form of fragments and suggestions), these techniques require us to forsake the purely formal or abstract mode of analysis, and instead to rely in some sense on “concrete representations” or “interpretations” or “modes of action” which are said to “satisfy” the postulates in question. In short, they are all interpretational methods—and here is the crux of the problem. “Is it possible,” queries Weiss, “that the only way we can determine whether a set is consistent is by seeing all the postulates actually exemplified in some object? If so, we must arbitrarily assume that the object is self-consistent, so that the proof of consistency must ultimately rest on a dogma. As independence rests on consistency there are therefore no satisfactory proofs as yet of either independence or consistency.” The question “is thus answered,” according to Young, “only by reference to a concrete representation of the abstract ideas involved, and it is such concrete representations that we wished especially to avoid. At the present time, however, no absolute test for consistency is known.” Moreover, “suppose,” says the late Dr. Eaton, “that the system has no interpretation: how can the consistency (and) independence … of the postulates be shown? There should be some analytic way—purely in the realm of the abstract, without interpretation—of establishing these properties of a set of postulates. This is an important problem that awaits solution.”
1 For an excellent discussion and classification of various tests for consistency, cf. H. M. Sheffer's doctoral dissertation in the Harvard College Library, specifically the chapter on Consistency of Propositions.
2 According to Lewis, Sheffer has, in a privately circulated manuscript. General Theory of Notational Relativity, “offered a general method for testing consistency and independence without reference to any possible application.” Lewis adds that “the method of dealing with usual branches in this way must wait upon his further publication.” C. I. Lewis, Mind and the World Order, New York, 1929, p. 244n.
3 Paul Weiss, “The Nature of Systems,” The Monist, vol. xxxix (1929), p. 468.
4 J. W. Young, Lectures on Fundamental Concepts of Algebra and Geometry, New York, 1930, p. 144.
5 R. M. Eaton, General Logic, New York, 1931, p. 474n.
6 E. V. Huntington, “The Fundamental Laws of Addition and Multiplication in Elementary Algebra,” Annals of Math., 2nd series, vol. viii (1906), 1.
7 Principles of Mathematics, Camb. Univ. Press, 1903, or L. Couturat, Principes des Mathematiques, Paris, 1905.
8 E. V. Huntington, “The Fundamental Propositions of Algebra,” in J. W. A. Young, Monographs on Topics of Modern Mathematics, New York, 1932, pp. 164–172. Cf. also, G. Frege, “Über die Grundlagen der Geometrie” in Jahresbericht der deutschen Mathematiker-Vereinigung, vols. xii (1903), xv (1906), and A. N. Whitehead, The Axioms of Projective Geometry, Cambridge, 1913, chap. 1.
9 H. M. Sheffer, “Review of ‘Principia Mathematica,‘ ” Isis, vol. viii (1926), pp. 226–231.
10 A variable is said to be “free” when it is neither generalized nor specified by the assignment of a value to it.
11 I am indebted to Dr. William T. Parry of Harvard University for certain criticisms and suggestions in connection with this section.
12 “p materially implies q” means “it is not the case that p is true and q false.”
13 “p strictly implies q” means “it is not the case that it is possible that p should be true and q false.”
14 C. I. Lewis and C. H. Langford, Symbolic Logic, New York, 1932, p. 179. In symbols (∃p, q):~(p < q)·~(p < ~q).
15 J. Royce, “Principles of Logic,” in A. Ruge, Encyclopaedia of the Philosophical Sciences, London, 1913, vol. i, pp. 120–125.
16 G. Cardanus referred to it as “the most wonderful thing that has been devised since the creation of the world.” Cf. “The Logica Demonstrativa of Girolamo Saccheri,” by the present author, in Scripta Mathematica, vol. iii (1935), nos. 2 and 3.
17 C. I. Lewis, “The Structure of Logic and its Relation to Other Systems,” The Journal of Philosophy, vol. xviii (1921), no. 19, pp. 507–510.
18 Symbolic Logic, op. cit., pp. 209–211, 284.
19 Ibid., p. 176.
20 Ibid., pp. 24, 211–212.
21 For further discussion with respect to the inadequacy of strict implication as a relation to be used for purposes of inference, the significance of “p ∽ q” in the theory of deduction, and the calculus based upon it, cf. the present author's paper, entitled “Implication and Deducibility,” to be published in The Journal of Symbolic Logic.
22 In correspondence.
