§ 11 n. 1. By far the most complete account of the non-Peanesque methods will be found in the three volumes of Schröder, Vorlesungen über die Algebra der Logik, Leipzig, 1890, 1891, 1895. ↩

§ 11 n. 2. See Formulaire de Mathématique, Turin, 1895, with subsequent editions in later years; also Revue de Mathématiques, Vol. VII, No. 1 (1900). The editions of the Formulaire will be quoted as F. 1895 and so on. The Revue de Mathématiques, which was originally the Revisti di Matematica, will be referred to as R. d. M.. ↩

§ 11 n. 3. In what follows the main outlines are due to Professor Peano, except as regards relations; even in those cases where I depart from his views, the problems considered have been suggested to me by his works. ↩

§ 12 n. 1. I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses. ↩

§ 12 n. 2. See below, Part V, Chap. XXXVI. ↩

§ 13 n. 1. On the points where the duality breaks down, cf. Schröder, op. cit., Vol. II, Lecture 21. ↩

§ 13 n. 2. Cf. The Calculus of Equivalent Statements, Proceedings of the London Mathematical Society, Vol. IX and subsequent volumes; Symbolic Reasoning, Mind, Jan. 1880, Oct. 1897, and Jan. 1900; La Logique Symbolique et ses Applications, Bibliothèque du Congrès International de Philosophie, Vol. III (Paris, 1901). I shall in future quote the proceedings of the above Congress by the title Congrès. ↩

§ 13 n. 3. F. 1901, p. 2. ↩

§ 16 n. 1. The reader is recommended to observe that the main implications in these statements are formal, i.e. p implies q formally implies p’s truth implies q’s truth, while the subordinate implications are material. ↩

§ 16 n. 2. I may as well state once for all that the alternatives of a disjunction will never be considered as mutually exclusive unless expressly said to be so. ↩

§ 18 n. 1. Note that the implications denoted by if and then, in these axioms, are formal, while those denoted by implies are material. ↩

§ 18 n. 2. (7) and (8) cannot (I think) be deduced from the definition of the logical product, because they are required for passing from If p is a proposition, then q is a proposition implies etc. to If p and q are propositions, then etc. ↩

§ 19 n. 1. The principle that false propositions imply all propositions solves Lewis Carroll’s logical paradox in Mind, N. S. No. 11 (1894). The assertion made in that paradox is that, if p, q, r be propositions, and q implies r, while p implies that q implies not-r, then p must be false, on the supposed ground that q implies r and q implies not-r are incompatible. But in virtue of our definition of negation, if q be false both these implications will hold: the two together, in fact, whatever proposition r may be, are equivalent to not-q. Thus the only inference warranted by Lewis Carroll’s premisses is that if p be true, q must be false, i.e. that p implies not-q; and this is the conclusion, oddly enough, which common sense would have drawn in the particular case which he discusses. ↩

§ 21 n. 1. See his Begriffsschrift, Halle, 1879, and Grundgesetze der Arithmetik, Jena, 1893, p. 2. ↩

§ 22 n. 1. Verbs and adjectives occurring as such are distinguished by the fact that, if they be taken as variable, the resulting function is only a proposition for some values of the variable, i.e. for such as are verbs or adjectives respectively. See Chap. IV. ↩

§ 25 n. 1. Schröder, Algebra der Logik, Vol. II, pp. 258-9; McColl, Calculus of Equivalent Statements, fifth paper, Proc. Lond. Math. Soc. Vol. XXVIII, p. 182. ↩

§ 27 n. 1. Camb. Phil. Trans. Vol. X, On the Syllogism, No. IV, and on the Logic of Relations. Cf. ib. Vol. IX, p. 104; also his Formal Logic (London, 1847), p. 50. ↩

§ 27 n. 2. See especially his articles on the Algebra of Logic, American Journal of Mathematics, Vols. III and VII. The subject is treated at length by C. S. Peirce’s methods in Schröder, op. cit., Vol. III. ↩

§ 27 n. 3. See his article On the Nature of Judgment, Mind, N. S. No. 30. ↩

§ 27 n. 4. See my articles in R. d. M. Vol. VII, No. 2 and subsequent numbers. ↩

§ 30 n. 1. There is a difficulty in regard to this primitive proposition, discussed in §§ 53, 94 below. ↩

§ 31 n. 1. E.g. F. 1901, p. 6; F. 1897, Part I, pp. 62–63. ↩

§ 31 n. 2. See Chap. IV. ↩

§ 31 n. 3. See Chap. V. ↩

§ 32 n. 1. F. 1901 and R. d. M. Vol. VII, No. 1 (1900). ↩

§ 32 n. 2. F. 1897, Part I. ↩

§ 33 n. 1. In consequence of the criticisms of Padoa, R. d. M. Vol. VI, p. 112. ↩

§ 33 n. 2. R. d. M. Vol. VII, No. 1, p. 23; F. 1901, p. 21, § 2, Pro. 4. 0, Note. ↩

§ 34 n. 1. See e.g. F. 1901, Part I, § 1, Prop. 3. 3 (p. 10). ↩

§ 36 n. 1. E.g. F. 1901, Part I, § 10, Props. 1. 0. 01 (p. 33). ↩

§ 36 n. 2. See my article Sur la logique des relations, R. d. M. Vol. VII, 2 (1901). ↩