§ 152 n. 1. On the independence of these three properties, see Peano, Revue de Mathématique, VII, p. 22. The reflexive property is not strictly necessary; what is properly necessary and what is alone (at first sight at any rate) true of quantitative equality, is, that there exists at least one pair of terms having the relation in question. It follows then from the other two properties that each of these terms has to itself the relation in question. ↩

§ 153 n. 1. I shall never use the word unequal to mean merely not equal, but always to mean greater or less, i.e. not equal, though of the same kind of quantities. ↩

§ 153 n. 2. E.g. Mr Bradley, What do we mean by the Intensity of Psychical States? Mind, N. S. Vol. IV; see esp. p. 5. ↩

§ 153 ¶ 6. Compare, with the above discussion, Meinong, Ueber die Bedeutung des Weberschen Gesetzes, Hamburg and Leipzig, 1896; especially Chap. I, § 3. ↩

§ 156 n. 1. This does not follow from (c) and (d) alone, since they do not assert that A is ever equal to B. See Peano, loc. cit. ↩

§ 157 n. 1. The proof of these assertions is mathematical, and depends upon the Logic of Relations; it will be found in my article Sur la Logique des Relations, R. d. M. VII, No. 2, § 1, Propos. 6. 1, and 6. 2. ↩

§ 157 n. 2. The principle is proved by showing that, if R be a symmetrical transitive relation, and a a term of the field of R, a has, to the class of terms to which it has the relation R taken as a whole, a many-one relation which, relationally multiplied by its converse, is equal to R. Thus a magnitude may, so far as formal arguments are concerned, be identified with a class of equal quantities. ↩

§ 158 n. 1. It is not necessarily in (5) and (6) to add A, B, C being magnitudes, for the above relations of greater and less are what define magnitudes, and the addition would therefore be tautological. ↩

Ch. 19 n. 1 n. 1. Reine Vernunft, ed. Hartenstein (1867), p. 158. ↩