The Principles of Mathematics (1903)

Chapter V. Denoting

Table of Contents

  1. § 56. Definition of denoting
  2. § 57. Connection with subject-predicate propositions
  3. § 58. Denoting concepts obtained from predicates
  4. § 59. Extensional account of all, every, any, a and some
  5. § 60. Intensional account of the same
  6. § 61. Illustrations
  7. § 62. The difference between all, every, etc. lies in the objects denoted, not in the way of denoting them.
  8. § 63. The notion of the and definition
  9. § 64. The notion of the and identity
  10. § 65. Summary

§ 57 n. 2. There are two allied propositions expressed by the same words, namely Socrates is a-man and Socrates is-a man. The above remarks apply to the former; but in future, unless the contrary is indicated by a hyphen or otherwise, the latter will always be in question. The former expresses the identity of Socrates with an ambiguous individual; the latter expresses a relation of Socrates to the class-concept man.

§ 58 n. 1. I shall use the word object in a wider sense than term, to cover both singular and plural, and also cases of ambiguity, such as a man. The fact that a word can be framed with a wider meaning than term raises grave logical problems. Cf. § 47.

§ 58 n. 2. On the indefinite article, some good remarks are made by Meinong, Abstrahiren und Vergleichen, Zeitschrift für Psychologie und Physiologie der Sinnesorgane, Vol. XXIV, p. 63.

§ 59 n. 1. I intend to distinguish between a and some in a way not warranted by language; the distinction of all and every is also a straining of usage. Both are necessary to void circumlocution.

§ 64 n. 1. On relations of terms to themselves, v. inf. Chap. IX, § 95.

§ 64 n. 2. The word is is terribly ambiguous, and great care is necessary in order not to confound its various meanings. We have (1) the sense in which it asserts Being, as in A is; (2) the sense of identity; (3) the sense of predication, in A is human; (4) the sense of A is a-man (cf. § 57, note), which is very like identity. In addition to these there are less common uses, as to be good is to be happy, where a relation of assertions is meant, that relation, in fact, which, where it exists, gives rise to formal implication. Doubtless there are further meanings which have not occurred to me. On the meanings of is, cf. De Morgan, Formal Logic, pp. 49, 50.