The Principles of Mathematics (1903)

§ 139

It is to be observed that what we called classes as one may always, except where they contain one term or none, or are defined by quadratic propositional functions, be interpreted as aggregates. The logical product of two classes as one will be the common part (in the second of our three senses) of the two aggregates, and their sum will be the aggregate which is identical with or part of (again in the second sense) any aggregate of which the two given aggregates are parts, but is neither identical with nor part of any other aggregate[97]. The relation of whole and part, in the second of our three senses, is transitive and asymmetrical, but is distinguished from other such relations by the fact of allowing logical addition and multiplication. It is this peculiarity which forms the basis of the Logical Calculus as developed by writers previous to Peano and Frege (including Schröder)[98]. But wherever infinite wholes are concerned it is necessary, and in many other cases it is practically unavoidable, to begin with a class-concept or predicate or propositional function, and obtain the aggregate from this. Thus the theory of whole and part is less fundamental logically than that of predicates or class-concepts or propositional functions; and it is for this reason that the consideration of it has been postponed to so late a stage.(§ 139 ¶ 1)

§ 139 n. 1. Cf. Peano, F. 1901, § 2, Prop. 1 ·0 (p. 19).

§ 139 n. 2. See e.g. his Algebra der Logik, Vol. I (Leipzig, 1890).