The Principles of Mathematics (1903)

Appendix B. The Doctrine of Types

Table of Contents

  1. § 497. Statement of the doctrine
  2. § 498. Numbers and propositions as types
  3. § 499. Are propositional concepts individuals?
  4. § 500. Contradiction arising from the question whether there are more classes of propositions than propositions

§ 500 n. 1. It might be doubted whether the relation of ranges of propositions to their logical products is one-one or many-one. For example, does the logical product of p and q and r differ from that of pq and r? A reference to the definition of logical product (§ 25) will set this doubt at rest; for the two logical products in question, though equivalent, are by no means identical. Consequently there is a one-one relation of all ranges of propositions to some propositions, which is directly contradictory to Cantor's theorem.