The Principles of Mathematics (1903)

The Propositional Calculus

  1. § 14. Definition
  2. § 15. Distinction between implication and formal implication.
  3. § 16. Implication indefinable
  4. § 17. Two indefinables and ten primitive propositions in this calculus
  5. § 18. The ten primitive propositions
  6. § 19. Disjunction and negation defined

§ 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.