§ 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

while the subordinate implications are material. ↩`p`’s truth implies `q`’s truth,

§ 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

to `p` is a proposition, then

implies etc.`q` is a propositionIf

↩`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

and `q` implies `r`

are incompatible. But in virtue of our definition of negation, if `q` implies not-`r``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. ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.