The Principles of Mathematics (1903)

§ 19

Disjunction or logical addition is defined as follows: p or q is equivalent to p implies q implies q. It is easy to persuade ourselves of this equivalence, by remembering that a false proposition implies every other; for if p is false, p does imply q, and therefore, if p implies q implies q, it follows that q is true. But this argument again uses principles which have not yet been demonstrated, and is merely designed to elucidate the definition by anticipation. From this definition, by the help of reduction, we can prove that p or q is equivalent to q or q. An alternative definition, deducible from the above, is: Any proposition implied by p and implied by q is true, or, in other words, p implies s and q implies s together imply s, whatever s may be. Hence we proceed to the definition of negation: not-p is equivalent to the assertion that p implies all propositions, i.e. that r implies r implies p implies q whatever r may be[16]. From this point we can prove the laws of contradiction and excluded middle, double negation, and establish all the formal properties of logical multiplication and addition—the associative, commutative and distributive laws. Thus the logic of propositions is now complete.(§ 19 ¶ 1)

Philosophers will object to the above definitions of disjunction and negation on the ground that what we mean by these notions is something quite distinct from what the definitions assign as their meanings, and that the equivalences stated in the definitions are, as a matter of fact, significant propositions, not mere indications as to the way in which symbols are going to be used. Such an objection is, I think, well-founded, if the above account is advocated as giving the true philosophical analysis of the matter. But where a purely formal purpose is to be served, any equivalence in which a certain notion appears on one side but not on the other will do for a definition. And the advantage of having before our minds a strictly formal development is that it provides the data for philosophical analysis in a more definite shape than would be otherwise possible. Criticism of the procedure of formal logic, therefore, will be best postponed until the present brief account has been brought to an end.(§ 19 ¶ 2)

§ 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.