The Principles of Mathematics (1903)

§ 16

A definition of implication is quite impossible. If p implies q, then if p is true q is true, i.e. p’s truth implies q’s truth; also if q is false p is false, i.e. q’s falsehood implies p’s falsehood[12]. Thus truth and falsehood give us merely new implications, not a definition of implication. If p implies q, then both are false or both true, or p is false and q true; it is impossible to have q false and p true, and it is necessary to have q true or p false[13]. In fact, the assertion that q is true or p false turns out to be strictly equivalent to p implies q; but as equivalence means mutual implication, this still leaves implication fundamental, and not definable in terms of disjunction. Disjunction, on the other hand, is definable in terms of implication, as we shall shortly see. It follows from the above equivalence that of any two propositions there must be one which implies the other, that false propositions imply all propositions, and true propositions are implied by all propositions. But these are results to be demonstrated; the premisses of our subject deal exclusively with rules of inference.(§ 16 ¶ 1)

It may be observed that, although implication is indefinable, proposition can be defined. Every proposition implies itself, and whatever is not a proposition implies nothing. Hence to say p is a proposition is equivalent to saying p implies p; and this equivalence may be used to define propositions. As the mathematical sense of definition is widely different from that current among philosophers, it may be well to observe that, in the mathematical sense, a new propositional function is said to be defined when it is stated to be equivalent to (i.e. to imply and be implied by) a propositional function which has either been accepted as indefinable or has been defined in terms of indefinables. The definition of entities which are not propositional functions is derived from such as are in ways which will be explained in connection with classes and functions.(§ 16 ¶ 2)

§ 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.