The Principles of Mathematics (1903)

§ 134

Wherever we have a one-sided formal implication, it may be urged, if the two propositional functions involved are obtainable one from the other by the variation of a single constituent, then what is implied is simpler than what implies it. Thus Socrates is a man implies Socrates is a mortal, but the latter proposition does not imply the former: also the latter proposition is simpler than the former, since man is a concept of which mortal forms part. Again, if we take a proposition asserting a relation of two entities A and B, this proposition implies the being of A and the being of B, and the being of the relation, none of which implies the proposition, and each of which is simpler than the proposition. There will only be equal complexity, according to the theory that intension and extension vary inversely as one another—in cases of mutual implication, such as A is greater than B and B is less than A. Thus we might be tempted to set up the following definition: A is said to be part of B when B is implies A is, but A is does not imply B is. If this definition could be maintaind, whole and part would not be a new indefinable, but would be derivative from logical priority. There are, however, reasons why such an opinion is untenable.(§ 134 ¶ 1)

The first objection is, that logical priority is not a simple relation: implication is simple, but logical priority of A to B requires not only B implies A, but also A does not imply B. (For convenience, I shall say that A implies B when A is implies B is.) This state of things, it is true, is realized when A is part of B; but it seems necessary to regard the relation of whole to part as something simple, which must be different from any possible relation of one whole to another which is not part of it. This would not result from the above definition. For example, A is greater and better than B implies B is less than A, but the converse implication does not hold: yet the latter proposition is not a part of the former[92].(§ 134 ¶ 2)

Another objection is derived from such cases as redness and colour. These two concepts appear to be equally simple: there is no specification, other and simplre than redness itself, which can be added to colour to produce redness, in the way in which specifications will turn mortal into man. Hence A is red is no more complex than A is coloured, although there is here a one-sided implication. Redness, in fact, appears to be (when taken to mean one particular shade) a simple concept, which, although it implies colour, does not contain colour as a constituent. The inverse relation of extension and intension, therefore, does not hold in all cases. For these reasons, we must reject, in spite of their very close connection, the attempt to define whole and part by means of implication.(§ 134 ¶ 3)