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

and `A` is greater than `B`

Thus we might be tempted to set up the following definition: `B` is less than `A`.`A` is said to be part of `B` when * B is* implies

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

but also `B` implies `A`,

(For convenience, I shall say that `A` does not imply `B`.`A` implies `B` when * A is* implies

impliesAis greater and better thanB

but the converse implication does not hold: yet the latter proposition is not a part of the formerBis less thanA,

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

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