I come now to formal implication, which is a far more difficult notion than material implication. In order to avoid the general notion of propositional function, let us begin by the discussion of a particular instance, say

This proposition is equivalent to `x` is a man implies `x` is a mortal for all values of `x`.all men are mortal

every man is mortal

and any man is mortal.

But it seems highly doubtful whether it is the same propositionl. It is also connected with a purely intensional proposition in which *man* is asserted to be a complex notion of which *mortal* is a constituent, but this proposition is quite distinct from the one we are discussing. Indeed, such intensional propositions are not always present where one class is included in another: in general, either class may be defined by various different predicates, and it is by no means necessary that every predicate of the smaller class should contain every predicate of the larger class as a factor. Indeed, it may very well happen that both predicates are philosophically simple: thus *colour* and *existent* apear to be both simple, yet the class of colours is part of the class of existents. The intensional view, derived from predicates, is in the main irrelevant to Symbolic Logic and to Mathematics, and I shall not consider it further at present.(§ 40 ¶ 1)

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