The Principles of Mathematics (1903)

§ 42

It is to be observed that x is a man implies x is a mortal is not a relation of two propositional functions, but is itself a single propositional function having the elegant property of being always true. For x is a man is, as it stands, not a proposition at all, and does not imply anything; and we must not first vary our x in x is a man, and then independently vary it in x is a mortal, for this would lead to the proposition that everything is a man implies everything is a mortal, which, though true, is not what was meant. This proposition would have to be expressed, if the language of variables were retained, by two variables, as x is a man implies y is a mortal. But this formula too is unsatisfactory, for its natural meaning would be: If anything is a man, then everything is a mortal. The point to be emphasized is, of course, that our x, though variable, must be the same on both sides of the implication, and this requires that we should not obtain our formal implication by first varying (say) socrates in Socrates is a man, and then in Socrates is a mortal, but that we should start from the whole proposition Socrates is a man implies Socrates is a mortal, and vary Socrates in this proposition as a whole. Thus our formal implication asserts a class of implications, not a single implication at all. We do not, in a word, have one implication containing a variable, but rather a variable implication. We have a class of implications, no one of which contains a variable, and we assert that every member of this class is true. This is a first step towards the analysis of the mathematical notion of the variable.(§ 42 ¶ 1)

But, it may be asked, how comes it that Socrates may be varied in the proposition Socrates is a man implies Socrates is mortal? In virtue of the fact that true propositions are implied by all others, we have Socrates is a man implies Socrates is a philosopher; but in this proposition, alas, the variability of Socrates is sadly restricted. This seems to show that formal implication involves something over and above the relation of implication, and that some additional relation must hold where a term can be varied. In the case in question, it is natural to say that what is involved is the relation of inclusion between the classes men and mortals--the very relation which was to be defined and explained by our formal implication. But this view is too simple to meet all cases, and is therefore not required in any case. A larger number of cases, though still not all cases, can be dealt with by the notion of what I shall call assertions. This notion must now bebriefly explained, leaving its critical discussion to Chapter VII.(§ 42 ¶ 2)