The Principles of Mathematics (1903)

§ 41

It may be doubted, to begin with, whether x is a man implies x is a mortal is to be regarded as asserted strictly of all possible terms, or only of such terms as are men. Peano, though he is not explicit, appears to hold the latter view. But in this case, the hypothesis ceases to be significant, and becomes a mere definition of x: x is to mean any man. The hypothesis then becomes a mere assertion concerning the meaning of the symbol x, and the whole of what is asserted concerning the matter dealt with by our symbol is put into the conclusion. The premiss says: x is to mean any man. The conclusion says: x is mortal. But the implication is merely concerning the symbolism: since any man is mortal, if x denotes any man, x is mortal. Thus formal implication, on this view, has wholly disappeared, leaving us the proposition any man is mortal as expressing the whole of what is relevant in the proposition with a variable. It would now only remain to examine the proposition any man is mortal, and if possible to explain this proposition without reintroducing the variable and formal implication. It must be confessed that some grave difficulties are avoided by this view. Consider, for example, the simultaneous assertion of all the propositions of some class k: this is not expressed by x is a k implies x for all values of x. For as it stands, this proposition does not express what is meant, since, if x be not a proposition, x is a k cannot imply x; hence the range of variability of x must be confined to propositions unless we prefix (as above, § 39) the hypothesis x implies x. This remark applies generally, throughout the propositional calculus, to all cases where the conclusion is represented by a single letter: unless the letter does actually represent a proposition, the implication asserted will be false, since only propositions can be implied. The point is that, if x be our variable, x itself is a proposition for all values of x which are propositions, but not for other values. This makes it plain what the limitations are to which our variable is subject: it must vary only within the range of values for which the two sides of the principal implication are propositions, in other words, the two sides, when the variable is not replaced by a constant, must be genuine propositional functions. If this restrictionis not observed, fallacies quickly begin to appear. It should be noticed that there may be any number of subordinate implications which do not require that their terms should be propositions: it is only of the principal implication that this is required. Take, for example, the first principle of inference: If p implies q, then p implies q. This holds equally whether p and q be propositions or not; for if either is not a proposition, p implies q becomes false, but does not cease to be a proposition. In fact, in virtue of the definition of a proposition, our principle states that p implies q is a propositional function, i.e. that it is a proposition for all values of p and q. But if we apply the principle of importation to this proposition, so as to obtain p implies q, together with p, implies q, we have a formula which is only true when p and q are propositions: in order to make it true universally, we must preface it by the hypothesis p implies p and q implies q. In this way, in many cases, if not in all, the restriction on the variability of the variable can be removed; thus, in the assertion of the logical product of a class of propositions, the formula if x implies x, then x is a q implies x appears unobjectionable, and allows x to vary without restriction. Here the subordinate implications in the premiss and the conclusion are material: only the principal implication is formal.(§ 41 ¶ 1)

Returning now to x is a man implies x is a mortal, it is plain that no restriction is required in order to insure our having a genuine proposition. And it is plain that, although we might restrict the values of x to men, and although this seems to be done in the proposition all men are mortal, yet there is no reason, so far as the truth of our proposition is concerned, why we should so restrict our x. Whether x be a man or not, x is a man is always, when a constant is substituted for x, a proposition implying, for that value of x, the proposition x is a mortal. And unless we admit the hypothesis equally in the case where it is false, we shall find it impossible to deal satisfactorily with the null-class or with null propositional functions. We must, therefore, allow our x, wherever the truth of our formal implication is thereby unimpaired, to take all values without exception; and where any restriction on variability is required, the implication is not to be regarded as formal until the said restriction has been removed by being prefixed as hypothesis. (If ψx be a proposition whenever x satisfies ϕx, where ϕx is a propositional function, and if ψx, whenever it is a proposition, implies χx, then ψx implies χx is not a formal implication, but ϕx implies that ψx implies χx is a formal implication.)(§ 41 ¶ 2)