A difficult point arises as to the variation of the concept in a proposition. Consider, for example, all propositions of the type aRb, where a and b are fixed terms, and R is a variable relation. There seems no reason to doubt that the class-concept relation between a and b is legitimate, and that there is a corresponding class; but this requires the admission of such propositional functions as aRb, which, moreover, are frequently required in actual Mathematics, as, for example, in counting the number of many-one relations whose referents and relata are given classes. But if our variable is to have, as we normally require, an unrestricted field, it is necessary to substitute the propositional function R is a relation implies aRb. In this proposition the implication involved is material, not formal. If the implication were formal, the proposition would not be a function of R, but would be equivalent to the (necessarily false) proposition: All relations hold between a and b. Generally we have some such proposition as aRb implies ϕ(R) provided R is a relation, and we wish to turn this into a formal implication. If ϕ(R) is a proposition for all values of R, our object is effected by substituting If R is a relation implies aRb, then ϕ(R). Here R can take all values^[61], and the if and then is a formal implication, while the implies is a material implication. If ϕ(R) is not a propositional function, but is a proposition only when R satisfies ψ(R), where ψ(R) is a propositional function implied by R is a relation for all values of R, then our formal implication can be put in the form If R is a relation implies aRb, then, for all values of R, ψ(R) implies ϕ(R), where both the subordinate implications are material. As regards the material implication R is a relation implies aRb, this is always a proposition, whereas aRb is only a proposition when R is a relation. The new propositional function will only be true when R is a relation which does hold between a and b: when R is not a relation, the antecedent is false and the consequent is not a proposition, so that the implication is false; when R is a relation which does not hold between a and b, the antecedent is true and the consequent false, so that again the implication is false; only when both are true is the implication true. Thus in defining the class of relations holding between a and b, the antecedent is true and the consequent false, so that again the implication is false; only when both are true is the implication true. Thus in defining the class of relations holding between a and b, the formally correct course is to define them as the values satisfying R is a relation implies aRb--an implication which, though it contains a variable, is not formal, but material, being satisfied by some only of the possible values of R. The variable R in it is, in Peano's language, real and not apparent.(§ 83 ¶ 1)

The general principle involved is: If ϕx is only a proposition for some values of x, then ϕx implies ϕx implies ϕx is a proposition for all values of x, and is true when and only when ϕx is true. (The implications involved are both material.) In some cases, ϕx implies ϕx will be equivalent to some simpler propositional function ψx (such as R is a relation in the above instance), which may then be substituted for it^[62].(§ 83 ¶ 2)

Such a propositional function as R is a relation implies aRb appears even less capable than previous instances of analysis into R and an assertion about R, since we should have to assign a meaning to a...b, where the blank space may be filled by anything, not necessarily by a relation. There is here, however, a suggestion of an entity which has not yet been considered, namely the couple with sense. It may be doubted whether there is any such entity, and yet such phrases as R is a relation holding from a to b seem to show that its rejection would lead to paradoxes. This point, however, belongs to the theory of relations, and will be resumed in Chapter IX (§ 98).(§ 83 ¶ 3)

From what has been said, it appears that the propositional functions must be accepted as ultimate data. It follows that formal implications and the inclusion of classes cannot be generally explained by means of a relation between assertions, although, where a propositional function asserts a fixed relation to a fixed term, the analysis into subject and assertion is legitimate and not unimportant.(§ 83 ¶ 4)