A difficult point arises as to the variation of the concept in a proposition. Consider, for example, all propositions of the type `a``R``b`, 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

is legitimate, and that there is a corresponding class; but this requires the admission of such propositional functions as `a` and `b``a``R``b`, 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

In this proposition the implication involved is material, not formal. If the implication were formal, the proposition would not be a function of `R` is a relation implies `a``R``b`.`R`, but would be equivalent to the (necessarily false) proposition: All relations hold between

Generally we have some such proposition as `a` and `b`.

and we wish to turn this into a formal implication. If `a``R``b` implies `ϕ`(`R`) provided `R` is a relation,`ϕ`(`R`) is a proposition for all values of `R`, our object is effected by substituting If

Here

implies `R` is a relation

then `a``R``b`,`ϕ`(`R`).`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

for all values of `R` is a relation`R`, then our formal implication can be put in the form If

where both the subordinate implications are material. As regards the material implication

implies `R` is a relation`a``R``b`, then, for all values of `R`, `ψ`(`R`) implies `ϕ`(`R`),

this is always a proposition, whereas

implies `R` is a relation`a``R``b`,`a``R``b` 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

--an implication which, though it contains a variable, is not formal, but material, being satisfied by some only of the possible values of `R` is a relation implies `a``R``b``R`. The variable `R` in it is, in Peano's language, real and not apparent.(§ 83 ¶ 1)

The general principle involved is: If `ϕ``x` is only a proposition for some values of `x`, then

is a proposition for

implies `ϕ``x` implies `ϕ``x``ϕ``x`*all* values of `x`, and is true when and only when `ϕ``x` is true. (The implications involved are both material.) In some cases,

will be equivalent to some simpler propositional function `ϕ``x` implies `ϕ``x``ψ``x` (such as

in the above instance), which may then be substituted for it`R` is a relation^{[62]}.(§ 83 ¶ 2)

Such a propositional function as

appears even less capable than previous instances of analysis into `R` is a relation implies `a``R``b``R` and an assertion about `R`, since we should have to assign a meaning to

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 `a`...`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)`R` is a relation holding from `a` to `b`

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)

§ 83 n. 1. It is necessary to assign some meaning (other than a proposition) to `a``R``b` when `R` is not a relation. ↩

§ 83 n. 2. A propositional function, though for every value of the variable it is true or false, is not itself true or false, being what is denoted by any proposition of the type in question,

which is not itself a proposition. ↩

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