We may now briefly review the conclusions arrived at in Part I. Pure Mathematics was defined as the class of propositions asserting formal implications and containing no constants except logical constants. And logical constants are: Implication, the relation of a term to a class of which it is a member, the notion of *such that*, the notion of relation, and such further notions as are involved in formal implication, which we found (§ 93) to be the following: propositional function, class^{[70]}, denoting, and *any* or *every term*. This definition brought Mathematics into very close relation to Logic, and made it practically identical with Symbolic Logic. An examination of Symbolic Logic justified the above enumeration of mathematical indefinables. In Chapter III we distinguished implication and formal implication. The former holds between any two propositions provided the first be false or the second true. The latter is not a relation, but the assertion, for every value of the variable or variables, of a propositional function which, for every value of the variable or variables, asserts an implication. Chapter IV distinguished what may be called *things* from predicates and relations (including the *is* of predications among relations for this purpose). It was shown that the distinction is connected with the doctrine of substance and attributes, but does not lead to the traditional results. Chapters V nad VI developed the theory of predicates. In the former of these chapters it was shown that certain concepts, derived from predicates, occur in propositions not *about* themselves, but about combinations of terms, such as are indicated by *all*, *every*, *any*, *a*, *some*, and *the*. Concepts of this kind, we found, are fundamental in Mathematics and enable us to deal with infinite classes by means of propositions of finite complexity. In Chapter VI we distinguished predicates, class-concepts, concepts of classes, classes as many, and classes as one. We agreed that single terms, or such combinations as result from *and*, are classes, the latter being classes as many; and that classes as many are the objects denoted by concepts of classes, which are the plurals of class-concepts. But in the present chapter we decided that it is necessary to distinguish a single term from the class whose only member it is, and that consequently the null-class may be admitted.(§ 106 ¶ 1)

In Chapter VII we resumed the study of the verb. Subject-predicate propositions, and such as express a fixed relation to a fixed term, could be analyzed, we found, into a subject and an assertion; but this analysis becomes impossible when a given term enters into a proposition in a more complicated manner than as a referent of a relation. Hence it becomes necessary to take *propositional function* as a primitive notion. A propositional function of one variable is any proposition of a set defined by the variation of a single term, while the other terms remain constant. But in general it is impossible to define or isolate the constant element in a propositional function, since what remains, when a certain term, wherever it occurs, is left out of a proposition, is in general no discoverable kind of entity. Thus the term in question must be not simply omitted, but replaced by a *variable*.(§ 106 ¶ 2)

The notion of the variable, we found, is exceedingly complicated. The `x` is not simply *any* term, but any term with a certain individuality; for if not, any two variables would be indistinguishable. We agreed that a variable is any term quâ term in a certain propositional function, and that variables are distinguished by the propositional functions in which they occur, or, in the case of several variables, by the place they occupy in a given multiply variable propositional function. A variable, we said, is *the* term in *any* proposition of the set denoted by a given propositional function.(§ 106 ¶ 3)

Chapter IX pointed out that relational propositions are ultimate, and that they all have *sense*; i.e. the relation being the concept as such in a proposition with two terms, there is another proposition containing the same terms and the same concept as such, as in

and `A` is greater than `B`

These two propositions, though different, contain precisely the same constituents. This is a characteristic of relations, and an instance of the loss resulting from analysis. Relations, we agreed, are to be taken intensionally, not as classes of couples`B` is greater than `A`.^{[71]}.(§ 106 ¶ 4)

Finally, in the present chapter, we examined the contradiction resulting from the apparent fact that, if `w` be the class of all classes which as single terms are not members of themselves as many, then `w` as one can be proved both to be and not to be a member of itself as many. The solution suggested was that it is necessary to distinguish various types of objects, namely terms, classes of terms, classes of classes, classes of couples of terms, and so on; and that a propositional function `ϕ``x` in general requires, if it is to have any meaning, that `x` should belong to some one type. Thus `x`∈`x` was held to be meaningless, because ∈ requires that the relatum should be a class composed of objects which are of the type of the referent. The class as one, where it exists, is, we said, of the same type as its constituents; but a quadratic propositional function in general appears to define only a class as many, and the contradiction proves that the class as one, if it ever exists, is certainly sometimes absent.(§ 106 ¶ 5)

§ 106 n. 1. The notion of *class* in general, we decided, could be replaced, as an indefinable, by that of the class of propositions defined by a propositional function. ↩

§ 106 n. 2. On this point, however, see Appendix. ↩

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