BY BERTRAND RUSSELL
Published in: American Journal of Mathematics, vol.30(1908), pp. 222-262
Transcribed into hypertext by Burtzev B.I., Jul., 07, 2003
e-mail: firstname.lastname@example.org, site: http://bbi-math.narod.ru/
EP.The functions hitherto considered have been propositional functions, with the exception of a few particular functions such R S. But the ordinary functions of mathematics, such as x2, sin x, log x, are not propositional. Functions of this kind always mean Уthe term having such-and-such a relation to x.Ф For this reason they may be called descriptive functions, because they describe a certain term by means of its relation to their argument. Thus
ER.The general definition of a descriptive function is
that is, УR*yФ is to mean Уthe term which has the relation R to y.Ф If there are several terms or none having the relation R to y, all propositions about R*y will be false. We put
Here УE ! ( #x) (φx)Ф may be read Уthere is such a term as the x which satisfies φx,Ф or Уthe x which satisfies φx exists.Ф We have
ES.From the above it appears that descriptive functions are obtained from relations. The relations now to be defined are chiefly important on account of the descriptive functions to which they give rise.
Here Cnv is short for Уconverse.Ф It is the relation of a relation to its converse; e.g., of greater to less, of parentage to sonship, of preceding to following, etc. We have
For a shorter notation, often more convenient, we put
We want next a notation for the class of terms which have the relation R to y. For this purpose, we put
Similarly we put whence
ET.We want next the domain of R (i.e., the class of terms which have the relation R to something), the converse domain of R (i.e., the class of terms to which something has the relation R), and the field of R, which is the sum of the domain and the converse domain. For this purpose we define the relations of the domain, converse domain, and field, to R. The definitions are:
EU.We have, in virtue of the above definitions,
EV.We want next a notation for the relation, to a class α a contained in the domain of R, of the class of terms to which some member of α has the relation R, and also for the relation, to a class β contained in the converse domain of R, of the class of terms which have the relation R to some member of β. For the second of these we put
Thus if R is the relation of father to son, and β is the class of Etonians, Rε*β will be the class Уfathers of Etonians,Ф if R is the relation Уless than,Ф and β is the class of proper fractions of the form 1 Ч 2-n for integral values of n, Rε*β will be the class of fractions less than some fraction of the form 1 Ч 2-n; i.e., Rε*β will be the class of proper fractions. The other relation mentioned above is (Rº )ε.
EW.We put, as an alternative notation often more convenient,
We put also
EY.The product and sum of a class of classes are often required. They are defined as follows:
EZ.We need a notation for the class whose only member is x. Peano uses ιx, hence we shall use ι*x. Peano showed (what Frege also had emphasized) that this class can not be identified with x. With the usual view of classes, the need for such a distinction remains a mystery; but with the view set forth above, it becomes obvious.
i.e., if α is a class which has only one member, then ιº*α is that one
FB.For the class of classes contained in a given class, we put
FC.We can now proceed to the consideration of cardinal and ordinal numbers, and of how they are affected by the doctrine of types.
24) See the above-mentioned article "On Denoting," where the reasons for this view are given at length. back to text24
25) This contradiction was suggested to me by
I. The Contradictions.
II. All and Any
III. The Meaning and Range of Generalized Propositions.
IV. The Hierarchy of Types.
V. The Axiom of Reducibility.
VI. Primitive Ideas and Propositions of Symbolic Logic.
VII. Elementary Theory of Classes and Relations.
VIII. Descriptive Functions.
IX. Cardinal Numbers.
X. Ordinal Numbers.
II. “¬се, каждый” и “любой, произвольный”
III. —мысл и область применимости обобще...