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Published online by Cambridge University Press: 14 March 2022
To bring clearly before the mind what is meant by class and to distinguish this notion from all the notions to which it is allied, is one of the most difficult and important problems of mathematical philosophy.”
When Russell wrote this in 1903, he could illustrate the difficulty of the problem by his own confusing attempt at a solution. He was able to demonstrate the importance of classes for mathematical philosophy in his later work: the definition of cardinal number as a class of classes similar to a given class served him as a cornerstone for the derivation of mathematics from logic.
1 B. Russell, The Principles of Mathematics, 2nd Ed., 1938, W. W. Norton, Col., Pbb.
2 Cf. Footnote 9.
3 Cf. Russell and Whitehead, Principia Mathematica, 2nd Ed., pp. 23-30.
4 The term “function” will be used henceforth for “propositional function”.
5 Cf. A. Ushenko, The Problems of Logic, Ch. IV, Allen & Unwin, 1941. The American edition of the book is published by Princeton University Press.
6 These examples show that a class is usually designated by a noun (in the singular or in the plural) or by a descriptive phrase beginning with the word “the” (used in the plural). A definite description, such as “the man in the iron mask,” must not be confused with a unit-class such as “man in the iron-mask”; the former refers to an individual, the latter to a class. On the other hand, a phrase of the form “the so and so” is not always a definite description. Thus in “The triangle is a plane figure” the phrase “the triangle” designates a type of things, so that the whole sentence is nearly equivalent to “The class triangle is included in the class plane figure.” For the difference between class and type see J. Wisdom, Logical Constructions, Mind, 1931, p. 189 f.
7 According to the theory of types classes of individuals, classes of the latter kind, etc. constitute an infinite heirarchy of orders of classes. Given a class of a certain order its members must be classes of a lower order, unless it is the class of individuals.
8 It must be mentioned, however, that logicians of Zermelo's school allow for counting individuals and classes together, because they do not accept the original form of Russell's theory of types. The conceptual unity of a class is not an organization. The members of an organization cooperate through the performance of different functions, unlike the members of a class which are conditioned by the same function. Thus “The Ode to the Grecian Urn” is an organization and not a class of words chosen by Keats in writing the poem.
9 In the first draft of his theory of classes, in “The Principles of Mathematics,” Russell hesitatingly distinguished between the class-concept and the corresponding function (p. 80). He then contrasted the class-concept with the class itself, arriving (1) at the untenable conclusion that “the null-class, which has no terms, is a fiction, though there are null class-concepts,” and (2) at a confusion of classes with enumerations (pp. 69 f). These defects have been remedied in Russell's later works. For the distinction between class and enumeration see W. E. Johnson, Logic, V. 1, Ch. VIII.
10 Intr. to Math. Phil., p. 188.
11 A peculiar treatment of class-membership is found in the “Symbolic Logic” by C. I. Lewis and C. H. Lanford: “Thus we may say that the class of Presidents of the U. S. up to the present time comprises some thirty-members. But there cannot possibly be any such groups or collection of particular persons; nor, of course, was there even a time at which such a concrete group or aggregate existed. We shall have to revise this crude conception of the class of Presidents; and perhaps the following will serve this purpose as well as anything else. It is a fact that there was a President satisfying the description ”Lincoln“ and it is also a fact that there was a President satisfying the description ‘Washington’ etc; and since the number of facts of this kind is the number we want, they can be held to be what we are counting when we enumerate the Presidents.” (p. 329). If this statement means that the Presidents of the U. S. are facts and not people, it must be dismissed as a reductio ad absurdum.
12 The Problems of Logic, ch. IV, Allen & Unwin, 1941.
13 In his interesting article “The Metaphysics and Logic of Classes” (The Monist, 1932) Professor Paul Weiss treats classes as conceptual complexes to be understood within the context of a proposition. His theory is that the subject of a proposition is the extension (of a class) relatively to the predicate, while the predicate is the extension relatively to the subject, and vice versa. But this view is at variance with the usual (as well as with the Boolean) interpretation of propositions according to which each term of a proposition can designate its own class so that the subject and the predicate often stand for two entirely different classes.
14 Cf. E. Cassierer, Substance and Function, ch. 2.
15 Cf. Principia Mathematica, “54.02.
16 Borrowing from the doctrine of Husserl and Meinong, the origins of which may be traced back to William of Ockham, some writers believe that through concepts thought within discourse intends to be and is about things outside discourse. But when things are absent, an intention to reach them can not replace transportation or communication by messages. When you are not in a zoo, your thought of leopards is about “the members of the class leopard” and does not reach any actual leopard. “They”, i. e. actual leopards, cannot figure even as objects of thought-intention, unless “they” is a demonstrative symbol used when the leopards are in sight. Even then one can argue that “they” designates a complex of sense-data, which are appearances of leopards, rather than the leopards themselves.
17 This account of a meaningful use of the judgment “This is a chair” must not be taken as an attempt to define or even explain the meaning of the word “chair.”