The Principles of Mathematics (1903)

§ 497

The doctrine of types is here put forward tentatively, as affording a possible solution of the contradiction; but it requires, in all probability, to be transformed into some subtler shape before it can answer all difficulties. In case, however, it should be found to be a first step towards the truth, I shall endeavour in this Appendix to set forth its main outlines, as well as some problems which it fails to solve.(§ 497 ¶ 1)

Every propositional function φ(x)—so it is contended—has, in addition to its range of truth, a range of significance, i.e. a range in which x must lie if φ(x) is to be a proposition at all, whether true or false. This is the first point in the theory of types; the second point is that ranges of significance form types, i.e. if x belongs to the range of significance of φ(x), then there is a class of objects, the type of x, all of which must also belong to the range of significance of φ(x), however φ may be varied; and the range of significance is always either a single type or a sum of several whole types. The second point is less precise than the first, and the case of numbers introduces difficulties; but in what follows its importance and meaning will, I hope, become plainer.(§ 497 ¶ 2)

A term or individual is any object which is not a range. This is the lowest type of object. If such an object—say a certain point in space—occurs in a proposition, any other individual may always be substituted without loss of significance. What we called, in Chapter VI, the class as one, is an individual, provided its members are individuals: the objects of daily life, persons, tables, chairs, apples, etc. are classes as one. (A person is a class of psychical existents, the others are classes of material points, with perhaps some reference to secondary qualities.) These objects, therefore, are of the same type as simple individuals. It would seem that all objects designated by single words, whether things or concepts, are of this type. Thus e.g. the relations that occur in actual relational propositions are of the same type as things, though relations in extension, which are what Symbolic Logic employs, are of a different type. (The intensional relations which occur in ordinary relational propositions are not determinate when their extensions are given, but the extensional relations of Symbolic Logic are classes of couples.) Individuals are the only objects of which numbers cannot be significantly asserted.(§ 497 ¶ 3)

The next type consists of ranges or classes of individuals. (No ordinal ideas are to be associated with the word range.) Thus Brown and Jones is an object of this type, and will in general not yield a significant proposition if substituted for Brown in any true or false proposition of which Brown is a constituent. (This constitutes, in a kind of way, a justification for the grammatical distinction of singular and plural; but the analogy is not close, since a range may have one term or more, and where it has many, it may yet appear as singular in certain propositions.) If u be a range determined by a propositional function φ(x), not-u will consist of all objects for which φ(x) is false, so that not-u is contained in the range of significance of φ(x), and contains only objects of the same type as the members of u. There is a difficulty in this connection, arising from the fact that two propositional functions φ(x), ψ(x) may have the same range of truth u, while their ranges of significance may be different; thus not-u becomes ambiguous. There will always be a minimum type within which u is contained, and not-u may be defined as the rest of this type. (The sum of two or more types is a type; a minimum type is one which is not such a sum.) In view of the Contradiction, this view seems to be the best; for not-u must be the range of falsehood of x is a u, and x is an x must be in general meaningless; consequently, x is a u must require that x and u should be of different types. It is doubtful whether this result can be insured except by confining ourselves, in this connection to minimum types.(§ 497 ¶ 4)

There is an unavoidable conflict with common sense in the necessity for denying that a mixed class (i.e. one whose members are not all of the same minimum type) can ever be of the same type as one of its members. Consider, for example, such phrases as Heine and the French. If this is to be a class consisting of two individuals, the French must be understood as the French nation, i.e. the class as one. If we are speaking of the French as many, we get a class consisting not of two members, but of one more than there are Frenchmen. Whether it is possible to form a class of which one member is Heine, while the other is the French as many, is a point to which I shall return later; for the present it is enough to remark that, if there be such a class, it must, if the Contradiction is to be avoided, be of a different type both from classes of individuals and from classes of classes of individuals.(§ 497 ¶ 5)

The next type after classes of individuals consists of classes of classes of individuals. Such are, for example, associations of clubs; the members of such associations, the clubs, are themselves classes of individuals. It will be convenient to speak of classes only where we have classes of individuals, of classes of classes only where we have classes of classes of individuals, and so on. For the general notion, I shall use the word range. There is a progression of such types, since a range may be formed of objects of any given type, and the result is a range of higher type than its members.(§ 497 ¶ 6)

A new series of types begins with the couple with sense. A range of such types is what Symbolic Logic treats as a relation: this is the extensional view of relations. We may then form ranges of relations, or relations of relations, or relations of couples (such as separation in Projective Geometry[140]), or relations of individuals to couples, and so on; and in this way we get, not merely a single progression, but a whole infinite series of progressions. We have also the types formed of trios, which are the members of triple relations taken in extension as ranges; but of trios there are several kinds that are reducible to previous types. Thus if φ(x, y, z) be a propositional function, it may be a product of propositions φ1(x) . φ2(y) . φ3(z) or a product φ1(x) . φ2(y, z), or a proposition about x and the couple (y, z), or it may be analyzable in other analogous ways. In such cases, a new type does not arise. But if our proposition is not so analyzable—and there seems no à priori reason why it should always be so—then we obtain a new type, namely the trio. We can form ranges of trios, couples of trios, trios of rios, couples of a trio and an individual, and so on. All these yield new types. Thus we obtain an immense hierarchy of types, and it is difficult to be sure how many there may be; but the method of obtaining new types suggests that the total number is only α0 (the number of finite integers), since the series obtained more or less resembles the series of rationals in the order 1, 2, ..., n, ..., 1/2, 1/3, ... 1/n, ..., 2/3, ..., 2/5, ...2/(2n + 1), ... This, however, is only a conjecture.(§ 497 ¶ 7)

Each of the types above enumerated is a minimum type; i.e., if φ(x) be a propositional function which is significant for one value of x belonging to one of the above types, then φ(x) is significant for every value of x belonging to the said type. But it would seem—though of this I am doubtful—that the sum of any number of minimum types is a type, i.e. is a range of significance for certain propositional functions. Whether or not this is universally true, all ranges certainly form a type, since every range has a number; and so do all objects, since every object is identical with itself.(§ 497 ¶ 8)

Outside the above series of types lies the type proposition; and from this as starting-point a new hierarchy, one might suppose, could be started; but there are certain difficulties in the way of such a view, which render it doubtful whether propositions can be treated like other objects.(§ 497 ¶ 9)