There is no difficulty in manufacturing similar contradictions ad lib. The solution of such contradictions by the theory of types is set forth fully in Principia Mathematica,[27] and also, more briefly, in articles by the present author in the American Journal of Mathematics,[28] and in the Revue de Metaphysique et de Morale.[29] For the present an outline of the solution must suffice.
[27]Vol. I., Introduction, chap. II., * 12 and * 20; vol. II., Prefatory Statement
[28]"Mathematical Logic as based on the Theory of Types," vol. XXX., 1908, pp. 222-262.
[29]"Les paradoxes de la logique," 1906, pp. 627-650.
The fallacy consists in the formation of what we may call "impure" classes, i.e. classes which are not pure as to "type." As we shall see in a later chapter, classes are logical fictions, and a statement which appears to be about a class will only be significant if it is capable of translation into a form in which no mention is made of the class. This places a limitation upon the ways in which what are nominally, though not really, names for classes can occur significantly: a sentence or set of symbols in which such pseudo-names occur in wrong ways is not false, but strictly devoid of meaning. The supposition that a class is, or that it is not, a member of itself is meaningless in just this way. And more generally, to suppose that one class of individuals is a member, or is not a member, of another class of individuals will be to suppose nonsense; and to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything.
Thus if there are
individuals in the world, and
classes of individuals, we cannot form a new class, consisting of both individuals and classes and having