Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

240 MATHEMATICAL PHIIOSOPHY
sense-giving terms,-constitute a class of terms —the
range of x. Let us denote the class by C. The question
is: Do the nonsense-giving terms,-the inadmissible
terms,-constitute a class? The answer is No; there are
such terms, but they do not together constitute a class.
The correctness of the answer may be shown as follows:
If the terms in question constitute a class, denote it by
C'; C', being itself a term, is either r in C or in C'; to see
that C' is not a term in C, consider any simple example of
>(x)-say, x is a man; in this case C is composed of ail
the terms such that it is significant,-true or false,-to say
that any one of them is a man; our hypothetical C' consists of all the terms such that it is neither true nor false,
but is nonsense to say that any one of them is a man;
evidently it is neither true nor false, but is nonsense to
say, "The class of ail the terms such that it is nonsense
to say they are men " is a man; hence C', if there be such
a class, is not a term in C, but is a term in C'; it is, however, foolish to say that C' is a term in C', in itself-as
foolish as to say, for example, that a class of apples or of
points is an apple or a point, or that the class of featherless
bipeds is a two-legged thing without feathers. And thus
you see that the inadmissible terms for a given propositional function do not constitute a class. The question
I have thus summarily treated is of the kind of questions
which have led Messrs. Russell and Whitehead-or rather
have driven them-to the theory of Types in the Principia.
In its present state the theory is far from being entirely
satisfactory, but it is exceedingly helpful and it undoubtedly faces in the right direction. I desire to recomtnend
it to you for consideration and for improvement.
Examples of Fariables.-Let us now turn to the task of
illustrating the mathematical notion of a variable by