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

BASIC CONCEPTS 55
constants, assign to each of them a unique and definite
meaning? The answer is No: each of the terms admits
of many, infinitely many, different definite meanings
satisfying the postulates. The answer will be justified
at a later stage of our discussion. For the present, I ask
you to assume its correctness. We may, therefore, now
state, in answer to our main question, that the six terms
are not constants, but variables, and that, accordingly,
the postulates are not propositions, as they are wont to
be called, but are propositional functions. As you reflect
upon this fact, you will find that its importance is immeasurable, not only for philosophy in its narrower sense,
but for Criticism 1 in the widest sense, in all its fields.
In a future lecture, I shall return to the matter of estimating the fact's general importance. For the present, let
us follow its strictly logical and philosophical leading.
We have to say at once that the postulates of the
system we are examining as a representative specimen of
postulate systems in general, are neither true nor false,
being propositional functions. The same must, of course,
be said of all the theorems deduced or deducible from
them as their logical consequences or implicates, for all
such theorems, being statements involving the same
variables as are present in the postulates, are propositional
functions and are, therefore, neither true nor false. At
this point, I cannot refrain from pausing long enough to
point out how the most vitally fundamental fact in
logical theory appears here with startling vividness in
new light. Suppose that in the postulates we replace the
seven terms-" point," " straight line," "plane," etc.respectively, by any meaningless vocables whatever, as
x In this connection the reader should consult Professor F. C. S.
Schiller's very suggestive article "Doctrinal Functions" in The Journal
of Philosophy, Psychology and Scientific Methods, Vol. XVI., 1919.