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

VARIABLES AND LIMITS 241
means of various examples. Every example of a great
idea gives a little light and casts a big shadow; we must
try to see the idea in the mingled lights and not lose it in
the composite dark of the shadows. Again consider the
propositional function O(x). The class C-the range of
the function's variable x-is, as you know, the logical
sum of two sub-classes: C1, composed of the verifiers of
4(x); and C2, composed of the falsifiers of c(x). We
may chance to be interested only in the true propositions
derivable from c(x) or only in the false ones; accordingly
we then restrict our thought to the verifiers or else to the
falsifiers; if to the former, then the variable x represents,
not any one of the terms in C, but any one of the verifiers
-x's range being, not C, but Ci; if to the latter, then
x's range is C2. In these cases what determines the
variable's range? The answer is: neither the function
alone nor our restrictive decision alone, but the two
things taken together. The range of a variable is in every
case either the class of admissible terms for some propositional function or a sub-class of such a class, the subclass being determined by some restriction which the
function as such does not impose; observe that, if some
symbol x is to be a variable, it is we who decide what its
range is to be, for it is we who choose the function and,
if any restriction is added, it is we who impose it. Is our
freedom in the matter absolute? No; there is no such
thing as absolute freedom; in the matter in question, as
in all other purely intellectual matters, we have all the
freedom there is, but it is not absolute; we have just seen
that we can not have a variable representing " any one"
of the inadmissible terms for a given propositional function, for the supposition that we can leads, as we saw, to
contradiction. Freedom of thought,-intellectual free