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

ESSENTIAL DISCRIMINATIONS 115
of the question, the last property is always specified; the
first two, seldom or never, though they are evidently
essential, as you will presently see.
In saying that the collection of propositional functions
must be pregnant, or fertile, I mean that it must be such
that one or more consequences, or theorems, can be
logically deduced from its component functions. In
other words, it must be capable of giving rise to a doctrinal function containing one or more propositional
functions besides those serving as postulates. No one,
I imagine, would deliberately call a barren collection of
propositional functions a postulate system.
In saying that a collection of propositional functions,
if it is to be a postulate system, must be compendent, or
connected, I mean something sufficiently easy to grasp,
once it is perceived, but not very easy to state precisely
and clearly. I will try to be intelligible. You know that
owing to the presence of variables in the postulates of a
postulate system, the latter has no specific subject
matter; we may say, however, that, since the postulates
talk about the variables as about subject-matter, the
system has "apparent" subject-matter, or, better, we
may say the system has undetermined subject-matter
represented by the variable-symbols or variable-names.
Now, if you will examine the postulates of some postulate
system, say those of the Hilbert system, you will discover-what you may not have before noticed consciously-that the variable-symbols are each connected
with every other; that is to say, the via's are in V2', which
are in v3's, which are in V4, R1 is a relation of vl's, R2 a relation of segments composed of via's and of angles composed
of half-rays (or half-v2's), and so on. This connectedness
gives the undetermined subject-matter of the system