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

BASIC CONCEPTS 53
to say, in mathematics-form is all-important-so imiportant indeed that some critical thinkers have ventured to
call mathematics the science of Form.
The constants that convert a given propositional function into nonsense may be called inadmissible constants
for that function; all other constants may be called
admissible constants for the function since they convert
it into propositions. It is worthy of note, in passing,
that the line of cleavage between the admissible and the
inadmissible constants for a given furiction is not always
sharply defined. You can readily construct or find functions of x in respect of which it may be doubtful whether
certain constants-the sweetness of sugar, for example,
or the glory of renown-are admissible or not. You
stand here before an open and inviting field for research,
thé problem being to determine criteria for deciding, in
the case of any propositional function, what constants
in the universe of constants are admissible and what ones
are not. The situation may be likened to that of physical
organisms, for there are plants and there are animals, but
in the case of some living organisms there is at present
no means of deciding to which division of the kingdom
they belong.
The admissible constants for a given function fall into
two classes: those converting it into true propositions and
those converting it into false ones. It is convenient to
call the constants of the former class verifiers of the function; and those of the latter class falsifiers of it. The
verifiers of a function are said to satisfy it aid are called
the values of its variables; and the propositions derived
from a function by substituting values of its variables
for these are called values of the function. Thus, you
see that a propositional function is itself a variable