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

VARIABLES AND LIMITS 7
variable x in an equation, say, 3x2+2zx-9=O, appears as
a variable in a propositional function, for, as you know,
such an equation is such a function; and, for another
example, when the mathematician says, "I will let x
represent any point in a certain line L," thereby indicating
that he will use x as a variable, he virtually (implicitly)
says, " I will let x represent any one of the verifiers of the
propositional function-x is a point in the line L."
Now let o(x) denote some given propositional
function involving one and only one of the things called
variables. I am going to speak of 4>(x); while 1 am doing
so, you may find it helpful to attach what is said to some
simple specific function such as " x is a man " or " 2 =4"
or; x is a member of this audience." Our function
0(x) contains, we say, one variable, namely, x; x, we say,
is a symbol; notice that, when speaking precisely, we do
not say that the symbol denotes the variable, we say that
the symbol is the variable. The question is: What is
meant by saying that as here used the symbol x is a
variable? Before attempting to answer, let us reflect
that there are terrns such that if any one of them be
substituted for x in p(x) the resulting statement is nonsensical,-non-significant,-and hence neither true nor
false; and that there are other terms which, on being thus
substituted, yield significant statements-that is, propositions (true or false). You will recall that terms of the
former kind,-nonsense-giving terms,-were described
in a previous lecture as inadmissible for p(x) and that the
latter kind,-sense-giving terms,-were described as admissible terms for <(x). Now, it is significant statements,
-statements that are true or false,-propositions,-and
only such that we are concerned with when using propositional functions as instruments in research or in exposi