Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION B]

THE AXIOM OF REDUCIBILITY

169

It is unnecessary, in practice, to know what objects belong to the lowest
type, or even whether the lowest type of variable occurring in a given context
is that of individuals or some other. For in practice only the relative types
of variables are relevant; thus the lowest type occurring in a given context
may be called that of individuals, so far as that context is concerned. It
follows that the above account of individuals is not essential to the truth
of what follows; all that is essential is the way in which other types are
generated from individuals, however the type of individuals may be constituted.
By applying the process of generalization to individuals occurring in
elementary propositions, we obtain new propositions. The legitimacy of this
process requires only that no individuals should be propositions. That this
is so, is to be secured by the meaning we give to the word individual. We
may explain an individual as something which exists on its own account;
it is then obviously not a proposition, since propositions, as explained in
Chapter II of the Introduction (p. 46), are incomplete symbols, having no
meaning except in use. Hence in applying the process of generalization to
individuals we run no risk of incurring reflexive fallacies. We will give the
name of first-order propositions to such as contain one or more apparent
variables whose possible values are individuals, but contain no other apparent variables. First-order propositions are not all of the same type, since,
as was explained in *9, two propositions which do not contain the same
number of apparent variables cannot be of the same type. But owing to
the systematic ambiguity of negation and disjunction, their differences of
type may usually be ignored in practice. No reflexive fallacies will result,
since no first-order proposition involves any totality except that of individuals.
Let us denote by "'! " or "'! (, y)" or etc. an elementary function
whose argument or arguments are individual. We will call such a function
a predicative function of an individual. Such functions, together with those
derived from them by generalization, will be called first-order functions. In
practice we may without risk of reflexive fallacies treat first-order functions
as a type, since the only totality they involve is that of individuals, and,
by means of the systematic ambiguity of negation and disjunction, any
function of a first-order function which will concern us will be significant
whatever first-order function is taken as argument, provided the right
meanings are given to the negations and disjunctions involved.
For the sake of clearness, we will repeat in somewhat different terms our
account of what is meant by a first-order function. Let us give the name of
matrix to any function, of however many variables, which does not involve
any apparent variables. Then any possible function other than a matrix is
derived from a matrix by means of generalization, i.e. by considering the
proposition which asserts that the function in question is true with all