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

172

MATHEMATICAL LOGIC

[PART I

In this way we can proceed indefinitely to matrices, functions and
propositions of higher and higher orders. We introduce the following
definition:
A function is said to be predicative when it is a matrix. It will be
observed that, in a hierarchy in which all the variables are individuals or
matrices, a matrix is the same thing as an elementary function (cf. pp.
132, 133).
" Matrix " or " predicative function " is a primitive idea.
The fact that a function is predicative is indicated, as above, by a note of
exclamation after the functional letter.
The variables occurring in the present work, from this point onwards, will
all be either individuals or matrices of some order in the above hierarchy.
Propositions, which have occurred hitherto as variables, will no longer do so
except in a few isolated cases of which no subsequent use is made. In
practice, for the reasons explained on p. 169, a function of a matrix may be
regarded as capable of any argument which is a function of the same order
and takes arguments of the same type.
In practice, we never need to know the absolute types of our variables,
but only their relative types. That is to say, if we prove any proposition on
the assumption that one of our variables is an individual, and another is a
finction of order n, the proof will still hold if, in place of an individual, we
take a function of order m, and in place of our function of order n we take a
function of order n + m, with corresponding changes for any other variables
that may be involved. This results from the assumption that our primitive
propositions are to apply to variables of any order.
We shall use small Latin letters (other than p, q, r, s) for variables of the
lowest type concerned in any context. For functions, we shall use the letters
P, *, x, 0, f, g, F (except that, at a later stage, F will be defined as a constant
relation, and 0 will be defined as the order-type of the continuum).
We shall explain later a different hierarchy, that of classes and relations,
which is derived from the functional hierarchy explained above, but is more
convenient in practice.
When any predicative function, say! 2, occurs as apparent variable, it
would be strictly more correct to indicate the fact by placing " (p! z)" before
what follows, as thus: "(! ).f(O! )." But for the sake of brevity we
write simply "() " instead of "((! S)." Since what follows the 0 in brackets
must always contain 0 with arguments supplied, no confusion can result from
this practice.
It should be observed that, in virtue of the manner in which our hierarchy
of functions was generated, non-predicative functions always result from such
as are predicative by means of generalization. Hence it is unnecessary to
introduce a special notation for non-predicative functions of a given order