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

136

MATHEMATICAL LOGIC

[PART I

(The definitions *9'07'08 are to apply also when f and s are not both
elementary functions.)
In virtue of these definitions, the true scope of an apparent variable is
always the whole of the asserted proposition in which it occurs, even when,
typographically, its scope appears to be only part of the asserted proposition.
Thus when (:qx). Ox or (x). Ox appears as part of an asserted proposition, it
does not really occur, since the scope of the apparent variable really extends
to the whole asserted proposition. It will be shown, however, that, so far
as the theory of deduction is concerned, (gax). fx and (x).,Ox behave like
propositions not containing apparent variables.
The definitions of implication, the logical product, and equivalence are to
be transferred unchanged to (x). fOx and (ax). fOx.
The above definitions can be repeated for successive types, and thus reach
propositions of any type.
Primitive Propositions. The primitive propositions required are six in
number, and may be divided into three sets of two. We have first two
propositions which effect the passage from elementary to first-order propositions, namely
*91. F: Ox. D. (z). Oz Pp
*9 11. F: Ox v y. D. (aZ). kz Pp
Of these, the first states that, if Ox is true, then there is a value of fxz
which is true; i.e. if we can find an instance of a function which is true, then
the function is " sometimes true." (When we speak of a function as " sometimes" true, we do not mean to assert that there is more than one argument
for which it is true, but only that there is at least one.) Practically, the
above primitive proposition gives the only method of proving "existencetheorems": in order to prove such theorems, it is necessary (and sufficient) to
find some instance in which an object possesses the property in question. If
we were to assume what may be called "existence-axioms," i.e. axioms stating
(3z). pz for some particular p, these axioms would give other methods of
proving existence. Instances of such axioms are the multiplicative axiom
(*88) and the axiom of infinity (defined in *12003). But we have not
assumed any such axioms in the present work.
The second of the above primitive propositions is only used once, in
proving (az). Oz. v. (az). fz: ). (gz). fz, which is the analogue of *1 2
(namely p vp. D. p) when p is replaced by (gz). ^bz. The effect of this
primitive proposition is to emphasize the ambiguity of the z required in
order to secure (az). fz. We have, of course, in virtue of *9-1,
Ox. ). (gz). Oz and Oy. D. (az). Oz.
But if we try to infer from these that Ox v Oy. D. (z). fz, we must use the