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

62

INTRODUCTION

LCHAP.

the above assumption could be substituted for the axiom of reducibility
in symbolic deductions, since its use would require the explicit introduction
of the further assumption that by a finite number of downward steps we
can pass from any function to a predicative function, and this assumption
could not well be made without developments that are scarcely possible at
an early stage. But on the above grounds it seems plain that in fact,
if the above alternative axiom is true, so is the axiom of reducibility. The
converse, which completes the proof of equivalence, is of course evident.
VII. Reasons for Accepting the Axiom of Reducibility.
That the axiom of reducibility is self-evident is a proposition which can
hardly be maintained. But in fact self-evidence is never more than a part
of the reason for accepting an axiom, and is never indispensable. The reason
for accepting an axiom, as for accepting any other proposition, is always
largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is
known by which these propositions could be true if the axiom were false,
and nothing which is probably false can be deduced from it. If the axiom
is apparently self-evident, that only means, practically, that it is nearly
indubitable; for things have been thought to be self-evident and have yet
turned out to be false. And if the axiom itself is nearly indubitable, that
merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a
radically different kind. Infallibility is never attainable, and therefore some
element of doubt should always attach to every axiom and to all its
consequences. In formal logic, the element of doubt is less than in most
sciences, but it is not absent, as appears from the fact that the paradoxes
followed from premisses which were not previously known to require limitations. In the case of the axiom of reducibility, the inductive evidence
in its favour is very strong, since the reasonings which it permits and the
results to which it leads are all such as appear valid. But although it
seems very improbable that the axiom should turn out to be false, it is by
no means improbable that it should be found to be deducible frolm some
other more fundamental and more evident axiom. It is possible that the
use of the vicious-circle principle, as embodied in the above hierarchy of
types, is more drastic than it need be, and that by a less drastic use the
necessity for the axiom might be avoided. Such changes, however, would
not render anything false which had been asserted on the basis of the
principles explained above: they would merely provide easier proofs of the
same theorems. There would seem, therefore, to be but the slenderest ground
for fearing that the use of the axiom of reducibility may lead us into error.
j.