VARIOUS USES OF FACTORS 39

proof harder, not easier : like every young student, he
yearns to show that he is proving something unexpected,
something that had hitherto seemed improbable a priori-—
at least to all except himself. The paradoxical character of
his conclusion, however, on which he has laid so much
stress, springs simply from its sweeping form: even Mill
could have warned him that " the precariousness of the
method of simple enumeration is in an inverse ratio to the
largeness of the generalization " (loc. cit., Bk. Ill, chap, xxi,
§ 3). Had he narrowed his ' inferable positive analogy '
to something that had a fairly high a priori probability
(e.g. explicitly limited his conclusion to children of a
certain age, certain intelligence, certain fundamental
acquirements, and to lessons of a certain type, instead of
generalizing about all children and all school subjects),
his inductive proof could have proceeded quite plausibly
on a much narrower basis than I have proposed.1

But in any case, at some stage of the work, an elimination
of competing hypotheses is essential. The difficulty is to
be sure that our enumeration of the possible competitors is
exhaustive. " Rejection or exclusion," as Bacon observes,
" is quickly said ; but the way to come at it is intricate
and winding." 2 There would seem to be at least three
ways which, separately or simultaneously employed, may
in some measure increase its efficacy. All of them depend
on much the same principle—namely, on so planning the
experimental and statistical procedure that the unknown
influences may be legitimately treated as ' chance factors.*

First, by employing an appropriate method of multiple
factor-analysis, we can resolve the given table into a series of

1 We need not, I think, altogether accept the arguments advanced by the
Oxford logicians against the traditional notion of induction, viz. that it is
an entirely indirect and negative procedure, that its sole principle " in all its
forms is elimination "—the ** exclusion of all alternatives but one " (Joseph,
loc. cit.y p. 430; Cook Wilson, Statement and Inference, vol. II, p. 595).
Their criticisms, I take it, are valid only against that kind of empirical
induction which, like Mill's, aimed at universal certainties, instead of being
content with merely probable conclusions, i,e, with verifying hypotheses
which themselves already possess a reasonably high a priori certainty.

a Novum Organon, Bk. II, Aphorism xvi (beginning; " The first task of
induction is rejection or exclusion . . .")*