APPROXIMATE GENERALISATIONS, AND PROBABLE EVIDENCE.
The inferences called _probable_ rest on approximate generalisations.
Such generalisations, besides the inferior assurance with which they
can be applied to individual cases, are _generally_ almost useless as
premisses in a deduction; and therefore in _Science_ they are valuable
chiefly as steps towards universal truths, the discovery of which is its
proper end. But in _practice_ we are forced to use them--1, when we have
no others, in consequence of not knowing what general property
distinguishes the portion of the class which have the attribute
predicated, from the portion which have it not (though it is true that
we can, in such a case, usually obtain a collection of exactly true
propositions by subdividing the class into smaller classes); and, 2,
when we _do_ know this, but cannot examine whether that general property
is present or not in the individual case; that is, when (as usually in
_moral_ inquiries) we could get universal majors, but not minors to
correspond to them. In any case an approximate generalisation can never
be more than an empirical law. Its authority, however, is less when it
composes the whole of our knowledge of the subject, than when it is
merely the most available form of our knowledge for practical guidance,
and the causes, or some certain mark of the attribute predicated, being
known to us as well as the effects, the proposition can be tested by our
trying to deduce it from the causes or mark. Thus, our belief that most
Scotchmen can read, rests on our knowledge, not merely that most
Scotchmen that we have known about could read, but also that most have
been at efficient schools.
Either a single approximate generalisation may be applied to an
individual instance, or several to the same instance. In the former
case, the proposition, as stating a general average, must be applied
only to average cases; it is, therefore, generally useless for guidance
in affairs which do not concern large numbers, and simply supplies, as
it were, the first term in a series of approximations. In the latter
case, when two or more approximations (not connected with each other)
are _separately_ applicable to the instance, it is said that two (or
more) _probabilities are joined by addition_, or, that there is a
_self-corroborative chain_ of evidence. Its type is: Most A are B; most
C are B; this is both an A and a C; therefore it is probably a B. On the
other hand, when the subsequent approximation or approximations is or
are applicable only by virtue of the application of the first, this is
joining two (or more) probabilities, _by way of Deduction_, which
produces a _self-infirmative chain_; and the type is: Most A are B; most
C are A; this is a C; therefore it is probably an A; therefore it is
probably a B. As, in the former case, the probability increases at each
step, so, in the latter, it progressively dwindles. It is measured by
the probability arising from the first of the propositions, abated in
the ratio of that arising from the subsequent; and the error of the
conclusion amounts to the aggregate of the errors of all the premisses.
In two classes of cases (exceptions which prove the rule) approximate
can be employed in deduction as usefully as complete generalisations.
Thus, first, we stop at them sometimes, from the inconvenience, not the
impossibility, of going further; and, by adding provisos, we might
change the approximate into an universal proposition; the sum of the
provisos being then the sum of the errors liable to affect the
conclusion. Secondly, they are used in Social Science with reference to
masses with _absolute_ certainty, even without the addition of such
provisos. Although the premisses in the Moral and Social Sciences are
only probable, these sciences differ from the exact only in that we
cannot decipher so many of the laws, and not in the conclusions that we
do arrive at being less scientific or trustworthy.