Chapter 38

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.