APPROXIMATE GENERALIZATIONS. 423 



B are C, was arrived at in a manner leaving no suspicion that the probabili- 

 ty arising from it is otherwise than fairly distributed over the section of B 

 which belongs to A. For though the instances which are A may be all in 

 the minority, they may, also, be all in the majority ; and the one possibility 

 is to be set against the other. On the whole, the probability arising from 

 the two propositions taken together, will be correctly measured by the prob- 

 ability arising from the one, abr.ted in the ratio of that arising from the; 

 other. If nine out of ten Swedes have light hair, and eight out of nine in- 

 habitants of Stockholm are Swedes, the probability arising from these two 

 propositions, that any given inhabitant of Stockholm is light-haired, will 

 amount to eight in ten ; though it is rigorously possible that the whole 

 Swedish population of Stockholm might belong to that tenth section of the 

 people of Sweden who are an exception to the rest. 



If the premises are known to be true not of a bare majority, but of near- 

 ly the whole, of their respective subjects, Ave may go on joining one such 

 proposition to another for several steps, before we reach a conclusion not 

 presumably true even of a majority. The error of the conclusion will 

 amount to the aggregate of the errors of all the premises. Let the prop- 

 osition, most A are B, be true of nine in ten ; Most B are C, of eight in 

 nine ; then not only will one A in ten not be C, because not B, but even of 

 the nine-tenths which are B, only eight-ninths will be C; that is, the cases 

 of A which are C will be only |- of -j^, or four-fifths. Let us now add 

 Most C are D, and suppose this to be ti'ue of seven cases out of eight ; 

 the proportion of A which is D will be only I- of f of y^^, or -j^. Thus the 

 probability progressively dwindles. The experience, however, on which our 

 approximate generalizations are grounded, has so rarely been subjected to, 

 or admits of, accurate numerical estimation, that we can not in general ap- 

 ply any measurement to the diminution of probability which takes place at 

 each illation ; but must be content with remembering that it does diminish 

 at every step, and that unless the premises approach very nearly indeed to 

 being universally true, the conclusion after a very few steps is worth noth- 

 ing. A hearsay of a hearsay, or an argument from presumptive evidence 

 depending not on immediate marks but on marks of marks, is worthless at 

 a very few removes from the first stage. 



§ 7. There are, however, two cases in which reasonings depending on ap- 

 proximate genei-alizations may be carried to any length we please with as 

 much assurance, and are as strictly scientific, as if they were composed of 

 universal laws of nature. But these cases are exceptions of the sort which 

 are currently said to prove the rule. The approximate generalizations are 

 as suitable, in the cases in question, for purposes of ratiocination, as if they 

 were complete generalizations, because they are capable of being transform- 

 ed into complete generalizations exactly equivalent. 



First : If the approximate generalization is of the class in which our rea- 

 son for stopping at the approximation is not the impossibility, but only the 

 inconvenience, of going further; if we are cognizant of the character which 

 distinguishes the cases that accord with the generalization from those 

 which are exceptions to it; we may then substitute for the approximate 

 proposition, a universal proposition with a proviso. The proposition, 

 Most persons who have uncontrolled power employ it ill, is a generalization 

 of this class, and may be transformed into the following : All persons who 

 have uncontrolled power employ it ill, provided they are not persons of un- 

 usual strength of judgment and rectitude of purpose. The proposition, 



