358 INDUCTION. 



part of it. Still, the two propositions in question afford an appreciable 

 probability that any given A is C, provided the average, on which the 

 second proposition is grounded, Was taken fairly Avith reference to the 

 first ; provided the proposition Most B are C was anived at in a man- 

 ner leaving no suspicion that the probability arising from it is other- 

 wise than fairly distributed over the section of B which belongs to A. 

 For although the instances which are A 7nay 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 fi-om the 

 two propositions taken together will be correctly measured by the 

 probability arising from the one, abated in the ratio of that arising 

 from the other. If nine out of ten Swedes have light hair, and eight 

 out of nine inhabitants of Stockholm are Swedes, the probability 

 arising from these two propositions, that any given inhabitant of Stock- 

 holm is light-haired, will amount to eight in ten ; although it is rigor- 

 ously possible (however improbable) that the whole Swedish popula- 

 lation of Stockholm may belong to that tenth section of the people of 

 Sweden who are an exception to the rest. 



If the premisses are known to be true not of a bare majority, but of 

 nearly the whole, of their respective subjects, we 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 en'or of the 

 conclusion will amount to the aggregate of the eiTors of all the prem- 

 isses. Let the proposition. 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 

 I of y9j, or fovir-fifths. Let us now add Most C are D, and suppose 

 this to be true of seven cases out of eight ; the proportion of A which 

 is D will be only |- of -f of /^ > ^^ Vo • Thus the probability progressively 

 dwindles. The experience, however, on which our approximate gen- 

 eralizations are groimded,has so rarely laeen subjected to, or admits of, 

 accurate numerical estimation, that we cannot in general apply 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 premisses approach veiy 

 nearly indeed to being universal truths, the conclusion after a very few 

 steps is worth nothing. A hearsay of a hearsay, or an argument from 

 presumptive evidence depending not upon immediate marks but upon 

 marks of marks, is worthless at a very few removes from the first stage. 



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

 upon approximate generalizations 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. Both these cases are ex- 

 ceptions 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 transformed into complete general- 

 izations exactly equivalent. 



First : If the approximate generalization is of the class in which nur 

 reason for stopping at the approximation is not the impossibility, but 

 only the inconvenience, of going further ; if we are cognizant of tho 



