APPROXIMATE GENERALISATIONS. 



393 



it will therefore belong to the other, 

 or even have a greater chance of 

 doing so. Otherwise the not-Bs which 

 are Cs may be, most or even all of 

 them, identical with the not-Bs which 

 are As ; in which last case the proba- 

 bility arising from A and C together 

 will be no greater than that arising 

 from A alone. 



When approximate generalisations 

 are joined together in the other mode, 

 that of deduction, the degree of pro- 

 bability of the inference, instead of 

 increasing, diminishes at each step. 

 From two such premises as Most A 

 are B, Most B are C, we cannot with 

 certainty conclude that even a single 

 A is C ; for the whole of the portion 

 of A which in any way falls under B 

 may perhaps be comprised in the ex- 

 ceptional part of it. Still, the two 

 propositions in question aflFord an ap- 

 preciable probability that any given 

 A is C, provided the average on which 

 the second proposition is grounded 

 was taken fairly with reference to 

 the first ; provided the proposition, 

 Most B are C, was arrived at in a 

 manner 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 though the instances which 

 are A may be all in the minority, 

 they may also be all in the majo- 

 rity ; 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 

 probability arising from the ouq, 

 abated in the ratio of that arising 

 three. Now suppose twelve cases which 

 are both As and Cs. The whole twelve are 

 uow under the operation of both sets of 

 causes. One set is sufficient to prevail in 

 eight of the twelve cases, the other in 

 nine. The analysis of the cases shows that 

 tiix of the twelve will be Bs through the 

 operation of both sets of causes ; two more 

 in virtue of the causes operating on A ; 

 and three more through those operating 

 on C, and that there will be only one case 

 in which all the causes will be inoperative. 

 The total number therefore which are Bs 

 will be eleven in twelve, and the evalua- 

 tion in the text is correct. 



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 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 

 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 ma- 

 jority. The error of the conclusion 

 will amount to the aggregate of the 

 errors of all the premises. Let the 

 proposition. Most A are B, be true 

 of nine in ten ; Most B are 0, 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 f of yV, or four-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 ^\, or ^. 

 Thus the probability proi^ressively 

 dwindles. The experience, however, 

 on which our approximate generalisa- 

 tions are grounded has so rarely been 

 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 un- 

 less the premises approach very nearly 

 indeed to being universally true, the 

 conclusion after a very few steps is 

 worth nothing. A hearsay of a hear- 

 say, 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. 



