APPROXIMATE GENERALISATIONS. 



391 



by the proposition, Nine out of every 

 ten A are B, only in cases of which 

 we know nothing except that they 

 fall within the class A. For if we 

 know of any particular instance r, 

 not only that it falls under A, but to 

 what species or variety of A it be- 

 longs, we shall generally err in apply- 

 ing to i the average struck for the 

 whole genus, from which the average 

 corresponding to that species alone 

 would, in all probability, materially 

 differ. And so if t, instead of being 

 a particular sort of instance, is an 

 instance known to be under the in- 

 fluence of a particular set of circum- 

 stances, the presumption drawn from 

 the numerical proportions in the whole 

 genus would probably, in such a case, 

 only mislead. A general average 

 should only be applied to cases which 

 are neither known nor can be pre- 

 sumed to be other than average cases. 

 Such averages, therefore, are com- 

 monly of little use for the practical 

 guidance of any affairs but those 

 which concern large numbers. Tables 

 of the chances of life are useful to in- 

 surance offices, but they go a very 

 little way towards informing any one 

 of the chances of his own life, or any 

 other life in which he is interested, 

 since almost every life is either better 

 or worse than the average. Such 

 averages can only be considered as 

 supplying the first term in a series of 

 approximations, the subsequent terms 

 proceeding on an appreciation of the 

 circumstances belonging to the parti- 

 cular case. 



§ 6. From the application of a single 

 approximate generalisation to indi- 

 vidual cases, we proceed to the appli- 

 cation of two or more of them together 

 to the same case. 



When a judgment applied to an 

 individual instance is grounded on 

 two approximate generalisations taken 

 in conjunction, the propositions may 

 co-operate towards the result in two 

 different ways. In the one, each pro- 

 position is separately applicable to 

 the case in hand, and our object in 



combining them is to give to the con- 

 clusion in that particular case the 

 double probability arising from the 

 two propositions separately. This 

 may be called joining two probabili- 

 ties by way of Addition ; and the 

 result is a probability greater than 

 either. The other mode is, when only 

 one of the propositions is directly ap- 

 plicable to the case, the second being 

 only applicable to it by virtue of the 

 application of the first. This is join- 

 ing two probabilities by way of Ratio- 

 cination or Deduction ; the result of 

 which is a less probability than either. 

 The type of the first argument is, 

 Most A are B ; most C are B ; this 

 thing is both an A and a C ; there- 

 fore it is probably a B. The type of 

 the second is, Most A are B ; most 

 C are A ; this is a C ; therefore it is 

 probably an A, therefore it is pro- 

 bably a B. The first is exemplified 

 when we prove a fact by the testi- 

 mony of two unconnected witnesses ; 

 the second, when we adduce only the 

 testimony of one witness that he has 

 heard the thing asserted by another. 

 Or again, in the first mode it may be 

 argued that the accused committed 

 the crime because he concealed him- 

 self, and because his clothes were 

 stained with blood ; in the second, 

 that he committed it because he 

 washed or destroyed his clothes, 

 which is supposed to render it pro- 

 bable that they were stained with 

 blood. Instead of only two links, as 

 in these instances, we may suppose 

 chains of any length. A chain of the 

 former kind was termed by Bentham * 

 a self -corroborative chain of evidence ; 

 the second, a self-infirmative chain. 



When approximate generalisations 

 are joined by way of addition, we 

 may deduce from the theory of pro- 

 babilities laid down in a former chap- 

 ter, in what manner each of them 

 adds to the probability of a conclusion 

 which has the warrant of them all. 



If, on an average, two of every 

 three As are Bs, and three of every 



* Rationale of Judicial Evidence, vol. iii. 

 p. 224. 



