APPROXIMATE GENERALIZATIONS. 421 



§ 6. From the application of a single approximate generalization to in- 

 dividual cases, we proceed to the application of two or more of them to- 

 gether to the same case. 



When a judgment applied to an individual instance is grounded on two 

 approximate generalizations taken in conjunction, the propositions may co- 

 operate toward the result in two different ways. In the one, each proposi- 

 tion is separately applicable to the case in hand, and our object in combin- 

 ing them is to give to the conclusion in that particular case the double 

 probability arising from the two propositions separately. This may be 

 called joining two probabilities 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 applicable to the case, the second being only appli- 

 cable to it by virtue of the application of the first. This is joining two 

 probabilities by way of Ratiocination 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 ; therefore 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 probably a B. 

 The first is exemplified when we prove a fact by the testimony of two un- 

 connected 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 himself, 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 probable 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 generalizations are joined by way of addition, we may 

 deduce from the theory of probabilities laid down in a former chapter, in 

 wiiat 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 four 

 Cs are Bs, the probability that something which is both an A and a C is a 

 B, will be more than two in three, or than three in four. Of every twelve 

 things which are As, all except four are Bs by the supposition ; and if the 

 whole twelve, and consequently those four, have the characters of C like- 

 wise, three of these will be Bs on that ground. Therefore, out of twelve 

 which are both As and Cs, eleven are Bs. To state the argument in anoth- 

 er way ; a thing which is both an A and a C, but which is not a B, is found 

 in only one of three sections of the class A, and in only one of four sections 

 of the class C ; but this fourth of C being spread over the whole ol A in- 

 discriminately, only one-third part of it (or one-twelfth of the whole num- 

 ber) belongs to the third section of A ; therefore a thing which is not a B 

 occurs only once, among twelve things which are both As and Cs. The 

 argument would, in the language of the doctrine of chances, be thus ex- 

 pressed : the chance that an A is not a B is ^, the chance that a C is not a 

 B is :|^ ; hence if the thing be both an A and a C, the chance is ^ of ^ =TV-t 



* Rationale of Judicial Evidence, vol. iii., p. 224. 



t The evaluation of the chances in this statement has been objected to by a mathematical 

 Mend. The coiTect mode, in his opinion, of setting out the possibilities is as follows. If 

 the thing (let us call it T) which is both an A and a C, is a B, something is true which is 

 only true twice in every thrice, and something else which is only true thrice in every four 



