APPROXIMATE GENERALIZATIONS. 357 



type of llio iii-st argument is, Most A are B ; most C are 13 ; 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 ai-e A ; this is a C ; therefore 

 it is probably an A, therefore it is probably a B. The first is exem- 

 plified when we prove a fact by the testimony 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 bo argued that the accused committed the crime, 

 because he concealed himself, and because his clothes wore stained 

 with blood ; in the second, that he committed it because he washed or 

 burnt his clothes, which is supposed to render it probable that they 

 were stained with blood. Instead only oY two links, as in these 

 instances, we may suppose chains of any length. A chain of tho 

 former kind was termed by Mr. Bentham* a self-corroborative chain 

 of evidence ; the second, a self-infiiTnative chain. 



When approximate generalizations are joined by way of addition, it 

 is easily seen from the theory of probabilities laid down in a former 

 chapter, in what manner each of them adds to the probability of a con- 

 clusion which has the warrant of them all. If two of every three A are B, 

 and three of every four C are B, 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 A, all except four 

 are B, by the supposition ; and if the whole twelve, and consequently 

 those four, have the characters of C likewise, three more will be B on 

 that ground. Therefore, out of twelve which are both A and C, eleven 

 are B. To state the argument in another way ; a thing which is both 

 A and C, but which is not B, is found in only one of three sections 

 of the class A, and in only one of foui- sections of the class C ; but this 

 fourth of C being spread over the whole of A indiscriminately, only 

 one-third part of it (or one-twelfth of the whole number) belongs to the 

 third section of A ; therefore a thing which is not B occurs only once, 

 among twelve things which are both A and C The argument would, 

 in the language of the doctrine of chances, be thus expressed : — the 

 chance that an A is not B is i, the chance that a C is not B is ^, hence 

 if the thing be both an A and a C the chance is i of i =:= j\. 



This aigument presupposes (as the reader will doubtless have re- 

 marked) that the probabilities arising from A and C are independent 

 of one another. There must not be any such connexion between A 

 and C, that when a thing belongs to the one class it will therefore 

 belong to the other, or even have a greater chance of doing so. Else 

 the fourth section of C, instead of being equally distributed over the 

 three sections of A, might be comprised in greater proportion, or even 

 wholly, in the third section ; in which last case the probability arising 

 from A and C together would be no greater than that arising from A 

 alone. 



Wlien approximate generalizations are joined together in the other 

 mode, that of deduction, the degree of probability of the inference, in- 

 stead of increasing, diminishes at each step. From two such prem- 

 isses 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 exceptional 



* Rationale of Judicial Evidence. Book v. Circumstantial. 



