392 



INDUCTION. 



four Cs are Bs, the probablity 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 conse- 

 sequently those four, have the char- 

 acters of likewise, three of these 

 will be Bs on that ground. There- 

 fore, out of twelve which are both 

 As and Cs, eleven are Bs. To state 

 the argument in another 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 of A indis- 

 criminately, 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 a B 

 occurs only once among twelve things 

 which are both As and Cs. The ar- 

 gument 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 J ; hence if the thing be both an 

 A and a C, the chance is ^ of | = ^V-* 



* The evaluation of the chances in this 

 statement has been objected to by a ma- 

 thematical friend. The correct 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, some- 

 thing is true which is only true twice in 

 every thrice, and something else which is 

 only true thrice in every four times. The 

 first fact being true eight times in twelve, 

 and the second being true six times in 

 every eight, and consequently six times in 

 those eight ; both facts will be true only 

 six limes in twelve. On the other hand, if 

 T, although it is both an A and a C, is not 

 a IJ, something is true which is only true 

 once in every thrice, and something else 

 which is only true once in every four 

 times. The former being true four times 

 out of twelve, and the latter once in every 

 four, and therefore once in those four; 

 both are only true in one case out of twelve. 

 So that T is a B six times in twelve, and T 

 is not a B only once; making the com- 

 parative probabilities, not eleven to one, as 

 I had previously made them, but six to 

 •n«. 



In the aeventh edition I accepted this 



In this computation it is of course 

 supposed that the probabilities arising 

 from A and C are independent of each 

 other. There must not • be any such 

 connection between A and C, that 

 when a thing belongs to the one class 



reasoning as conclusive. More attentive 

 consideration, however, has convinced me 

 that it contains a fallacy. 



The objector argues, that the fact of A's 

 being a B is true eight times in twelve, 

 and the fact of Cs being a B six times in 

 eight, and consequently six times in those 

 eight; both facts therefore are true only 

 six times in every twelve. That is, he 

 concludes that because among As taken 

 indiscriminarely only eight out of twelve 

 are Bs and the remaining four are not, it 

 must equally hold that four out of twelve 

 are not Bs when the twelve are taken from 

 the select portion of As which are also Cs. 

 And by this assumption he arrives at the 

 strange result, that there are fewer Bs 

 among things which are both As and Cs 

 than there are among either As or Cs taken 

 indiscriminately ; so that a thing which 

 has both chances of being a B is less likely 

 to be so than if it had only the one chance 

 or only the other. 



The objector (as has been acutely re- 

 marked by another correspondent) applies 

 to the problem under consideration a 

 mode of calculation only suited to the re- 

 verse problem. Had the question been — 

 If two of every three Bs are As and three 

 out of every four Bs are Cs, how many Bs 

 will be both As and Cs, his reasoning would 

 have been correct. For the Bs that are 

 both As and Cs must be fewer than either 

 the Bs that are As or the Bs that are Cs, 

 and to find their number we must abate 

 either of these numbers in the ratio due to 

 the other. But when the problem is to 

 find, not how many Bs are both As and Cs, 

 but how many things that are both As and 

 Cs are Bs, it is evident that among these 

 the proportion of Bs must be not less, but 

 greater, than among things which are only 

 A, or among things which are only B. 



Tlie true theory of the chaTices is best 

 found by going back to the scientific 

 grounds on which the proportions rest. 

 The degree of frequency of a coincidence 

 depends on, and is a measure of, tiie fre- 

 quency, combined with the efl&cacy, of the 

 causes in operation that are favourable 

 to it. If out of every twelve As taken in- 

 discriminately eight are Bs and four are 

 not it is implied that there are causes 

 operating on A which tend to make it a B, 

 and that these causea are sufficiently con- 

 stant and suflSciently powerful to succeed 

 in eight out of twelve cases, but fail in the 

 remaining four. So if of twelve Cs, nine 

 are Bs and three are not, there must be 

 causes of the same tendency operating on 

 C, which succeed in nine cases and fail in 



