422 INDUCTION. 



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 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 probability 

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

 alone. 



When approximate generalizations are joined together in the other mode, 

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

 creasing, diminishes at each step. From two such premises as Most A are 

 B, Most B are C, we can not with certainty conclude that even a single A 

 is C ; for the whole of the portion of A Avhich in any Avay falls under B, 

 may perhaps be comprised in the exceptional j^art of it. Still, the two 

 propositions in question afford an appreciable 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 



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 times 

 in twelve. On the other hand, if T, although it is both an A and a C, is not a B, something 

 is true which is only true once in every thrice, and something else wliich 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 tAvelve, and T is not a B, only once : making the comparative prob- 

 abilities, not eleven to one, as I had previously made them, but six to one. 



In the last edition I accepted this 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 C's 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 indiscriminately 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 por- 

 tion 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 remarked by another correspondent) applies to the prob- 

 lem under consideration, a mode of calculation only suited to the reverse 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. Eor 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. 



The true theory of the chances 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, the frequency, combined with the efficacy, of the causes in operation that are fa- 

 vorable to it. If out of every twelve As taken indiscriminately 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 

 causes are sufficiently constant and sufficiently 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 

 three. Now suppose twelve cases which are both As and Cs. The whole twelve are now 

 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 six 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 evaluation in the text is correct. 



