developed in Professor Boole's " Laws of Thought." 475 



treated as independent ; fii'st, in the case in whicli E does not 

 happen ; secondly, in the case where it is not observed whether 

 E does or does not happen. 



Though the data of the problem, together with the equations 

 derived from the assumed independence of the simple events, are 

 always enough to determine the unknown quantities x, y, &c., 

 and consequently to determine the chances of the compound 

 events represented above by the Greek letters 8, e, &c., there are 

 cases in which the required chance cannot be exactly expressed 

 in a series of the terms S, e, &c. In these cases the problem 

 remains indeterminate, notwithstanding the assumptions. Of 

 this nature are Examples 1, 4, 7 of Chapter XVIII. In Ex. 1, 

 for instance, the absolute chances of the four events there repre- 

 sented by ux, u{\—x), {\—u)x, {\—u){l—x) may be found, 

 but the chance of the required event cannot be expressed in a 

 series of these chances, for it comprises all cases which come 

 under the event ux, but only part, an unknown part, of those 

 which come under (1— m)(1— *•). 



^Vhat, now, is the practical value of Professor Boole's logical 

 method as applied to the theory of chances ? In cases determi- 

 nable by ordinary algebraical processes, his book gives a system- 

 atic and uniform method of solving the questions, though very 

 commonly a longer one than we should otherwise use ; at least 

 it appears to me that the really determinate problems solved in 

 the book, as 2 and 3 of Chap. XVIIL, might be moi-e shortly 

 solved mthout the logical equations. In these cases the ori- 

 ginally assumed independence of the simple events is unneces- 

 sary, none of the equations implied thereby consisting wholly of 

 terms comprised in V. The disadvantage of Professor Boole's 

 method in such cases is, that it does not show us whether the 

 problem is really determinate or requires further assumptions, — 

 whether, in fact, the assumptions made are necessai-y or not. 

 On the other hand, in cases not determinable by ordinary algebra, 

 his system is this ; he takes a general indeterminate problem, 

 applies to it particular assumptions not definitely stated in his 

 book, but which may be shown, as I have done, to be implied 

 in his method, and with these assumptions solves it ; that is to 

 say, he solves a j-urticular determinate case of an indeterminate 

 problem, while his book may mislead the reader by making him 

 suppose that it is the general problem which is being treated of. 

 The question arises. Is the particular case thus solved a pecu- 

 liarly valuable one, or one more worthy than any other of being 

 solved ? It is clearly not an assumption which must in all cases 

 be true ; nor is it one which, without knowing the connexion 

 among the simple events, wc can suppose more likely than any 

 other to represent that connexion ; for if we examined the assurap- 



