361 On a Question in the Theory of Probabilities, 



Srdly. I assume not only 



Prob. A = *, Prob. B = j3, Prob. AB = «£, 



but also as 1st above stated; and I consider that, inasmuch as 

 the result of the investigation is to show that the conditions 

 q — up<£ } p—/3q<£ are necessary and sufficient conditions, it 

 is also a result of the investigation that these are the conditions 

 of consistency among the data, viz. the conditions in order that 

 the data may be consistent with the above assumptions as to the 

 independence of the causes. It is clear that since, as just stated, 

 I do assume something beyond the last-mentioned three equa- 

 tions, the conditions of consistency ought to be narrower than 

 those in Prof. Boole's 3rdly. 



4thly. I had not overlooked, but I ought to have stated, that 

 Prof. Boole's conditions were actuallv the conditions of consist- 

 ency in the data. 



5thly. I contend that the conception of A and B as causes 

 does alter the nature of the problem. For when A and B are 

 conceived of as causes, there is a definite notion of the efficient 

 or inefficient action of A or B ; and in particular when they both 

 act, one of them, say A, may act inefficiently. But according to 

 the concomitance statement, then either there is no such notion 

 as that of the efficient or non-efficient happening of A or B (I 

 believe this to be so), or else the only notion of efficient or ineffi- 

 cient happening is happening in concomitance or in non-conco- 

 mitance with E ; but in this view, if A, B, E all happen, then A 

 and B each of them happens efficiently. The argument is to me 

 conclusive as to the diversity of the two problems. 



6thly. I do not in anywise assert, or even suppose, that the ideal 

 problem is arbitrary, or that its connexion with the real problem 

 is arbitrary. I simply do not know what the ideal problem is ; 

 I do not know the point of view, or the assumed mental state of 

 knowledge or ignorance according to which x, y, s, t are the 

 probabilities of A, B, AE, BE. It is to be borne in mind that 

 x } y, 5, t are, in Prof. Boole's solution, determined as numerical 

 quantities included between the limits and 1, i. e. as quantities 

 which are or may be actual probabilities. What I desiderate is, 

 that Prof. Boole should give for his auxiliary quantities x, y, s, t 

 such an explanation of the meaning as I have given for my 

 auxiliary quantities X, (ju. I do not find any such explanation 

 in the memoir referred to. 



7thly and 8thly. No remark is necessary. 



March 29, 1862. 



Prof. Boole, in his reply, dated April 2, writes, " No such ex- 

 planation as you desiderate of the interpretation of the auxiliary 



