606 PROFESSOR BOOLE ON THE COMBINATION 



Determination of the Conditions of Possible Experience. 



13. To explain the method of effecting this object, by an example, I will first 

 symbolically express the problem of Art. 11. 



Let us then represent rain by x, snow by y, and wind by z. The problem in 

 question then takes the following form: — 



Given Prob. xy=p, Prob. yz = q, Prob. xz — r . . (1) 



Required Prob. xyz (2) 



The value required we shall represent by u. It is our present object, not to solve 

 this problem, but to ascertain the conditions which must connect p % q, and r, 

 in order that the data may be possible, with the corresponding limitations of u. 

 For if u were itself determined by experience, it would be subject to conditions 

 of possibility similar to those which govern ]?, q, and r. 



Now let us write, resolving the events in the problem into the possible alter- 

 nations out of which they are formed, 



Prob. xyz=u, Prob. xyz — X, Prcb. xzy = fx, Prob. yzx=v- 

 We have then 



u + X=p, u + u=q, u + fx=r • • • (3) 



The first of these equations only expresses that the probability of the concurrence 

 of x and y is equal to the probability of the concurrence of x, y, and z, and the 

 probability of the concurrence of x and y without z. To the equations (3) we 

 must now add the inequations 



«>0, A>0, M>0, v^zO, 



_ (4) 



U + X + /JL + V< 1 V ' 



expressing the conditions to which u, \, //, v, 1st, as probabilities, and, 2dly, as 

 probabilities which do not altogether make up certainty, are subject. 

 First, we will eliminate X, ji, and v. Their values found from (3) are 



X—p — u ji = r — u i/ = q — u. 



Substituting these in (4) we have 



Whence, 



p- 



-u >0 





q — u 



>0 r- 



u > 



p 



+ q + r- 



-2 



m<1» 









u<p, 





u<.q> 



«<r, 



\ 

 1 





w>0, 





u > 



p+q+r— 1 

 2 



(5) 



Such are the conditions to which the quantity u is subject, conditions which the 

 value of Prob. xyz must a priori satisfy. 



To determine the conditions connecting^, q, and r, we must from (5) eliminate 

 u. Now, if we have any two inequations of the form 



u < a 



