438 Prof. Boole on a General Method in 



The above, together with the general principles of symbolical 

 algebra, suffice for the ground of the following demonstration, 

 which differs from that contained in the Laws of Thought only 

 in applying throughout the familiar illustration of the um. 



Demonstration, 



Let those events, which in the actual language of the problem 

 appear as simple events, be represented by the logical symbols 

 Xy ijy z . . Any event whose probability is given or sought may 

 then be represented upon the principles of the Calculus of Logic 

 by 2i function of those symbols. Thus the event which consists 

 in the concurrence of x and y jointly with the absence of z will 

 be represented by xy[\ —z) ; and the event which consists in the 

 happening of some one alone of the events Xy y, z will be repre- 

 sented by the function 



x[\-y){\^z)-Vy{\-^x){\^z)+z{\^x){\^y)'y 



of which function, it is to be observed, that the several terms 

 connected by the sign 4- are called constituents. If we express 

 generally functions of the above description by the ordinaiy 

 functional symbols (^, '^/r, 6y F, &c., we may thus express the 

 problem which we have to consider in the following manner. 

 Probabilities given : — 



Prob. (i>{xy y, z) =;?, Prob. •>^{xy y, z) = 9, &c. . . (1 ) 



Annexed absolute conditions : — 



(9(a7,y,^..)=0, &c (2) 



Qusesitum, or probability sought : — 



Prob. F(^,2/,r..) (3) 



Now the most obvious mode of procedure is to seek to express 

 the event whose probability is sought, explicitly in terms of the 

 events whose probabilities are given. To do this, we must, in 

 accordance with Principle II., regard all these as simple events, 

 expressing them by new logical symbols Wy s, t, &c. Let then 



<i>{xyyyZ ,.)=zsy '\lr{xyy,z,.)=ty ¥{XyyyZ..) = w. (4) 



From the logical equations (2) and (4) we can now determine 

 w in terms of s, /, &c. The solution will be of the form 



w;=A + 0B+5c + iD (5) 



Here A, B, C, D are functions of s, t, &c., and the several terms 

 of the development are, by means of their coefficients, thus in- 

 terpretable. 



Ist. A represents those combinations of the events s, t, &c. 

 which must happen if w happen. 



