118 OF REDUCTION. [CHAP. VIII. 



3. The following example will serve to illustrate the above 

 method. 



Ex. 1. Suppose that an analysis of the proper ties of a parti- 

 cular class of substances has led to the following general conclu- 

 sions, viz. : 



1st. That wherever the properties A and B are combined, 

 either the property C, or the property D, is present also ; but 

 they are not jointly present. 



2nd. That wherever the properties B and C are combined, 

 the properties A and D are either both present with them, or 

 both absent. 



3rd. That wherever the properties A and B are both absent, 

 the properties C and D are both absent also; and vice versa, where 

 the properties C and D are both absent, A and B are both absent 

 also. 



Let it then be required from the above to determine what 

 may be concluded in any particular instance from the presence of 

 the property A with respect to the presence or absence of the 

 properties B and C, paying no regard to the property D. 



Represent the property A by x ; 

 the property B by y ; 

 the property C by z ; 

 ,, the property D by w. 



Then the symbolical expression of the premises will be 



xy = v [w (1 - z) + z (1 - w)} ; 

 yz = v {xw + (I- x) (I- w)}; 

 (\-x)(l-y) = (l-z)(l-w). 



From the first two of these equations, separately eliminating the 

 indefinite class symbol v, we have 



xy{l-w(l-z)-z(l-w)} = 0; 

 yz{l-xw-(l-x)(l- w)} = 0. 

 Now if we observe that by development 



1 - w (1 - z) - z (I - w) = wz + (1 - w) (1 - z), 

 and 



1 - xw - (1 - x) ( 1 - w) = x (1 - w) + w (1 - x), 



