238 ARISTOTELIAN LOGIC. [CHAP. XV. 



Suppose it then required to determine how the pieces marked 

 with green stood affected with reference to the colours white, 

 black, red, orange, and blue. 



Here if we assume v = white, x = green, v = black, y - yellow, 

 w = red, z = orange, w' = blue, the expression of our premises will 

 be 



vx = v'y, 

 wz = w'y, 



agreeing with the system (1) (2). The equation (I.) then leads 

 to the following conclusion : 



Pieces striped with green are either striped with orange, 

 white, black, red, and blue, together, all pieces possessing which 

 character are included in those striped with green ; or they are 

 striped with orange, white, and black, but not with red or blue ; 

 or they are striped with orange, red, and blue, but not with white 

 or black ; or they are striped with orange, but not with white or 

 red ; or they are striped with white and black, but not with blue 

 or orange ; or they are striped neither with white nor orange. 



Considering the nature of this conclusion, neither the sym- 

 bolical expression (I.) by which it is conveyed, nor the analysis 

 by which that expression is deduced, can be considered as need- 

 lessly complex. 



9. The form in which the doctrine of syllogism has been 

 presented in this chapter affords ground for an important obser- 

 vation. We have seen that in each of its two great divisions the 

 entire discussion is reducible, so far, at least, as concerns the de- 

 termination of rules and methods, to the analysis of a pair of 

 equations, viz., of the system (1), (2), when the premises have 

 like middle terms, and of the system (4), (5), when the middle 

 terms are unlike. Moreover, that analysis has been actually 

 conducted by a method founded upon certain general laws de- 

 duced immediately from the constitution of language, Chap, n., 

 confirmed by the study of the operations of the human mind, 

 Chap, in., and proved to be applicable to the analysis of all sys- 

 tems of equations whatever, by which propositions, or combina- 

 tions of propositions, can be represented, Chap. vm. Here, then, 

 we have the means of definitely resolving the question, whether 

 syllogism is indeed the fundamental type of reasoning, whether 



