CHAP. XV.] ARISTOTELIAN LOGIC. 235 



1 - x = [WWV + V (1 - V 7 ) (1 - w) + - (WW (1 - V) 



+ (1 - 17) (1 - I/) (1 - to) + t/(l - w) (1 - j ] z 



w') + (l-v)(l-v')}](l-z). (V.) 



u# = {uu'(l - 20)10' + - vv f (I - w) (I - w')} z 



+ (vv'w' + ? W (1 - /)} (1 - z). (VI.) 



Now the second member of (VI.) reduces to a single term rela- 

 tively to z 9 if w = 1, giving 



v# = { w'w' + - vv (1 - w) } (1 - z) ; 



the second member of which is opposite, both in quantity and 

 quality, to the corresponding extreme, wz, in the second premiss. 

 For since w = 1, wz is universal. But the factor vv' indicates 

 that the term to which it is attached is particular, since by hypo- 

 thesis v and v are not both equal to 1. Hence we deduce the 

 following law of inference in the case of like middle terms : 



FIRST CONDITION OF INFERENCE. At least one universal 

 extreme. 



RULE OF INFERENCE. Change the quantity and quality of 

 that extreme, and equate the result to the other extreme. 



Moreover, the second member of (V.) reduces to a single term 

 if v' = 1, w' = 1 ; it then gives 



1 - x = {vw + - (1 - v) w} z. 



Now since t/=l, w' = 1, the middle terms of the premises are 

 both universal, therefore the extremes vx 9 wz 9 are particular. 

 But in the conclusion the extreme involving x is opposite, both 

 in quantity and quality, to the extreme vx in the first premiss, 

 while the extreme involving z agrees both in quantity and qua- 

 lity with the corresponding extreme wz in the second premiss. 

 Hence the following general law : 



