234 ARISTOTELIAN LOGIC. [CHAP. XV. 



We must, therefore, in order to determine the rules of that 

 species of inference, ascertain under what conditions the second 

 members of any of our equations are reducible to a single term. 



The simplest form is (III.), an( i ft is reducible to a single 

 term if w = 1 . The equation then becomes 



vx = vv'wz, (3) 



the first member is identical with the extreme in the first pre- 

 miss; the second is of the same quantity and quality as the extreme 

 in the second premiss. For since w' = 1, the second member of 

 (2), involving the middle term y, is universal ; therefore, by the 

 hypothesis, the first member is particular, and therefore, the se- 

 cond member of (3), involving the same symbol w in its coeffi- 

 cient, is particular also. Hence we deduce the following law. 



CONDITION OF INFERENCE. One middle term, at least, uni- 

 versal. 



RULE OF INFERENCE. Equate the extremes. 



From an analysis of the equations (I.) and (II.), it will further 

 appear, that the above is the only condition of syllogistic in- 

 ference when the middle terms are of like quality. Thus the 

 second member of (I.) reduces to a single term, if w = \ and 

 v = I ; and the second member of (II.) reduces to a single term, 

 if w/ = 1, v=\ 9 w = 1. In each of these cases, it is necessary that 

 w' = 1 , the solely sufficient condition before assigned. 



Consider, secondly, the case in which the middle terms are 

 of unlike quality. The premises may then be represented un- 

 der the forms 



vx = v'y, (4) 



wz = w ( 1 y) ; (5) 



and if, as before, we eliminate y, and determine the expressions 

 of #, 1 - ar, and vx, we get 



(1 -w) w r + [ww'(\ -) + (! -v) (1 -v*) (1 -w) 



")}-]z 



a - *) 



