CHAP. XV.] ARISTOTELIAN LOGIC. 233 



represented. Thus, if we take v-l 9 and represent by v a class 

 indefinite, the equation (1) will represent a universal proposition 

 according to the ordinary sense of that term, i. e., a proposition 

 with universal subject and particular predicate. We may, in 

 fact, give to subject and predicate in either premiss whatever 

 quantities (using this term in the scholastic sense) we please, ex- 

 cept that by hypothesis, they must not both be universal. The 

 system (1), (2), represents, therefore, with perfect generality, 

 the possible combinations of premises which have like middle 

 terms. 



7. That our analysis may be as general as the equations to 

 which it is applied, let us, by the method of this work, elimi- 

 nate y from (1) and (2), and seek the expressions for #, 1 - x, and 

 vx 9 in terms of z and of the symbols v 9 v' 9 w, w f . The above will 

 include all the possible forms of the subject of the conclusion. 

 The form v (1 -x) is excluded, inasmuch as we cannot from the 

 interpretation vx = Some X's, given in the premises, interpret 

 v (1 - x) as Some not-X's. The symbol v, when used in the sense 

 of "some," applies to that term only with which it is connected 

 in the premises. 



The results of the analysis are as follows : 



x = [w'ww'+ 5 {vv'(l-w) (l-w')+ww'(l-v)(l-v')+(l-v)(l-w)}]z 

 + |j{ttf(l-t0+l -*}(!-*) (I.) 



1 -x= [v (1 -v) {ww +(\ -t0) (1 -0) +v(l -w)w 



z) ) (II.) 



vv(l -to) (1 -O) z+ (l _^') (i - z ). (HI.) 



Each of these expressions involves in its second member two 

 terms, of one of which z is a factor, of the other 1 - z. But 

 syllogistic inference does not, as a matter of form, admit of con- 

 trary classes in its conclusion, as of Z'a and not-^'s together. 



