CHAP. XV.] ARISTOTELIAN LOGIC. 237 



3. All y s are A"s. 

 All not- Y's area's. 



Here also the middle terms are unlike in quality. The premises 

 therefore belong to Case 2, and there being two universal middle 

 terms, the second condition of inference is satisfied. If by the 

 rule we change the quantity and quality of the first extreme, 

 (some) X's, we obtain All not-X's, which, equated with the 

 other extreme, gives 



All not-A r 's are Z's. 



The reverse order of procedure would give the equivalent result, 

 All not-^'s are X's. 



The conclusions of the two last examples would not be recog- 

 nised as valid in the scholastic system of Logic, which virtually 

 requires that the subject of ti proposition should be affirmative. 

 They are, however, perfectly legitimate in themselves, and the 

 rules by which they are determined form undoubtedly the most 

 general canons of syllogistic inference. The process of investi- 

 gation by which they are deduced will probably appear to be of 

 needless complexity ; and it is certain that they might have been 

 obtained with greater facility, and without the aid of any sym- 

 bolical instrument whatever. It was, however, my object to 

 conduct the investigation in the most general manner, and by an 

 analysis thoroughly exhaustive. With this end in view, the 

 brevity or prolixity of the method employed is a matter of indif- 

 ference. Indeed the analysis is not properly that of the syllogism, 

 but of a much more general combination of propositions ; for we 

 are permitted to assign to the symbols v 9 v' 9 w, w', any class-in- 

 terpretations that we please. To illustrate this remark, I will 

 apply the solution (I.) to the following imaginary case : 



Suppose that a number of pieces of cloth striped with diffe- 

 rent colours were submitted to inspection, and that the two fol- 

 lowing observations were made upon them : 



1st. That every piece striped with white and green was also 

 striped with black and yellow, and vice versa. 



2nd. That every piece striped with red and orange was also 

 striped with blue and yellow, and vice versa. 



