232 ARISTOTELIAN LOGIC. [CHAP. XV. 



whence x = - z 9 or, All X'B are Z's. 



And in the same way may any other case be treated. 



6. Quitting, however, the consideration of special examples, 

 let us examine the general forms to which all syllogism may be 

 reduced. 



PROPOSITION I. 

 To deduce the general rules of Syllogism. 



By the general rules of Syllogism, I here mean the rules appli- 

 cable to premises admitting of every variety both of quantity 

 and of quality in their subjects and predicates, except the com- 

 bination of two universal terms in the same proposition. The 

 admissible forms of propositions are therefore those of which a 

 tabular view is given in (1). 



Let X and Fbe the elements or things entering into the first 

 premiss, Z and Y those involved in the second. Two cases, fun- 

 damentally different in character, will then present themselves. 

 The terms involving Y will either be of like or of unlike quality, 

 those terms being regarded as of like quality when they both 

 speak of " Y's," or both of" Not- Y's," as of unlike quality when 

 one of them speaks of " F's," and the other of " Not- YV' Any 

 pair of premises, in which the former condition is satisfied, may 

 be represented by the equations 



vx = v'y, (1) 



wz = wy ; (2) 



for we can employ the symbol y to represent either " All F's," 

 or " All not- Y' s," since the interpretation of the symbol is purely 

 conventional. If we employ y in the sense of "All not-Y's," 

 then 1 - y will represent " All F's," and no other change will 

 be introduced. An equal freedom is permitted with respect 

 to the symbols x and z, so that the equations (1) and (2) may, 

 by properly assigning the interpretations of x, y, and z, be made 

 to represent all varieties in the combination of premises depen- 

 dent upon the quality of the respective terms. Again, by as- 

 suming proper interpretations to the symbols v, v', w, w\ in those 

 equations, all varieties with reference to quantity may also be 



