EATIOCINATION, OR SYLLOGISM. 115 



A is B ; wliich, though it does not express. the whole of what is assert- 

 ed in the proposition, All B is A, expresses, as was formerly sliown, part 

 of it, and must therefore lie true if the whole is true. We have, then, 

 as the result of the reduction, the following syllogism in the third mode 

 of the lirst figiu'e : — 



All B is C 

 Some A is B, 

 from which it obviously follows, that 



Some A is C. 

 In the same manner, or in a manner on which, after these examples, 

 it is not necessary to enlarge, every rrtode of the second, third, and 

 fourth figures may be reduced to some one of the four modes of the 

 first. In other words, every conclusion which can be proved in any of 

 the last three figures, may be proved in the first figure from the same 

 premisses, with a slight alteration in the mere manner of expnissiug 

 them. Eveiy valid ratiocination,. therefore, may be stated in the first 

 figure, that is, in one of the following forms : — 

 Every B is C No B is C 



All A ) • -p All A ) • T5 



Some A } ''^' Some A J ^^ ^ 



therefore therefore 



All A. I is c No A is I C 



Some A ) ■ Some A-is not ) 



•Or, if more significant symtjols are preferred,—^ 



To prove an affii:;rnative, the argument must %dmit of being stated 

 in this form : — y 



All animals are mortal ; 



All men ^ 



Some men > arc animals ; 



Socrates. . ) ' 



therefore 

 All men ") 



Some men > are mortal. 

 Socrates ) 



To prove a negative, the argument must be capable of being 

 expressed in this form : — 



No one who is capable of self-control is necessarily vicious ; 

 All negi-oes ) 



Some negroes > are capable of self-control ; 

 Mr. A's negro ) 



therefore 

 No negroes are i 



Some negroes are not > necessarily vicious. 

 Mr. A's negro ,is not ) 

 Although all ratiocination admits of being thrown into one or the 

 other of these fontis, and sometimes gains considerably by the trans- 

 formation, both in clearness and in the obviousness of its consequence ; 

 there axe, no doubt, cases in which the argument falls more naturally 

 into one of the other three figures, and in which its conclusiveness is 

 more apparent at the first glance in those figures, than when reduced 

 into the first. Thus, if the proposition were that pagans may be vir- 



