RATIOCINATION, OR SYLLOGISM. 129 



previously admitted, other propositions equally or less general are inferred ; 

 may be exhibited in some of the above forms. The whole of Euclid, for 

 example, might be thrown without difficulty into a series of syllogisms, 

 regular in mood and figure. 



Though a syllogism framed according to any of these formula) is a valid 

 argument, all correct ratiocination admits of being stated in syllogisms of 

 the first figure alone. The rules for throwing an argument in any of the 

 other figures into the first figure, are called rules for the reduction of syl- 

 logisms. It is done by the conversion of one or other, or both, of the prem- 

 ises. Thus an argument in the first mood of the second figure, as — 



No C is B 



All A is B 



therefore 



No A is C, 



may be reduced as follows. The proposition. No C is B, being a universal 

 negative, admits of simple conversion, and may be changed into No B is 

 C, which, as we showed, is the very same assertion in other words — the 

 same fact differently expressed. This transformation having been effected, 

 the argument assumes the following form : 



No B is C 



All A is B 



therefore 



No A is C, 



which is a good syllogism in the second mood of the first figure. Again, 

 an argument in the first mood of the. third figure must resemble the fol- 

 lowing : 



All B is C 



All B is A 



therefore 



Some A is C, 



where the minor premise. All B is A, conformably to what was laid down 

 in the last chapter respecting universal affirmatives, does not admit of sim- 

 ple conversion, but may be converted per accidens, thus. Some A is B ; 

 which, though it does not express the whole of what is asserted in the 

 proposition All B is A, expresses, as was formerly shown, part of it, and 

 must therefore be true if the whole is true. We have, then, as the re- 

 sult of the reduction, the following syllogism in the third mood of the first 

 figure : 



All B is C 



Some A is B, 



Prom Avhich it obviously follows, that 



Some A is C. 



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

 lot necessary to enlarge, every mood of the second, third, and fourth fig- 

 ires may be reduced to some one of the four moods of the first. In other 

 vords, every conclusion which can be proved in any of the last three fig- 

 u'es, may be proved in the first figure from the same premises, with a 

 light alteration in the mere manner of expressing them. Every valid ra- 

 iocination, therefore, may be stated in the first figure, that is, in one of the 

 : oUowing forms : 



9 / 



