THE DOCTRINE OF REDUCTION 343 



Ferio are exactly the same as the &quot; Some S s &quot; referred to in the 

 minor premiss. 1 If direct reduction be attempted, it will be found 

 to be impossible : the indirect process, or reductio ad impossibile, 

 is the only one by which we can &quot; reduce &quot; Barbara to Darii (or 

 vice versa), or Celarent to Ferio (or vice versa}. Thus, we may 

 indirectly prove the validity of a syllogism in Barbara by means 

 of a syllogism in Darii as follows : If S a P does not follow 

 from the premisses M a P, S a M, then its contradictory, 5 o P, 

 may follow from them as true. But if 5 o P is true, so is its 

 partial contrapositive P i S ; and 5 a M is also given true. Com 

 bining the latter we have this syllogism in Darii : 



S aM 



Pi S 



.-. P i M = M iP = M o P 



But M o P cannot be true, for it contradicts M a P which was 

 given true ; therefore the premiss P i S must be false ; and there 

 fore also S o P. Hence the original conclusion, S a P, must follow 

 as true from its premisses. 



Since all the moods of the first figure cannot be reduced directly to one 

 another, it is not correct to say that the reasoning in any mood of any other 

 figure than the first may be expressed in any mood of the first figure. What 

 is true is this, that the validity of any valid mood of any figure other than the 

 first (or of the first itself) may be proved by means of a syllogism in any 

 mood of the first figure. Of course, a person who will not admit the validity 

 of a syllogism in a mood of the first figure will not admit the validity of the 

 validating syllogism that may be brought forward in any other mood of that 

 figure. Indeed, even where there is question of the second and third figures, 

 whenever an argument falls more naturally into one of these than into the 

 first, its cogency may be seen more clearly in that figure than in the mood of 

 the first figure to which the traditional doctrine would have us reduce it. 

 That there are such arguments will appear presently from a closer study of 

 each figure. 



170. CHARACTERISTICS OF THE FIRST FIGURE. The first 

 figure (a) embodies the most usual and scientific form of syllogistic 

 inference, viz. the application of some abstract, necessary truth, or 

 general law, to concrete, particular cases subsumed under it : 

 hence it was called by Aristotle the perfect figure of the syllogism. 

 (&) It is the only figure which can prove all four forms of the 

 categorial judgment, A, E, I, and O. (c) It is the only figure in 

 which the universal affirmative (A) can be proved. This makes it 

 all-important ; for it is A propositions that all the sciences aim at 

 1 C/. KBYNBS, pp. 336-7, 



