THE DOCTRINE OF REDUCTION 339 



P i S, which we convert simply, and then obvert (sk\ in order to 

 obtain the original conclusion, S o P. 



169. INDIRECT REDUCTION : EXTENSION OF THE DOCTRINE 

 OF REDUCTION. Besides the process of Direct Reduction , which 

 has just been described, there is another method by which we 

 may prove the validity of a syllogism belonging to a figure other 

 than the first 



We may establish a given proposition by proving its con 

 tradictory to be false ; and we may do this latter by proving 

 that were this contradictory true something self-contradictory, and 

 therefore impossible, would follow. Euclid often makes use of 

 this method. It is called variously Indirect Proof (cf. infra, 254, b\ 

 Reductio ad Impossibile (dTraywyrj et? TO dSvvarov Aristotle) Re- 

 ductio per Impossibile, Reductio ad Absurdum, Deductio ad 

 Absurdum. It always runs on these lines : &quot; The proposition P 

 let us say is true ; for, if not, tken P is true ; but ifP be true, Q 

 must be true ; Q, however, cannot be true : we know it to be false ; 

 therefore P, from which it follows, must be false ; therefore its 

 contradictory, P, must be true; Q.E.D.&quot; We have referred 

 more than once already (148; 156, Cor. 2) to the principle 

 underlying this process : that if the conclusion of a valid inference 

 be false the premisses or antecedent from which it necessarily 

 follows must also be false a principle which is involved in the very 

 nature of logical inference. 



Now this principle, involved in the process of Indirect Proof, 

 may be employed for the purpose of proving to a person who 

 admits the validity of the moods of the first figure only, that the 

 moods of the other figures are also valid. Utilizing this principle, 

 we can force such a person, by means of a syllogism in the first 

 figure whose cogency he admits, to admit also the validity of the 

 moods of the other figures i.e. that if their premisses be true 

 their conclusions must be true under pain of contradicting him 

 self. It was in this way that the Scholastics, following Aristotle, 

 proved the validity of the moods Baroco and Bocardo. 



We may apply it to Baroco as follows : 



If it is true that All P is M, and that Some S is not M, then 

 it is true that Some S is not P ; for if this latter is not true, then, 

 by the principle of excluded middle, its contradictory, All S is P, 

 is true ; and, ex hypothesi, the two original premisses, All P is M, 

 and Some S is not M, are granted to be true. But if it be true 

 that All P is M, and that All S is P, then it must be true that 



