340 THE SCIENCE OF LOGIC 



All S is M by the admittedly cogent reasoning of the first 

 figure. But, by the principle of contradiction, All S is M can 

 not be true : for it is the contradictory of the other original 

 premiss, Some S is not M, which is already admitted to be true ; 

 hence All S is M must be false ; hence either of the premisses 

 from which it follows must be false. But the premiss All P is 

 M was originally given true ; hence it must be the premiss, All 

 S is P, that is false. Therefore its contradictory, Some S is not P, 

 must be true. That is, the conclusion of the original syllogism 

 (Baroco) has been proved by a syllogism in the first figure (Bar- 

 bard) to be necessarily involved in the truth of its premisses, and 

 to follow necessarily therefrom. 



The original syllogism and the one that proves its validity 

 are respectively : 



P a M &amp;gt; P a M 



S o P ^ S a M 



Similarly, the indirect proof of the validity of Bocardo may be 

 represented thus : 



M o P^ JX S a P 



Ma S X ^M a S 



S o P / ^ M a P 



The process here illustrated is not &quot;Reduction &quot; in the same 

 sense as when we speak of &quot;Direct Reduction &quot; : for in the latter 

 the premisses of the new syllogism are either identical with, or 

 are eductions from, the premisses of the original syllogism ; while 

 in the former the new syllogism is quite a different syllogism from 

 the original one. The new syllogism is, of course, involved in, or 

 implied by, the original one ; for every valid syllogism involves two 

 other valid syllogisms formed by combining the contradictory of the 

 original conclusion with each of the original premisses successively , 

 so as to yield for conclusion in each case the contradictory of the other 

 original premiss} In indirect reduction we take the contradictory 



1 C/. KEYNES, Formal Logic, p. 304, 214, where it is proved &quot;that if three 

 propositions involving three terms (each of which occurs in two of the propositions) 

 are together incompatible, then (a) each term is distributed at least once, and (b) one 

 and only one of the propositions is negative &quot; ; and &quot; that these rules are equivalent 

 to the rules of the syllogism &quot;. Of three such propositions, any two are said to be incom 

 patible with the third when from their truth the contradictory of the third is a neces 

 sary consequence. Three such propositions form what is called an Antilogism (ibid., 



