THE DOCTRINE OF REDUCTION 341 



of the conclusion that has been denied : this contradictory we 

 combine with whichever premiss of the original syllogism will 

 give us a syllogism in the first figure, having for its conclusion 

 the contradictory of the omitted premiss. 



The INDIRECT REDUCTION of a syllogism might therefore be 

 defined as the process of proving, by means of a syllogism in the first 

 figure, that if the given syllogism were not valid, some self -contradic 

 tory consequence would follow. 



The process may be applied to the other moods of the second and third 

 figures, as well as to Baroco and Bocardo, and its application to all the moods 

 of these figures will be found to shed a very instructive light on the relations 

 between the first three figures. It will be found, for instance, that if any 

 given syllogism belongs to any one of the first three figures, each of the 

 other syllogisms implied by it will be in each of the remaining two of those 

 three figures. 1 From this it follows that &quot; there must be an equal number 

 of valid syllogisms in each of the first three figures, and that they may be 

 arranged in sets of equivalent trios. These equivalent trios will be found to 

 be as follows (sets containing strengthened premisses or weakened con 

 clusions, being enclosed in square brackets) ; 



(I) (II) (III) 



Barbara, Baroco, Bocardo ; 



[A A I, A E O, Felapton ;] 



Celarent, Festino, Disamis ; 



[E A O, E A O, Darapti ;] 



Darii, Camestres, Ferison ; 



Ferio, Cesare, Datisi. 



The corresponding antilogisms are A A O, [A A E,] E A I, [E A A,] 

 A I E, E I A. 



. . . Figure four is ... self-contained in the sense that if we start with 

 a syllogism in this figure, both the other syllogisms will be in the same figure. 

 ... It follows that in figure four the number of valid syllogisms must 

 be some multiple of three. The number is, as we know, six. There are 

 therefore two equivalent trios ; and they will be found to be as follows ; 

 [Bramantip, A E O, Fesapo ;] 

 Camenes, Fresison, Dimaris.&quot; 2 



Since every valid syllogism in the second or third figure im 

 plies two others, only one of which is in the first figure, how are 

 we to know with which premiss of the original we are to com 

 bine the contradictory of the conclusion, in order to obtain the 

 syllogism in the first figure, needed in indirect reduction ? The 



p. 332). Any such trio, therefore, yields three valid syllogisms, each having a pair 

 of the propositions as premisses and the contradictory of the third for its conclusion. 

 From this it also follows that every valid syllogism implies two other valid syllogisms : 

 each of these will take one of the original premisses, and the contradictory of the 

 original conclusion, to prove the contradictory of the other original premiss. 

 1 KEYNES, op. cit., pp. 333-4. ibid., pp. 334-5. 



