HYPOTHETICAL AND DISJUNCTIVE SYLLOGISMS 3^5 



that X and F are mutually exclusive, we are justified in inferring 

 the denial of either alternative from the affirmation of the other. 

 For instance, the inference &quot; He was either first or second in the 

 race ; but he was first ; therefore he was not second &quot; is obviously 

 valid ; its validity, however, depends not on the alternation as 

 such, but on the mutually exclusive character of the alternatives. 

 If the major premiss were stated in the strict disjunctive form 

 11 He could not be both first and second in the race&quot; (180), the 

 reasoning might be called and has been called * the Modus 

 Ponendo Tollens of the mixed &quot; disjunctive &quot; syllogism, taking 

 &quot; disjunctive &quot; in its strict, etymological sense. 



182. REDUCTION OF HYPOTHETICAL AND DISJUNCTIVE SYL 

 LOGISMS TO CATEGORICAL FORM. We have already seen that 

 sometimes one and the same judgment may be expressed equally 

 well either in the categorical or in the conditional form (135), and 

 that the general principle under which the narrower case is sub 

 sumed in the first figure of syllogism may be expressed in either 

 of these forms (170), thus showing the fundamental identity of 

 the categorical, with the mixed hypothetical, syllogism. The two 

 identical syllogisms would be : 



Whatever is M is P ; \ (If anything is M it is P ; 



S is M ; I and 4 S is M ; 



Therefore, 5 is P ; } (Therefore, 5 is P. 



The first of these is a categorical, the second a mixed hypotheti 

 cal, syllogism. 



Though we may hold that the pure hypothetical proposition 

 cannot be reduced to the categorical, owing to an existential 

 element implied in the latter and left doubtful in the former 

 (134), we have to remember that this element of doubt is removed 

 by the categorical minor in the case of the mixed hypothetical 

 syllogism : so that there is no syllogism in the latter form which 

 may not be reduced to the categorical. Such reduction is always 

 possible, although it may often involve considerable circumlocu 

 tion and verbal change. For instance, the syllogism : 

 If A is B, Cis D ; 

 But A is B ; 

 therefore, C is D ; 

 may be stated thus : 



All cases of A being B are cases of C being D. 



1 C/. KKYNES, op. cit., p. 362. 



