312 THE SCIENCE OF LOGIC 



The rule therefore holds, in all cases, that if we have two pro 

 positions which are really premisses of a syllogism as they stand, 

 i.e. which connect two terms with a common third in order to 

 draw a conclusion about either of these extremes in terms of the 

 other, one of the premisses at least must be affirmative. If, as 

 they stand, they are two negatives, they cannot yield a conclu 

 sion about either actual extreme, from comparison of these with 

 the single actual middle (if there be such). 



The premisses of any valid syllogism can be made to stand 

 as two negatives by the simple process of obversion ; but in that 

 negative form they do not yield the conclusion. Thus, the pre 

 misses of the syllogism : &quot; All M is P ; All S is M ; therefore, All 

 S is P&quot; : may be written : No M is not-P ; No S is not-M ; 

 but it is only by obverting them back to their original form that 

 we get the original conclusion from them. 



This whole difficulty is an old one. It was considered by the 

 mediaeval Scholastic logicians ; and it was probably their con 

 sideration of it that led to their treatment of the process now 

 called Obversion, under the name of Aequipollence of Judgments 



(117). 



But it is a curious fact, to which Dr. Keynes calls attention, 1 that those 

 logicians who have found this difficulty so troublesome &quot; do not appear to 

 have observed that, as soon as we admit more than three terms, other appar 

 ent breaches of the syllogistic rules may occur in what are perfectly valid 

 reasonings. Thus, the premisses All P is M and All S is M, in which M 

 is not distributed, yield the conclusion Some not-S is not-P ; 2 and hence 

 we might argue that undistributed middle does not invalidate an argument. 

 Again, from the premisses All M is P, All not-M is 5, we may infer Some 

 S is not P, 3 although there is apparently illicit process of the major. . . . 

 But of course none of the above examples really invalidate the syllogistic 

 rules ; for these rules have been formulated solely with reference to reason 

 ings of a certain form, namely, those which contain three and only three 

 terms. In every case the reasoning inevitably conforms to the rule which 

 it appears to violate, as soon as, by the aid of immediate inferences, the 

 superfluous number of terms has been eliminated. 1 



A practical corollary from all this is that, when we are asked whether any 

 thing can be inferred from two given propositions containing four terms, we 

 should see whether these latter might not be reduced, by the aid of processes 

 of immediate inference, to three, and so, perhaps, yield a valid conclusion. 



l op. cit., pp. 297-8. 



2 By the contraposition of both premisses, this reasoning is reduced to the valid 

 syllogistic form, All not-M is not-P, All not-M is not-S, therefore Some not- 

 S is not-P. 



3 By inversion of the first premisses, this reasoning is reduced to the valid syl 

 logistic form, Some not-M is net P, All not-M is S, therefore, Some S is not P. 



