316 THE SCIENCE OF LOGIC 



The above results have been reached by the employment of the principle 

 referred to in the proof of Corollary 2, above (156). The principle will be 

 again invoked in a subsequent chapter, in connexion with the process of 

 Indirect Reduction ( 1 69). The following proofs, borrowed from Dr. Keynes, 1 

 will illustrate its application in the present context. 



(a) &quot; Any syllogism involving directly an illicit process of major or 

 minor involves indirectly a fallacy of undistributed middle, and vice versa? 



&quot; Let P and Q be the premisses and R the conclusion of a syllogism 

 involving illicit major or minor, a term X which is undistributed in P being 

 distributed in R. Then the contradictory of R combined with P must prove 

 the contradictory of Q. But any term distributed in a proposition is undis 

 tributed in its contradictory. X is therefore undistributed in the contradictory 

 of R, and by hypothesis it is undistributed in P. But X is the middle term 

 of the new syllogism, which is therefore guilty of the fallacy of undistributed 

 middle. It is thus shown that any syllogism involving directly a fallacy of 

 illicit major or minor involves indirectly a fallacy of undistributed middle.&quot; 



&quot; Adopting a similar line of argument, 3 we might also proceed in the 

 opposite direction, and exhibit the rule relating to the distribution of the 

 middle term as a corollary from the rule relating to the distribution of the 

 major and minor terms.&quot; 



(b] Rule (5) is a corollary from Rule (3). &quot; This is shown by De Morgan 

 (Formal Logic , p. 13). He takes two universal negative premisses, E E. In 

 whatever figure they may be, they can be reduced by conversion to No P is 

 M, No S is M. Then by obversion they become (without losing any of their 

 force) All P is M, All S is M, and we have undistributed middle. Hence 

 rule [5] is exhibited as a corollary from rule [3]. For if any connexion 

 between 5 and P can be inferred from the first pair of premisses, it must 

 also be inferable from the second pair. 



&quot; The case in which one of the premisses in particular . . . may ... be 

 disposed of by saying that if we cannot infer anything from two negative 

 premisses both of which are universal, a fortiori we cannot from two negative 

 premisses one of which is particular.&quot; 



In this latter case, if M is the predicate of the particular premiss, the syl 

 logism may be shown to imply undistributed middle, as in the case of E E 

 above. If, however, M is subject of the O premiss we cannot show that the 

 syllogism indirectly involves undistributed middle, but we can show that 

 it indirectly involves four terms, S, P, M and M. This latter we can do 

 either by obverting the E premiss after making M its predicate, as Professor 

 Welton does, 4 or by also contraposing the O premiss as De Morgan does. 



1 op. cit., pp. 291-4. 



2 The &quot;invalid syllogism All M is P, No S is M, therefore No S is P, does not 

 directly involve [undistributed middle]. But if this syllogism is valid, then must also 

 the following syllogism be valid : All M is P (original major), Some S is P (contra 

 dictory of original conclusion), therefore Some S is M (contradictory of original 

 minor) ; and here we have undistributed middle. Hence the rule relating to the 

 distribution of the middle term establishes indirectly the invalidity of the syllogism 

 in question. The principle involved is the same as that on which we shall find the 

 process of indirect reduction to be based.&quot; Ibid., p. 294. n. 2. 



3 Cf. WELTON, op. cit., p. 301. 4 ibid., pp. 299, 300. 



