3 14 THE SCIENCE OF LOGIC 



over and above the distributed extreme or extremes, the premisses 

 must distribute the middle term at least once. Now, if both 

 premisses be O propositions, Rule 5 forbids a conclusion. If 

 both be I propositions, they distribute no term at all, and hence 

 we have the fallacy of Undistributed Middle. If one be I and 

 the other O, then, as we saw in De Morgan s proof, the O premiss 

 must have M for predicate in order that the syllogism may 

 avoid Undistributed Middle. But it avoids &amp;gt; this fallacy only by 

 falling into the other fallacy of Illicit Major ; for I and O distri 

 bute only the one term, M, between them, thus leaving P undis 

 tributed, whereas the conclusion, being negative, will necessarily 

 distribute it. 



Cor. 2. If both premisses are affirmative, one being particu 

 lar, they will distribute only one term, namely, the subject of 

 the other or universal premiss, and this distributed term must be 

 the middle term. Hence, 5 and P will be undistributed in the 

 premisses. Therefore the conclusion must be particular (and 

 affirmative). 



Secondly, if one premiss be affirmative and the other negative 

 [both cannot be negative (Rule 5)], the premisses distribute two 

 terms, namely, the subject of the universal premiss and the 

 predicate of the negative premiss. One of these distributed 

 terms must be the middle term (Rule 3) ; and the other must be 

 the major extreme, since this is distributed in the negative con 

 clusion (Rules 4 and 6). Hence the minor extreme, remaining 

 undistributed in the premisses, must be undistributed in the con 

 clusion ; that is, the conclusion must be particular (and negative). 

 Therefore, in every case when one premiss is particular the 

 conclusion must be particular. 



Another brief demonstration of the present corollary is based on the 

 previous corollary, combined with a consideration put forward in the preceding 

 chapter (148). Representing the propositions of a valid syllogism by A, B, 

 and C, respectively, we pointed out that the force of the syllogism may be 

 expressed by the hypothetical &quot; If A and B, then C&quot;. This may also be 

 expressed : &quot; Given A, if B then C,&quot; which yields the contrapositive &quot; Given 

 A, if C then B &quot;. In other words, if two propositions, A and B, prove a third, 

 C, then either of the two (A) and the denial of the third (Q will prove the 

 denial of the remaining one (B}. &quot; Now, if possible, let A (particular) and 

 B (universal) prove C (universal), then A (particular) and the denial of C 

 (particular) prove the denial of B. But two particulars can prove nothing.&quot; ] 

 Hence a universal and a particular can prove only a particular. 



1 DE MORGAN, op. cit., p. 14, using P, Q and R as symbols. 



