132 Cambridge Philosophical Society. 



of the conclusion does not alter the syllogism. Neither of these 

 forms are in the Aristotelian list. One of them is 



X)Y+Z)Y-xz; 



or if every X be aY, and also every Z, then there are things which are 

 neither X nor Z ; namely, all which are not Ys. 



2. A set of six, in which the interchange produces really different 

 syllogisms of the same form, and in which both premises and con- 

 clusion can be expressed in terms of three names, without the con- 

 trary of either. This set includes the whole Aristotelian list, except 

 those in which a weaker premise will give as strong a conclusion, or 

 the one in which the same premises will give a stronger conclusion. 



3. A set of six resembling the last in everything but this, that no 

 one of them is expressible without the new forms e and i ; that is, 

 requiring three names and the contraries of one or more of them. 



Those of the third set are not reducible to Aristotelian syllogisms, 

 as long as the eight standard forms of assertion are adhered to. 



The second theory of the syllogism has its principles laid down in 

 the memoir before us ; but those principles are only applied to the 

 evolution of the cases which are not admitted into the Aristotelian 

 system. The formal statement of the manner in which the ordinary 

 cases of syllogism are connected with those peculiar to this second 

 system is contained in an Addition. 



In providing that premises shall certainly furnish a conclusion, 

 the common system requires that one at least of the premises shall 

 speak universally of the middle term ; that is, shall make its asser- 

 tion or denial of every object of thought which is named by the middle 

 term. Mr. De Morgan points out that this is not necessary : m 

 being the fraction of all the cases of the middle term mentioned in one 

 premise, and n in the other, all that is necessary is that m + n should 

 be greater than unity. In such case, the real middle term, being 

 the collection of all the cases by comparison of which with other 

 things inference arises, is the fraction m + w — 1 of all the possible 

 cases of the middle term. Thus, from the premises ' most Ys are 

 Xs ' and ' most Ys are Zs,' it can be inferred that some Xs are Zs, 

 since m and n are both greater than one-half. The assignment of 

 definite quantity to the middle term in both premises, gives a canon 

 of inference, of which the Aristotelian rule is only a particular case. 



In the addition above alluded to, this same canon, namely ' that 

 more Ys in number than there exist separate Ys shall be spoken of 

 in both premises together,' is made to take the following form : — If 

 in an affii'raation or negation, in ' As are Bs ' and ' As are not Bs,' 

 definite numerical quantity be given to both subject and predicate, if 

 it be stated how many As are spoken of and how many Bs — the 

 number of effective cases of the middle term is seen to be the nnici- 

 her oi subjects in an affirmative proposition, whether the middle term 

 be subject or predicate. Hence, defining the effective number of a 

 premise to be the number of subjects if the proposition be affirmative, 

 and the number of cases of the middle term if it be negative, all 

 that is necessary for inference (over and above the usual condition 



