THE THEORY OF SYLLOGISM, ETC. 



101 



A' 



4 



o; 



A 



i 



A' 



a 



E' 



) ( Any one X is any one Y giving 



(.) Some one X is not some one F 



) ) Any one X is some one Y 



(.( Some one X is not any one Y 



( ( Some one X is any one Y 



).) Any one X is not some one Y 



).( Any one X is not any one Y 



( ) Some one X is some one Y 



giving 



giving 

 giving 

 giving 

 giving 

 giving 

 giving 



There is but one X and one Y, and X 



is Y. 

 There can be found some one X and some 



one Y which are not the same. 

 All Xs are Fs. 

 Some Xs are not Fs. 

 Every Fis X. 

 Some Fs are not Xs. 

 No X is F. 

 Some Xs are Fs. 



The detailed notation needs no explanation. The form given to the old notation may be 

 explained thus. In the system of contraries, the accents refer to the relation of standard terms 

 and contraries. Thus O t being X(.(Y, O' is a?(.(y or Y(.(X. In the exemplar system, the 

 accent refers to the example, sw&accent to subject, superaccent* to predicate. Every accent 

 whicli occurs shews that its term is named after the proposition. Thus in /' both subject and 

 predicate are like the proposition, particular or not wholly indefinite : but in A only the 

 subject is, like the proposition, universal or wholly indefinite. The E' and /' of the system of 

 contraries cannot exist here, but E' and /' have double accents, and we have the new proposi- 

 tions A' and O', which have similar selective forms in both terms. 



The non-entrance of contraries keeps the following rules of syllogism intact. The middle 

 term must be indefinite in one of the premises — negative premises give no conclusion. But on 

 the other hand, the system of contraries acknowledges the rule that particular premises give no 

 conclusion — which is not a rule of the exemplar system — and does not acknowledge the two 

 first named. 



Taking the 64 cases of combination of premises, in any one figure, we must then reject 

 16 cases of negative premises, and 12 others in which, the premises not being both negative, 

 both the middle terms are not indefinite. There remain 36 cases for examination, and all admit 

 of inference : so that the canon of validity is ; — one affirmative premise and one indefinite 

 middle term. This is also the case when exemplar forms are abandoned in favour of cumular 

 ones : but the conclusions in the second (or Sir William Hamilton's) form are not always the 

 same as those in the first. 



The validity of every case in which there is one indefinite middle and one premise is affirma- 

 tive, immediately appears : the any one Y of one premise may be taken to be the some one Y 

 of the other : and the same is then compared with two others, both giving agreement, or one 

 agreement and one disagreement. Further it will appear that in every case, the selective forms 

 of the terms of the conclusion are the same as they had in the premises. Thus ' any one X is 

 some one F, any one F is some one Z' gives ' any one X is some one Z,' for whatever some one 

 F the first premise can allow, the second premise makes it some one Z : and similarly of others. 



* Subject, as used in reference to a proposition, is liable to 

 some objections. He would be a bold man who would dare to 

 substitute subdict and superdict for subject and predicate : or, 



retaining subject, to call the predicate the superject : but a 

 word on the rashness of this imaginary individual may help 

 the reader in remembering the use of the accents. 



