AND ON LOGIC IN GENERAL. 195 



dislike to negation, and did his best to avoid subtraction : he used artifices contra vitium 

 negationis ; but algebra beat him. 



Philosophy, again, which scorned the idea of the el^os et^t/ccoraxoi' descending to the indi- 

 vidual, as savouring of the arithmetical whole, had to work in this whole through all the syllo- 

 gism, partly because she insisted on raising the yevo^ yeviKwraTov up to the whole universe of 

 thought. When all and some were selected as the exponents of quantity, no reason was ever 

 given for the exclusion of most and fewest, or of any of the signs of definite ratio of part to 

 whole called fractions. A reason could have been given, and I shall come to it ; but philo- 

 sophy never gave it, within my* reading: the light of second intention shone but dimly into 

 the arithmetical whole. The considerations which I shall proceed to offer are destructive of 

 the right of the numerical syllogism to a place in the logic of the two opposed wholes: but 

 they equally destroy the right of the common form of syllogism, under both quantifications, 

 natural and postulated. All go together to the arithmetical whole, in which all are formed, 

 whether the some be more or fewer, vague or definite, all, some perhaps all, some not all, most, 

 nearly all, two-sevenths exactly, two-sevenths or more, two-sevenths or thereabouts. I cannot 

 see how it is possible, under any effective understanding of the difference between first and 

 second intentions, to deny that the common proposition speaks by first intention ; though of 

 course those who use it may think of class or attribute. The arithmetical whole is subordi- 

 nated, though with different degrees of affinity, both to the mathematical and to the meta- 

 physical wholes : but the habit is to refer it to the former, and for a reason we shall see. 



XIII. Taking the mathematical form first, and dichotomising the universe in two 

 ways, by classes X and not-X (x), and Y and y, the onymatic relations of class to class 

 can be predicated without any explicit dichotomy of class, whether vague or definite. The 

 relations are inclusion and exclusion (inclusion in contrary); the judgments, assertion and 

 denial. 



Mathematical Form. Arithmetical Form. 



A I assert the inclusion of X in Y Every X is Y 



I deny the inclusion of X in Y Some Xs are not Ys 

 E I assert the exclusion of X from Y No X is Y 



1 I deny the exclusion of X from Y Some Xs are Ys. 



Here the classes X and Y are treated in their philosophical unity : the common reading repre- 

 sents objective verification. What is your right to deny the exclusion of X from Y ? Answer, 

 this X is a Y, and this, and this, &c. 



Quantity is here of no fundamental account : but if not a root, it must be a branch. The 

 objective verifications tell us in the common way the story of universal and particular terms. 

 But the objective and enumerative character has led to much extra-logical discussion on the 



* There is hardly an imaginahle speculation on thought 

 which is not to be found in the vast number of large volumes 

 by powerful authors which have descended to us. It is not an 

 uncommon mode of replying to a claim for the introduction 

 of this or that into logic, that some Optimus Albinus or 

 Pessimus Niger — as the case may be — mentioned the matter at 

 gome date preceding 1600. With this I have nothing to do. 



That an opinion has been held before now, and has not gained 

 room in the quod semper, &c. is no argument at all : and if it 

 were, it would come with no effect from those who are now 

 pressing the point that the whole of one side of logic, though 

 known to and hinted at by both the illustrious writers above 

 mentioned, has never been put in its proper place. 



25—2 



