476 



Mb DE morgan, ON THE SYLLOGISM, No. V. AND 



Suppose a universe of six individuals, of which the proper names, the representatives 

 of singular attributes, are 1, 2, 3, 4, 5, 6. Let us consider the class X, or 1, 2, 3. This class 

 has the parts, 1, 2, 3, 12, 23, 31, 123. It has the wholes, 123, 1234, 1235, 1236, 12345, 12356, 

 12346. But 123456, though a whole of 123, is not a term dividing the universe, which has six 

 lowest parts, 1, 2, &c, and six highest wholes 23456, 13456, &c. Every selection is to have its 

 name, which may equally designate the class and the attribute by which the class is distin- 

 guished ; being at once the instrument of cumulation and of distinction : of cumulation, when 

 one individual of the class is coupled with another; of distinction, when one in the class is 

 separated from one of the externals. 



Every point of correlation is seen, or wilP be seen, to be perfect. The individuals being 

 non-partient of each other, we may designate the class 1, 2, 3, by 1 +2 + 3. We have no 

 symbol in mathematics which may by analogy be employed to designate the attribute of this 

 class ; nothing which suggests ' the common mark of 1, 2, 3, and of them alone ' : except so far 

 as this, that A,23 is sometimes used in such a sense, inter alia; which may therefore denote 

 the common attribute. Suppose we describe the class 1+2 + 3 as the aggregate of 1+2 and 

 3: what is the correlative mode of describing its attribute in terms of the compound of two? 

 The answer is that a class may be described by its contrary, and ' that the alternative attribute 

 of As456 Ai2455 ' Is the description of the attribute required. The relations of aggregation and 

 composition are closely connected with those of direct and contrary : thus the propositions 

 'C is aggregate of A and B ' and ' not-C is compound of not-A and not-B' are convertible. 



The following is an example of the correlation of propositions. 



X) • (Y. No X is Y ; everything either 

 X or y : X and Y have no common part : but, 

 if not complements, have common wholes. 

 Every individual is in some of the parts either 

 of X or of y : and is either not in some whole 

 of X, or not in some whole of Y. That is, 

 no junction of a new attribute selects any 

 part of one out of the other: everything 

 Wants some attribute of one or the other. 



X(')Y. No X is y : everything either 

 X or Y : X and Y have no common^ whole: 

 but, if not externals, have common parts. 

 Every individual is in all the wholes either 

 of X or of Y, and is either not any part of x, 

 or not any part of y. That is, no dismissal 

 of an existing attribute makes any whole of 

 one a whole of the other : everything has all 

 the attributes of one or the other. 



' A qualification rendered necessary by the smallness of 

 the number who tliink of such distinctions. That esse is per- 

 cipi is especially true of the esse in anima. The logical eye of 

 the mathematician, and the mathematical eye of the logician, 

 are yet to be opened. The cultivators of beth the sides of 

 exact science seem to proceed upon the notion that distinct 

 vision is not possible with both eyes together. Some contend 

 for the right eye, some for the left: and the voice of mankind 

 finds no utterance ; for parmi les aveugles un bergne est roi, 

 let him have which eye he may. 



' One word more on this stumblingblock. All terms have 

 a common whole, the universe: but this is not a whole term. 

 The logician does not see why the universe is to be excluded, 

 nor can he see until his mathematical eye is open. But he 



excludes it in his own system, and very easily, by never in- 

 venting a name for it. ' Every thing that exists ', ' the omne 

 cogitabile', are opposed in thought to their contraries, the 

 non-existing, and the incogitabile. Where is the name that 

 includes both the existing and the non-existing, the thinkable 

 and the unthinkable ? Let this name be shown, and shown in 

 use, and then I shall be open to the charge of correcting the 

 old logic : but I think I have only imitated it. In a full work 

 on logic, the universal name might be discussed in the chapter 

 which treats of reslrictives and other extremes, not forgetting 

 vacuous names. But in the logic of the term — distinctive 

 name — the garter is as useless when all the thistles have it 

 as it would be if none had it. 



