AND ON THE LOGIC OP RELATIONS, r Z 



341 



now proceed to consider the formal laws of relation, so far as is necessary for the 

 treatment of the syllogism. Let the names X, Y, Z, be singular : not only will this 

 be sufHcient when class is considered as a unit, but it will be easy to extend conclusions 

 to quantified propositions. I do not use the mathematical symbols of functional relation, 

 <^, >|/, Sec : there are more reasons than one why mathematical examples are not well 

 suited for illustration. The most apposite instances are taken from the relations between 

 human beings : among which the relations which have almost monopolized the name, 

 those of consanguinity and affinity, are conspicuously convenient, as being in daily use. 



Just as in ordinary logic existence is implicitly predicated for all the terms, so in 

 this subject every relation employed will be considered as actually connecting the terms 

 of which it is predicated. Let X..LY signify that X is some one of the objects of 

 thought which stand to Y in the relation L, or is one of the Ls of Y. Let X.LY 

 signify that X is not any one of the Ls of Y. Here X and Y are subject and predicate: 

 these names having reference to the mode of entrance in the relation, not to order of 

 mention. Thus Y is the predicate in LY.X, as well as in X.LY. 



When the predicate is itself the subject of a relation, there may* be a composition: 

 thus if X..L(MY), if X be one of the Ls of one of the Ms of Y, we may think of X 

 as an ' L of M' of Y, expressed by X..(LM)Y, or simply by X..LMY. A wider treatment 

 of the subject would make it necessary to eiTect a symbolic distinction between ' X is not 

 any L of any M of Y' and ' X is not any L of some of the Ms of Y.' For my 

 present purpose this is not necessary : so that X.LMY may denote the first of the two. 

 Neither do I at present find it necessary to use relations which are aggregates of other 

 relations : as in X..(L,M)Y, X is either one of the Ls of Y or one of the Ms, or both. 



We cannot proceed further without attention to forms in which universal quantity 

 is an inherent part of the compound relation, as belonging to the notion of the relation itself, 

 intelligible in the compound, unintelligible in the separated component. 



First, let LM' signify t an L of every M, LM'X being an individual in the same 

 relation to many. Here the accent is a sign of universal quantity which forms part of 

 the description of the relation : LM' is not an aggregate of cases of LM. Next let L^M 

 signify an L of an M in every way in which it is an L at all : an L of none but Ms. 

 Here the accent is also a sign of universal quantity : and logic seems to dictate to 

 grammar that this should be read 'an every-L of M.' The dictation however is of 



* A mathematicinn may raise a moment's question as to 

 whether L and M are properly said to be compounded in the 

 sense in which X and Y are said to be compounded in the 

 term XY. In the phrase brother of parent, are brother and 

 parent compounded in the same manner as white and ball in 

 the term white ball. I hold the affirmative, so far as concerns 

 the distinction between composition and aggregation : not de- 

 nying the essential distinction between relation and attribute. 

 According to the conceptions by which man and brute are 

 aggregated in animal, while animal and reason are com- 

 pounded in man, one primary feature of the distinction is that 

 an impossible component puts the compound out of existence, 

 an impossible aggregant does not put the aggregate out of 



existence. In this particular the compound relation ' L of 

 M ' classes with the compound term ' both X and Y.' 



•f Simple as the connexion with the rest of what I now pro- 

 ceed to may appear, it was long before the quantified relation 

 suggested itself, and until this suggestion arrived, all my efforts 

 to make a scheme of syllogism were wholly unsuccessful. The 

 quantity was in my mind, but not carried to the account of 

 relation. Thus LX )) MY, or every L of X is an M of Y, 

 has the notion of universal quantity attached in the common 

 way to LX, not to L: its equivalents X..L-',MY, and 

 Y..M-' L'X, shew X and Y as singular terms, and though 

 expressing the same ideas of quantity as LX))MY, throw 

 the quantity entirely into the description of the relations. 



