112 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



copula 'is 1 partakes of the verb of identity on which, as above remarked, they test all their 

 inferences. When the copula of identity is employed, it is of necessity that one subject example 

 can only agree with one predicate example : any one man IS, merely because he IS one man, 

 only one animal. Extend the copula to a meaning short of perfect identity, and the copular 

 relation may belong, in an affirmative, to more than one example of the predicate. For instance, 

 when the copula is agreement in colour, ' Every X is F 1 may mean that each X is of the 

 colour of one or more of the Fs. The particular character of the predicate, that is, the 

 description of its extent as being ' one or more, possibly all ' may belong to it, independently 

 of the number of instances of the subject, in right of, and by derivation from, one instance of 



the subject. And the proposition • one X Every V may be true without our being 



obliged to say that there is therefore only one F in existence, by reason of the relation indi- 

 cated by existing between that one X and every F in existence. 



Hence follows an extended mode of interpreting Sir William Hamilton's system into con- 

 sistency. Let the quantity of the predicate be determined by the least number of instances of 

 the predicate which are declared to stand in the copular relation to each instance of the subject : 

 it being understood that the declaration of the copular relation existing between a certain X 

 and a certain Y does not deny that the relation may exist between that X and other Fs, or 



between other Xs and that Y. Thus, "All Xs all Fs," or X)(Y, now means that 



every X stands in the copular relation to each and every F. If there were ten Xs and ten Fs, 

 Sir William Hamilton's proposition would assert ten agreements : the form here proposed 



declares a hundred. And it is now contradicted by X(.)Y or " some Xs some Fs :" for 



either each X and each F agree, or some Xs do not agree with some of the Fs. 



This view of the subject meets another objection of mine to the system as proposed. It 

 can no longer be asserted that X) (F is a proposition compounded of two others of the system, 

 X))Y and Y))X: for each X may agree with some of the Fs, and each F with some of the 

 Xs, without every X agreeing with every Y. 



There is however a distinction to be drawn. When the quantity of the affirmative predi- 

 cate is determined solely by the number of instances which each instance of the subject is related 

 to, let us say that the predicate has exemplar quantity. But in the common system, in which 

 the quantity of the predicate is cumulated from all the instances of the subject, let us say that 

 it has cumular quantity : this second system permitting, like the first, each instance of the 

 subject to be related to more than one instance of the predicate. Thus, quantity being 

 exemplar, we must say of ' Every man is an animal ' that the predicate is singular : quantity 

 being cumular, that the predicate is arithmetically coextensive with the subject. But when it 

 is " Every man — agrees with in being carnivorous — a brute," the predicate is not singular, 

 even when its quantity is exemplar. 



[An extension of the numerically definite system here arises, which I shall not stop to 

 investigate the consequences of. If j- be the whole number of A"s in the universe, r\ the num- 

 ber of Fs, and n the number of Fs (at least) to which each of mXs is related ; them mn is the 

 least number of relations asserted. If mn be greater than (£ - l)rj, then mn - (£ - l)»y at least 

 of the r/ Fs must be related to £ Xs, that is, to all the Xs. Now as n cannot exceed r\, m must 

 be £ : and a necessarily convertible proposition may then exist, if n exceed t] - q -j- £.] 



