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De Morgan calls it the arithmetical whole, transferring the word 

 mathematical to what was called the logical whole. The common 

 mode of expression, as 'Every A is B/ &c, he considers as speaking 

 the language of the arithmetical whole, though the speaker may 

 attach the idea of either of the other wholes. • 



Extension predominates in the mathematical whole ; intension in 

 the metaphysical. The most usual mode of speech is the physical : 

 man is educated a mathematician as to the subject of his proposition, 

 a metaphysician as to the predicate. 



The most remarkable point at issue between Mr. De Morgan and 

 the logicians, is in his opposition to their notion of the whole attribute 

 being the sum of its components. The difference between aggregation 

 and composition is one of the turning-points of his whole system. 



The distinctions above drawn require differences of language to 

 express the relations which enter : the logicians have nothing but the 

 copula is. At the outset, however, we have the distinction which is 

 expressed by speaking of relations of terminal ambiguity and relations 

 of terminal precision : — the first seen in ' A is contained in B,' where 

 it is left unsaid whether or no A fills B ; the second seen in the case 

 in which it is implied that A is part only of B. 



By speaking in the arithmetical whole, the logicians have made a 

 system of syllogism from which the numerical syllogism cannot be 

 excluded. The propositions ' Some As are Bs,' and ' 50 As are 

 Bs,' are of the same kind : they are both referred to the arithmetical 

 whole. This whole is subordinate to both the mathematical and 

 metaphysical wholes ; though more prominent in the first than in 

 the second. 



When inclusion and exclusion are opposed to one another, and 

 combined with assertion and denial, the ordinary proposition takes 

 a form in which quantity is but an emergent incident, and not a 

 fundamental mode of discrimination. Thus the propositions A and 



are the assertion and denial of the inclusion of class in class ; E and 



1 are the assertion and denial of the exclusion of class from class. 



The opposition of the two kinds of quantity, extensive and inten- 

 sive, is not easy and natural, when the word quantity is used in 

 metaphysical reading. Mr. De Morgan proposes the word force 

 to express quantity in the second case. He finds this word in use. 

 Thus it is sometimes said that a term is or is not used in its com- 

 plete force, when the meaning is, that all the attributes of which 

 the term is compounded are or are not involved in the use made of 

 the term. This is, according to Mr. De Morgan, one of the cases 

 in which the logical system of the world at large has got beyond 

 that of the logicians. 



The spicular notation of the former papers is extended : the signs 

 ) and ] being used to distinguish mathematical and metaphysical 

 reading. Then X))Y signifies that the whole class X is contained 

 in the class Y; X]]Y signifies that the whole attribute Y is a com- 

 ponent of the attribute X. 



The syllogism is the deduction of a relation from the combination 

 of two others. By distinction of the mathematical and metaphysical, 



