ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 463 



to think of the middle term 'animal', the theologian of the middle term 'sinner'; but to 

 both it is enough for the conclusion that a middle term exists. This explicit reduction 

 of the middle term to mere existence is, I thinlc, essential to the formal consideration of 

 the syllogism. 



In such a proposition as )) )•( (•(, the spiculae being, 12, 34, &&, let 12, 56, be primary 

 relations, 34 the secondary relation, relation of the second order, or relation of relations. Let 

 the spiculae 3, 4, be means ; 2, 5, adjacents ; 1, 6, extremes. Take notice that the secondary 

 relation is the common identification, or its denial: thus 12))56 is not '2lX species of 56Y', 

 but 'Any 21 X is one 56Y, some 56 Y'. 



Of 512 secondary propositions, 256 are valid representations of unrestricted onymatic 

 forms : the remaining 256 are either assertions or denials of restrictives. The unrestricted 

 forms may' be obtained as follows: 32 of them are the forms of syllogism, with blank 

 middle terms, and the secondary (); 32 more are the contradictions which deny that a 

 middle term can be found, with the secondary )■(. Three other sets of 64 each are 

 found by varying the readings of the first 64 in the same manner as X()Y and X) • (Y are 

 varied by use of x and y for X and Y. Thus, the proposition X((Z being a necessary 

 consequence of X() Y)-(Z is an equivalent of X() 0)(Z, and of X() () )(Z. That is, 

 ' X genus of Z ' is an equivalent of ' Some complement of X is some external of Z '. The 

 denial is 'Any complement of X is not any external of Z', X() )( )(Z, which is denial 

 of X((Z, or an equivalent of X).)Z, or 'X deficient of Z '. The eight varieties, four of each 

 proposition, are as follows, relaxing the exemplar form into ordinary reading. 



X(') ( ) ) '(Z Some complement of X is external of Z 



X(*) (• ( ()Z Some complement of X is not partient of Z 



-^) ( ) ( 0^ Some class is neither coinadequate of X nor partient of Z 



X) ( )•) ) -(Z Some external of Z is not coinadequate of X. 



Here are four secondary ways of saying ' X is genus of Z '. Again, 



X(0 )■( )'(Z No complement of X is external of Z 



X(') ) ) ( )Z Every complement of X is partient of Z 



X) ( (') ()2 Every class is either coinadequate of X or partient of Z 



X)( ( ( )*(Z Every external of Z is coinadequate of X. 



Here are four secondary ways of saying ' X is deficient of Z\ 



We can now give meaning to the 32 compositions which fail to show valid conclusion : 

 they are all denials of restrictives. For instance X (•( Y )) Z gives no conclusion : and this 

 is X (•( ))Z. There is a term, says the proposition, which is both deficient of X and 

 species of Z. Of course there is, will be the first reply ; must every species of Z fill up X ? 

 Certainly not, unless every individual of Z be all X ; that is, unless Z and X be singular and 

 identical. Consequently, X("( Y ))Z has a conclusion; it denies 'Any X is any Z '; and 

 we have one of Hamilton's syllogisms, when the non-partitive ' some ' is used. The secondary 



> I did not obtain them so easily, for I worked through the i The reader may thug be made more sure of the completeness 

 512 cases separately and independently, before I saw what, of my investigation, 

 when seen, was aUo seen to be what ought to have been seen. | 



Vol. X. Part II, 69 



