464 Mr DE morgan, ON THE SYLLOGISM, No. V. AND 



form X(( )•( ))Z affirms that no deficient of X is a species of Z, and affirms 'any X is any 

 Z ', or denies that ' Some X is not some Z'. 



Again, X(.( ())-)Z expresses that some class is deficient both of X and Z. To deny 

 it is to say that every class is genus either of X or of Z, which gives only two individuals to 

 the universe, one X and one Z. 



The law which regulates these cases is very easily given. First, when the invalid form 

 is either a universal and a particular, or two particulars of unbalanced middle terms, as 

 )) (), )•()•), or )•))•)) ())(• Let there be two singular and identical terms, of course with 

 penultimate and identical contraries. When X is particular, let it be one of the singular 

 terms ; when X is universal, let it be one of the contraries : and the same for Z. The propo- 

 sition which the (hitherto) invalid form denies is then constructed. Thus, writing down the 

 instances of the universe, with their designations, we have four cases, under which are written 

 all the combinations which deny them. 



X X X X X... X XXXX... X XXXX,.. X x x x x.., 



Z z z z z... Z z z z z... z Z Z Z Z... z Z Z Z Z... 



(()•) (■)()' 

 (•()) ()(•) 



())•) (-(OJ 



))() )-()-)^ 

 t"'' )()) )•)(•) 

 )(() )•))•) 



))(•( )•()(! (()( (•)(•(] 

 f!"^ )■)(( )()-(^"y (((•()•( '''"y 



)()•) )■)){} ())( (-((-(J 



Thus X )) () Z, or 'a middle term is both genus of X and partient of Z' denies that Z 

 is singular and X its contrary: and the same of five others, )•( )•) &c. Secondly, when the 

 invalid form has two particulars with balanced middle terms, let terms and contraries be both 

 singular ; the cases in which X and Z have balanced quantities deny that X and Z are 

 contraries, the cases in which X and Z have unbalanced quantities deny that X and Z are 

 identical. Thus 



-or ^^ is denied by ()(), ) ( ) (, (•()•), )•)(■( 



:S or ^^ is denied by ()(■(, )()•), (•()(, )•)(). 



To produce the forms which affirm the restrictives, we must have recourse to the 

 secondary ) • ( . 



I return to the cases which are without restriction. There are three balances, which I 

 I shall call primary, secondary, and tertiary. The primary balance is even when the primary 

 relations are both universal or both particular; uneven in other cases. The secondary balance 

 is even when the spicula" of the secondary relation are both universal or both particular ; 

 uneven in other cases. The tertiary balance is even when the primary relations are both 

 Aristotelian', or both otherwise; uneven in other cases. 



1. When the primary and secondary balances are of the same name, both even or both 

 uneven, the primaries agree with their adjacent means or differ from them, according as the 



' These are species, exient, external, pattient ; )), (■(, )•(, (); lesser universal or greater paiticalar; the first spicula of 

 the same name as the proposition. 



