196 MEMOIR OF AUGUSTUS DE MORGAN. 



1846. subject, so far as their confusion on this point entitles one to say 

 they speak one way or the other, speak ideally, and not objectively. 

 Nay, more, they even admit contradictory propositions as ideally 

 enunciable, and subject to contradiction like others. Thus, 

 * every collection of two and two is five ' is properly convertible 

 into * some fives are collections of two and two.' Accordingly 

 they give and take no denial except contradiction ; nothing with 

 them overturns ' every A is B,' except ' some A's are not B's.' 

 But when we come to apply Logic to the working wants of the 

 mind we find another kind of denial, namely, denial by non- 

 existence ; necessary non-existence, or contingent, as the case 

 may be. When we speak objectively, there may be denial by 

 contingent non-existence perfectly distinct from denial by con- 

 tradiction. Thus objectively I deny that 'all unicorns are 

 animals,' not by saying that there are unicorns which are not 

 animals, but by saying that there are no such objects as uni- 

 corns ; and so far as a unicorn is not, so far it cannot be animal, 

 or anything else. Ideally, I admit, unicorns are animals ; my 

 notion is the notion of animal. 



I distinguish, then, denial of the terms from denial of the 

 copula. 



A is B ideally, objectively, or (say) x-itively. 

 No ! for A has no x-itive existence. 

 No ! for B has no x-itive existence. 



No ! for the x-itive existence of A and B belongs to is not, 

 not to is. 



Formal Logic usually is made only to treat of the copula. To 

 be strictly formal I need not introduce ideal and objective, more 

 than English and French, black and white, x and y. Two species 

 ot existence implied as belonging to the terms brought forward 

 would do as well. But ideal and objective is the important dis- 

 tinction in practice, and as to assertion or denial, so far as I 

 want it, is easy. 



I should now ask you to consider some phraseology. 



There are seven definite relations of term and term. I do not 

 call x ) y definite, for it consists equally well with y ) x and 



y- x - 



1, 2. Start with identical and contrary, complete co-existence 

 or complete mutual exclusion containing all things between them. 

 As (man being the universe) North Briton and Scotsman, or 

 Briton and alien. 



