92 



traries X(-(Y and X)(Y, each, so far as it affects the other, reduces 

 its probability. 



It appears that a quantified term has a quantified contrary : that 

 of ' Every X ' is ' some .rs,' &c. 



The symbolic canon of validity is ; — if both middle parentheses 

 turn the same way, there need be one universal proposition ; if dif- 

 ferent ways, two.' Thus )))( and (•))•( both have inferences; and 

 so has )•(')) ; but )•()( has none. The symbolic canon of inference 

 is ; — erase all signs of the middle term, and what is left (two nega- 

 tions, if there, counting as an affirmation) shows the inference. Thus 

 from X(-)Y)-(Z we infer X(*-(Z or X((Z : more simply, from (•))*( 

 we infer ((. 



Section IV. On the symbolic forms of the system in which all the 

 combinations of quantity are introduced by arbitrary invention of forms 

 of predication (Sir W. Hamilton's). 



The modes of predication peculiar to this system have the same 

 symbols, )( and (•), as the peculiar propositions of the system of 

 contraries ; but with very different significations, as follows : — 



Contraries, 



(•) Universal negative with 

 particular terms, and affirmative 

 form in common language. 



All things are either Xs or Ys. 



)( Particular affirmative with 

 universal terms, andnegative form 

 in common language. 

 Some things are neither Xsnor Ys. 



Mr. De Morgan argues that Sir William Hamilton's system cannot 

 be called an extension of that of Aristotle, in the sense in which that 

 word is used. 



The forms of predication are as follows : — 



Invention of predicates. 



(•) Particular negative with 

 particular terms, not used in 

 common language. 



Some Xs are not some Ys. 



)( Universal affirmative with 

 universal terms, being declaration 

 of identity in common language. 



All Xs are all Ys. 



A! + A0( AllXs are all Ys 

 I j () Some Xs are someYs 



Aj )) All Xs are some Ys 

 A 1 (( Some Xs are all Ys 



E, )•( No Xs are Ys 

 — (•) SomeXsarenotsomeYs 

 O 1 )■) No Xs are some Ys 

 O, (•( Some Xs are no Ys. 



Previously to entering upon the forms of syllogism, Mr. De Morgan 

 repeats and reinforces the objections brought forward in his Formal 

 Logic; namely, that )( is a compound of )) and ((, and has no sim- 

 ple contradiction in the system ; and that (•) not only has no simple 

 contradiction, but cannot be contradicted except when the terms are 

 singular and identical. He then proceeds to propose one mode of 

 remedying these defects. Calling the ordinary proposition annular, 

 he proposes to make it exemplar, as asserting or denying of one in- 

 stance only. In the universal proposition, the example is wholly in- 

 definite, any one ; in the particular proposition it is not wholly indefinite, 

 some one. The defects of contradiction are thus entirely removed, as 

 in the following list, in which each universal proposition is followed 

 by its contradiction. 



