148 



Cambridge Philosophical Society. 



Affirmative 



Negative 



Affirmative 



Negative 



{ 



A, 



A 1 



{a 



frequently distinguishes this system from Sir W. Hamilton's by calling 

 the former that of introduction of contraries, the latter that of inven- 

 tion of predicates. For distinctness, it may be stated that Mr. De 

 Morgan's other, or numerically definite system (the one concerned in 

 the discussion), does not appear in the present paper, except as matter 

 of allusion. 



The forms of predication in this system are as follows, with refer- 

 ence to the order XY, x and y being not X and not Y. 



Universalis. 

 X))Y Every X is Y 



x))y orX((Y Every Y is X 

 X))y orX)-(Y No X is Y 

 x))Y or X(-)Y Everything is X or Y or both. 



Particulars. 

 X()Y Some Xs are Ys 



cc()y or X) ( Y Some things are neither Xs nor Ys 

 XQy or X(-(Y Some Xs are not Ys 

 xQY or X)-)Y Some Ys are not Xs. 



Various rules of connexion are given, being all translations of 

 those in the work on Formal Logic, except a classification of the par- 

 ticulars by probability, answering to that of universals. Thus of 

 X))Y and X(«)Y, each makes the other impossible : of their con- 

 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 crs,' &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 thing save neither Xs nor Ys. 



| 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. 



