Cambridge Philosophical Society. 149 



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 : — 



Aj + AOC All Xs are all Ys 

 i! () SomeXsaresomeYs 



A l )) All Xs are some Ys 

 A 1 (( Some Xs are all Ys 



E, )-(NoXsare Ys 

 — (•) Some Xs are not some Ys 

 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 zvholly 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. 



)( Any one X is any* one Y 

 (•) Some one X is not some one Y 

 )) Any one X is some one Y 

 (•( Some one X is not any one Y 



(( Some one X is any one Y 

 )•) Any one X is not some one Y 

 )•( Any one X is not any one Y 

 () Some one X is some one Y 



In both systems there are thirty- six valid syllogisms, and in both 

 the canon of validity is, — one universal (or wholly indefinite) middle 

 term, and one affirmative proposition. But the symbolic canons of 

 inference differ as follows (with reference to the order XY, YZ, XZ). 



Exemplar system. — Erase the middle parentheses, and the symbol 

 of the conclusion is left : thus ())•) gives (•). 



Cumular system. — Erase the middle parentheses, and then, if both 

 the erased parentheses turn the same way, turn any universal paren- 

 thesis which turns the other way, unless it be protected by a mark 

 of negation. Thus )•(() gives )•), ())( gives (), and ())•( gives (•(. 



Section V. On the theory of the copula, and its connexion with 

 the doctrine of figure. — In his work on Formal Logic, Mr. De Morgan 

 had analysed the copula, and abstracted what he calls the copular 

 conditions of the relation connecting subject and predicate. These 

 are, transitiveness, seen in such copulas as support, govern, is greater 

 than, &c, ex. gr. if A govern B, and B govern C, A governs C : and 

 convertibility, seenin such copula? as is acquainted with, agrees with, &c. ; 

 ex. gr. if A agree with B, B agrees with A. Mr. De Morgan's position 

 is, that any mode of relation which satisfies both these conditions has 

 as much claim to be the copula as the usual one, is, which derives its 

 fitness entirely from satisfying the above conditions. So far the 

 work cited. In the present paper the correlative copula is introduced, 

 as is supported in opposition to supports, &c, and every system of 

 syllogism is thus extended. If a copula be taken which is only trans- 



* So that there can be but one X and one Y, and that X is Y. 



