'.)■■'> 



)( 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 someone 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 copula? as sup-port, govern, is greater 

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

 convertibility , seen in such copulse 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- 

 itive, but not convertible, every syllogism remains valid, provided 

 that the correlative of that copula be used instead of it, when needful. 

 And in this consists, according to Mr. De Morgan, the root of the 

 doctrine of figure. If -+- represent affirmative, and — negative, the 



four figures are connected with + +, H , (-, and (in the 



system of contraries, where negative premises may have a valid con- 

 clusion, the fourth figure has equal claims with the rest, though the 

 conditions of all the figures are singularly altered). These forms 



do not require the correlative copula : thus -\ in the second figure 



(as Cameslres and Baroho among the Aristotelian forms) are as valid 

 when the copula is 'supports' or 'is greater than,' as when 'is' is 

 employed. But in every other case the rule for the proper intro- 

 duction of the correlative copula is as follows : — The preceding being 

 called the primitive forms of the four figures, when one premise of a 

 primitive form is altered, the necessity of a correlative copula is 

 thrown upon the other ; when both, upon the conclusion. Thus, the 



primitive form of the second figure being -\ , and Cesare showing 



h , it is only valid with the copula ' governs,' by making ' is not 



governed by ' the copula of the conclusion, as follows : — 

 No Z governs any Y 

 Every X governs a Y 

 Therefore no X is governed by any Z. 

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



