150 Cambridge Philosophical Society. 



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 + + , -\ , f- > 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 Camestres and Barolco 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 



\- , 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. 



By an additional letter (g) introduced into the usual words of 

 syllogism, the places of the correlative copula may be remembered, 

 as in Barbara, Celagrent, &c. : a g being made to accompany any 

 member of the syllogism in which the correlative copula must be 

 employed. 



This theory is applied equally to the Aristotelian system, to Sir 

 William Hamilton's (though not of universal application in the cu- 

 mular form), and to Mr. De Morgan's system of contraries. The 

 extensions required by the use of a merely transitive copula, in the 

 last-mentioned system, are discussed ; and mention is made of the 

 tricopular system, in which the leading copula and its correlative 

 have an intermediate or middle relation, equally connected with 

 both ; as in > = and «< of the mathematicians. 



The next step is the assertion that it is not necessary that any 

 two of the three copulse of a syllogism should be the same ; all that 

 is requisite is that, in affirmative syllogisms, the copular relation in 

 the conclusion should be compounded of those in the premises. The 

 instrumental part of inference is described by Mr. De Morgan as the 

 elimination of a term by composition {including resolution) of relations, 

 which leads to the conclusion that whenever a negative premise occurs, 

 there is a resolution of a compound relation . This resolution is shown 

 in a case (among others) of the ordinary copula, in which, however, 

 it would hardly strike the mind more forcibly than would the pro- 

 perties of powers in algebra if every letter represented unity. Mr. 

 De Morgan shows (in an addition) that in some isolated cases of in- 

 ference which are not reducible to ordinary syllogism, logicians have 

 had recourse to what amounts to composition of relations. 



Mr. De Morgan next points out that the copular relation, 'in 

 affirmative propositions, need not be restricted as applying to one 



