352 Mr DE morgan, ON THE SYLLOGISM, No. IV, 



In the same way other cases may be treated. But the entrance of contrary relations 

 renders the method of decomposition useless for every purpose except historical com- 

 parison. 



Except when both premises are negative, the conclusion can always be expressed in 

 terms of the premising relations, without contraries. Thus among the concluding forms 

 of III. 2, we see L~'M'. The following rules may be collected. First, the relations 

 converted in the conclusions belong to those premises which must be converted when 

 reduction is made into the first figure. Secondly, the mark of inherent quantity appears 

 in the ordinary form of conclusion only when the premises are of different qualities. Thirdly, 

 when the conclusion is expressed without contraries, this mark is always attached to the rela- 

 tion of the affirmative premise. These rules would give mechanical canons of inference, if 

 such things were wanted : and it would be well to remember that in the second figure the 

 middle term usually comes second in both premises, and the second premise is converted 

 in reduction into the first figure. 



I shall now proceed to the consideration of the quantified proposition and its syllogism, 

 presuming the reader to be acquainted with the notation and classifications of my second 

 and third papers. If we take the proposition 'Every X is an L of one or more Ys' we may 

 denote it by X )) LY : and similarly LY )) X may denote ' Every L of any Y or Ys is an X.' 

 And similarly for the other parts of the notation. 



I enter on this part of the subject only so far as to illustrate the ancient or Aristotelian 

 syllogism. Though of necessity a part of logic, as involving possible forms and necessary 

 connexions, the quantified syllogism of relation is not of primary importance as an explanation 

 of actual thought : for by the time we arrive at the consideration of relation in general 

 we are clear of all necessity for quantification. And for this reason : quantification itself only 

 expresses a relation. Thus if we say that some Xs are connected with Ys, the relation 

 of the class X to the class Y is that of partial connexion : that some at least, all it may 

 be, are connected, is itself a connexion between the classes. Nevertheless, it may be useful to 

 exhibit the modifying quantification as a component, not as inseparably thought of in the 

 compound ; though in this we must confine ourselves to what may be called the Aristotelian 

 branch of the extended subject. If we would enter completely upon quantified forms, we 

 must examine not only the relation and its contrary, but the relation of a term in connexion 

 with the relation of the contrary term. And here we find that all universal connexion ceases. 

 The repugnance of X and not-X or x, which, joined with alternance, is the notion the 

 symbols X and x were invented to express, cannot be predicated of LX and Lx : for Y .. LX 

 and Y..Lx may coexist. The complete investigation would require subordinate notions of 

 form, effecting subdivisions of matter. 



This complete examination would also require the investigation of the manner in which 

 quantity of relation acts upon quantity of term : and this whether the quantity of relation be 

 inherent or not ; including an examination of all syllogisms in which inherent quantity of 

 relation appears in the premises. And thus in logic, as in mathematics, the horizon opens 

 with the height gained : generalisation suggests detail, which again suggests generalisation, 

 and so on ad infinitum. There is no more limit to the formulae of thought than to the 



