AND ON THE LOGIC OF RELATIONS. 



343 



Converses of contraries are contraries: thus L~' and (not-L)"' are contraries. For since 

 X..LY and X..not-LY are simple denials of each other, so are their converses Y..L~'X and 

 T..(not-L)~^X; whence L~' and (not-L)~^ are contraries. 



The contrary of a converse is the converse of the contrary : not-L~' is (not-L)"'. For 

 X..LYis identical with Y.not-L~'X and with X.(not-L)Y, which is also identical with 

 Y.(not-L)"*X. Hence the term not-L-verse is unambiguous in meaning, though ambiguous 

 in form. 



If a first relation be contained in a second, then the converse of the first is contained in 

 the converse of the second : but the contrary of the second in the contrary of the first. 



The conversion of a compound relation converts both components, and inverts their order. 

 If X be an L of an M of Y, then an M of Y is an L"' of X, and Y is an M"' of an L"' 

 of X. Or (LM)"' is M~'L~'. The mark of inherent quantity is also changed in place. 

 If X be* an L of every M of Y, then Y is an M~' of none but L~'s of X. And if X be 

 an L of none but Ms of Y, then Y is an M~' of every L~' of X. For X..LM'Y is 

 MY))L-iX or Y..MfiL-'X: and X..L MY is L'^X ))MY or Y..M-'L-''X. 



When there is a sign of inherent quantity, if each component be changed into its con- 

 trary, and the sign of quantity be shifted from one component to the other, there is no 

 change in the meaning of the symbol. Thus an L of every M is a not-L of none but 

 not-Ms; and vice versa: and an L of none but Ms is a not-L of every not-M ; and vice versa. 



When a compound has no inherent quantity, the contrary is found by taking the con- 

 trary of either component, and giving inherent quantity to the other. Thus, either L of an 

 M or not-L of every M : either L of an M or L of none but not-Ms. But if there be a 

 sign of inherent quantity in one component, the contrary is taken by dropping that sign, and 

 taking the contrary of the other component. Thus, either L of every M or not-L of an M; 

 either L of none but Ms, or L of a not-M. 



The following table containsj- all these theorems: 



* A good instance of the difficulty of abstract propositions : 

 it is easy enough on a concrete instance. If X be the superior 

 of every ancestor of V, then Y is the descendant of none but 

 inferiors of X. 



■|- The resultant relation in onymatic syllogism is identical 

 with the compound from which it results. Tims (.) ).) is ( ), 

 identically : every complement of a deficient is a partient ; 

 every partient is a complement of a deficient. The contraries 

 then are identical ; and this gives the key to the resulting 

 meaning of quantified compound relations : as ,(()), a genus 



Vol. X. Pakt IL 



of none but species ; or ( ( ) )', a genus of every species. The 

 complete rule of interpretation of such symbols is as follows. 

 Reject as incapable of meaning all cases in which two uni. 

 versals or two particulars have difi'erent middle quantities, or 

 in which a universal and particular have the accent upon 

 the particular. Thus there is no such thing as ) ) ) )', or ,(.( ( )', 

 or ((()': a species of every species of a given genus is non- 

 existing. In all other cases, invert the spicula nearest to the 

 accent, erase the middle spiculae, and the result shows the 

 relation identical with the given compound. Thus (.) {.Y 



44 



