Decembeb 31, inOO] 



SCIENCE 



957 



the other hand, up is the class of terms such 

 that, given any one y of them, we have, 

 for every x of ii, yRx; R is said to be 

 inclnded in R', R.iR', if and only if, for 

 all x's and y's, xRy implies xR'y; and 

 R and R' are equivalent when and only 

 when each of them includes the other; (2) 

 asserts that, given any R, there is a rela- 

 tion E'— called the converse of R and de- 

 noted by jB— such that xRy and yR'x are 

 equivalent functions; a relation R is said 

 to be symmetric when and only when 

 R^=R; (3) affirms that, given any two 

 terms x and y, there is between them a 

 relation that does not subsist between the 

 terms of any other pair of terms; the log- 

 ical sum, R^^Ro, of two relations i?i and 

 J?, is a relation such that the proposition 

 x{Ri^R„) is equivalent for all x's and y's 

 to the logical sum of the propositions xRiy, 

 xR^y ; the logical product, R^-R^, is such 

 that a;(72i„/?2)y is equivalent to the product 

 xR^y.xRoy, for all x's and y's; if S" be a 

 class of relations, their swn, ^K, affirmed 

 by (4) to be a relation, is a class of rela- 

 tions such that, given any one R of them 

 and any pair x, y for which xRy, there is 

 in JT a relation R' for which xR'y, and that, 

 given any R' of K and a pair x, y for which 

 xR'y, there is in the sum-class an R for 

 which xRy; similarly the product, -'K 

 assumed by (5) to be a relation, is the 

 class of relations such that, R being any 

 one of them and x and y being a pair for 

 which xRy, then, for every R' of K, xR'y, 

 and conversely, if x and i/ be a pair for 

 which xR'y holds for every E' of K, there 

 is in the product-class an R for which 

 xRy; Ri and R^ being relations, their rela- 

 tive product, R1R2, affirmed by (6) to be 

 a relation, is defined to be such that, if 

 XB1R2Z, there is a y for which xR^y and 

 yRiZ, and that, if xR,y and yR^z. then 

 xR^R^z; R' means RR; a relation R is 

 transitive if and onlv if R- Ls included in 



R, that is, if the product of xRy and yRz 

 implies xRz; R being a relation, its ncgor- 

 live, — R, affirmed by (7) to be a relation, 

 is defined to be such that, x — Ry is true 

 or false according as xRy is false or true; 

 if y is a class of classes, their sum 'y is 

 the class of terms x such that xe^y; 

 diversity, 0', is defined to be the negative 

 of identity, so that ' = — 1 ' ; /i is a uni- 

 form relation, Nc -f- 1, when and only 

 when, whatever a; of p be given, there is 

 one and but one y for which xRy; R is a, 

 CO uniform- relation, 1 -^ Nc, when R is 

 uniform: R is a hiuniform relation, 1 -h- 1, 

 when it is both uniform and couniform. 



Such are the chief of the concepts in the 

 superstructure of the logic of relations. 

 In the study of relations one is close to 

 reality. We do not say with Hegel "Das 

 Seyn ist das Nichts" but rather with Lotze 

 "Being consists in relations." The realm 

 of the thinkable is filled by a multidimen- 

 sional tissue of relations. These are finer 

 than gossamer but stronger than cables of 

 steel. Among the theorems of the general 

 theory the following, which are readily 

 proved by means of the symbolic machin- 

 ery, are cardinal. Each relation R has one 

 and but one converse relation R; the con- 

 verse of the converse of a relation is equiv- 

 valent to the relation, that is, R = R; if 

 Ri = R2, then p^=p„, and Pi=pi, and, if 

 the latter two equivalences subsist, then 

 R,=R2; also, if R^ = R., then Ei^J?,; 

 the converse of the relative product 

 of two relations is equivalent to the 

 relative product of their converses re- 

 versed in order, that is {R^R„) :^^R„R^; 

 if 72 is transitive and if xRz, there exists 

 a y such that xRy and yRz; the converse 

 of the negative of a relation is equivalent 

 to the negative of the converse of the rela- 

 tion ; a null-class is included in every other 

 class; if, for every x in the domain p of 



