956 



SCIENCE 



[N. S. Vol. XXX. No. 783 



such that uov.= -.xeu.o.xev. Two classes 

 w and V are (extensionally) identical, and 

 we write u = v, when and only when uov 

 and VDU. Two classes are disjoint if 

 neither includes a term of the other. It is 

 necessary to avoid confounding e with the 

 use of 3 in class logic, the former holds be- 

 tween an individual and a class but o holds 

 only between classes. Thus, if class uo 

 class V, and if individual otu, we can not 

 write aov. 



The important notions of class multipli- 

 cation and summation are definable as fol- 

 lows. The logical product of the classes u 

 and V, which is denoted by u~v, is such 

 that u-^v ^^ .X9{xeii.xev) ; while the log- 

 ical sum, u-v, u and v being disjoint or 

 not, is such that u^v.^^ .X3{xeu.-.Xiv). 

 Among cardinal theorems are the laws of 

 tautology, commutation, association, dis- 

 tribution and doiible negation: 



w~w ^ M = u^^u ; 

 u-^v = v-u, u^v = v^u ; 

 u—(v-'w) = {u-v)—w. u^{v~^i) = (m-v)-™ ; 

 M— (d-^o) = {u-v)-'(u^w), 



u^(v~^) = {u~v)^{u—w) ; 



and — ( — u) ^u, where — u, called the 

 negative of u, is, by definition, such that 

 — M. = .X9{x — ew). 



The foregoing sketch indicates how the 

 class logic sends its roots down into the 

 soil of the prepositional logic, and there is 

 at the same time exhibited a remarkable 

 parallelism between the two logics. It is 

 important, however, to note the fact, 

 pointed out by Schroder, that the parallel- 

 ism is not thoroughgoing. For example, if 

 p, q, r be propositions and a, b, c be classes, 

 we have 



pqor. = ipor.^.qor, 

 but not 



a—hoc. = •.aoc.^.bne. 



Explanations, Definitions and Theorems 

 in Relational Logic. — In its present form 



this calculus is mainly the creation of Mr. 

 Bertrand Russell. It was he who perceived 

 and demonstrated the advantage of adopt- 

 ing the extensional as distinguished from 

 the intensional view of relations. It was 

 he who perceived and demonstrated its 

 preeminent importance in and for mathe- 

 matics. Finally, it was he who cast its 

 general principles — primitive propositions, 

 fundamental definitions, theorems and 

 their proofs — in symbolic form (cf. Revue 

 de Mathematiques, vol. 7, 1900-1901). 



In order to understand the doctrine in- 

 cluding its primitive propositions above 

 given, it will be necessary to explain or 

 define the principal concepts involved in it 

 and to associate with them the symbols 

 (including those already explained) by 

 which they are denoted. These concepts 

 and symbols are as follows, the numbers 

 (1), (2), ••■ referring to primitive propo- 

 sitions. The writing xRy means to assert 

 that X has the relation R to y, so that a 

 relation has sense or direction ; the symbols 

 P and p, called respectively the domain 

 and the codomain of R, denote respect- 

 ively the classes of terms that may stand 

 before R and after R; the logical sum of 

 these classes is the field of R; if a; be a 

 term of p, px denotes the class of terms y 

 such that xRy, and if a; be a term of p, 

 px is the class of terms y such that yRx; 

 a class is said to exist unless it be a null- 

 class, and the existence of a class is affirmed 

 by writing S before its symbol, as in (3) ; 

 if i(. is a class of terms of p, pu is the class 

 of terms y such that, given any one of 

 them, there is in ii an x for which xRy; 

 on the other hand, m again being a class of 

 terms of p, up denotes the class of terms y 

 such that for every term x of u we have 

 xRy; if, now, « is a class of terms in the 

 codomain p, pu denotes the class of terms 

 such that, given any one y of them, there 

 is in M a term x for which yRx, while, on 



