958 



SGIENCJ£ 



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



R, xRy is equivalent to yex, then B = £; 

 if u and v are existent (not null) classes, 

 there exists a relation subsisting between 

 every term of u and every term of v but 

 not between other two terms; if u is an 

 existent class, there exists a relation B such 

 that xBu implies for every x both p = m 

 and xm, and, conversely, the product of 

 p = u and xeu implies xRii for every 

 x; identity is transitive; identity is 

 equivalent to its converse; the relative 

 product of identity by itself is equivalent 

 to identity; diversity is equivalent to the 

 converse of diversity; if B^B^ is included 

 in diversity, so is B^B^, and conversely; 

 identity is biuniform; if a relation is bi- 

 unif orm, so is its converse ; if a relation is 

 couniform, the relative product of it and 

 its converse is included in but is not always 

 identical with identity ; if two relations are 

 biuniform, so is their relative product; 

 given that B^ and Br, are uniform relations, 

 that u is a class included in pi, that pu is 

 included in p^ and that B^R., = B, then the 

 two classes, ~p„{p.^u) and p%i, are equivalent; 

 if Bi is uniform and if B^ = B^B^, then 

 R„ is transitive and symmetric ; conversely, 

 if an existent relation B^ is transitive and 

 symmetric, then there exists a uniform 

 relation B^ such that B^^BJR.^. 



So striking as well as important is the 

 theorem last stated that I can not refrain 

 from presenting its demonstration, which 

 runs as follows: i2, being given, p, is also 

 given: let x be a term of p.,, and denote by 

 u the class 'p^x; let 2?i be such that xR-^u 

 means icepj and u = p„x; then, if yR^u, 

 yep2 and u = jp2y ^p^x; but, if xB^^^u and 

 yB-^u, then, xB-^B^y; and, as B2 is transi- 

 tive and symmetric, xB^y; hence, as 

 xB-Jt^y implies xB^/y, B^B^ is included in 

 B^; again, as B. is transitive and sym- 

 metric, if xR„y then xepx, and so xB^y 

 implies xB^px and yR^px, and hence im- 



plies xRiR^y; hence B2 is included in 

 Bj^B-^; hence B2==B^Bj^; moreover, B^ is 

 uniform, its codomain consisting of the 

 single term it. Hence the theorem. 



As in the case of propositions and in that 

 of classes, so here, too, are valid the the- 

 orems of tautology, association, commuta- 

 tion, distribution and double negation : 

 -R~ix ^ R = R'^Ii ; 



R,~{R,-J?,) = iRrii,)H-li^-li,), 



RriJi.-R,) = {RrJR.)-{Rrii.); 



Awhile ago I promised to "explicate" 

 the thesis of modern logistic, to state it, 

 that is, explicitly in terms of the logical 

 .primitives upon which as the sufficient 

 foundation it asserts that the entire body 

 of mathematics, both actual and potential, 

 stands as a superstructure. The primitives 

 in question have been given ; so that, except 

 for a restatement of the thesis in terms of 

 them— which I shall omit as being now easy 

 and involving useless repetition — I may 

 claim to have done much more than fulfil 

 the promise; for I have given in addition 

 to the primitives, which were all that was 

 essential, a digest of modern logic. Indeed, 

 the concepts above defined and the theorems 

 above stated, though they are convention- 

 ally assigned to logic, are evidently, if the 

 thesis be true, genuine parts of mathe- 

 matics. 



How is the thesis, if true, to be estab- 

 lished? Obviously not, in the ordinary 

 sense, as the conclusion of a syllogism. 

 No, it affirms that a certain thing can be 

 done, namely, that all definable mathe- 

 matical ideas and all mathematical theo- 

 rems are respectively definable and demon- 

 strable in terms of the primitives given. 

 The only way to show that the deed is 



