Decembee 31, 1009] 



SCIENCE 



953 



And the primitive propositions are 



(1) poq.o.poq, 



(2) poq.o.pop, 



(3) poq.o qoq, 



(4) If poq and if p be trne, p may be 



dropped and q asserted, 



(5) pop.qoq.o.pqop, 



(6) poq.qsr.o.por, 



(7) qoq.ror.po.qor-.o.pqor, 



(8) pop.qdr.pqor-.o-.po.qor, 



(9) poq.-por.o.poqr, 



(10) pop.qoq o:{pdq)op.3p, 

 in which, as elsewhei-e, p, g and r denote 

 propositions, o (inverse of the letter c) 

 stands for the word implies, pq means "p 

 and q," while the points or dots serve the 

 double use of denoting the word and, like 

 the first dot in (5), or, like those in (1), 

 playing the role of parentheses in indi- 

 cating the relative ranks of the various 

 parts of a formula. Thus, for example, 

 (7) may be translated to read, the propo- 

 sition "q implies q and r implies r and 

 p implies that q implies r" implies the 

 proposition "p and q together imply r" ; 

 or, in hypothetic form, if q implies q, and 

 r implies r, and p implies that q implies r, 

 then p and q together imply r. 



The Logic of Classes.— The primitive no- 

 tions in this calculus are 



(1) Proportional Function, denoted bj* 



such S3'mbols as <f>(x), *(a;), etc., 



(2) The Eelation (denoted bj' c, read is 



or belongs to) of an individual to 

 a class (containing it), 



(3) The notion such that, denoted by ' 



(inverse of the Gi-eek letter «). 

 And the primitive propositions are 



(1) kc]x^<i>(x)\oi>{k), 



(2) <^(x) = *(2;).3:ar3<^(a;). = .X3<lr(x). 

 The Logic of Relations.— In this calculus, 



which Russell has shown to be the logic par 

 excellence of mathematics, the primitive 

 notions are 



(1) Relation, denoted hs a class by rel 



and as individuals by such capitals 

 as H, R', etc., 

 (2) Identity, denoted by the symbol 1'. 

 The primitive propositions are 



(1) Rerel.o-.xRy. = .x has the relation jR 



toy, 



(2) i?3rel o^rel-R'HxR'y. = .yRx), 



(3) 3r*\-R^(p = a-.p = '!/), 



(4) -'A'trel, 



(5) -'Kert'], 



(6) 7?,/?.,£rel, 



(7) -Riiel, 



(8) ££rel, 



(9) I'erel, 



(10) x^'x, 



(11) I'ai', 



(12) Rerel.xRy.y1 'z o.xRz. 



To the foregoing primitives must be 

 added the notion of denoting, which has 

 been made the topic of a mo-st subtle and 

 luminous discussion by Russell in the fifth 

 chapter of the work above cited. The 

 notion is that of the sense in which an 

 individual is denoted by a concept that 

 occurs in a proposition that is not a propo- 

 sition about the concept, as "She bought a 

 beautiful goiv7i"—t'he thing purchased be 

 nothing so tenuous and translucent as the 

 concept, a beautiful gown, but pre-sumably 

 a concrete thing reasonably opaque. 



By way of elucidating the foregoing and 

 further sketching out the three divisions of 

 logic, I shall now proceed to give some 

 explanation of the primitive terms and a 

 statement of the principal definitions and 

 theorems composing them. 



Definitions and Theorems in Proposi- 

 tional Tjogic.—lhe central term, proposi- 

 tion, is defined in terms of (material) 

 implication, namely, a proposition is that 

 which implies itself. The two varieties of 

 implication are often confused and the dis- 

 tinction between them, being difficult to 

 draw sharply and clearly, is to be acquired 

 very much as a child learns to distinguish 



