Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION D]

REFERENTS AND RELATA OF A GIVEN TERM

257

*3203. sg=AR(A=R) 1)f
*32-04. gs=AR(A=R) Df
*32-1. F: aRy.. a = x (xRy) [*213. (*32'01)]
4 -*32'101.: R/Rx. -./ = 9 (xRy) [*21'3. (*32-02)]
*3211. F. ^ (xRy) = R'y [*321. *30'3]
4 -*32 111. F. (xRy) = R'x [*32-101. *30-3]
*3212. F. E! R'y [*32-11. *1421]
*32-121.. E! R'x f[32-111. *14-21]
"E! R'y" must not be confounded with "! R'y." The former means
that there is such a class as R'y, which, as we have just seen, is always true;
the latter means that R'y is not null, which is only true if y is a term to
which some other term has the relation R. Note that, by *14'21, both g! R'y
and i.! R'y imply E! R'y. The contradictory of g! Ry is not g! R'y,
but -{[R'y]. a! R'y}. This last would not imply E! R'y, but for the fact
that E! R'y is always true.
*32-13. F. R'y= (xRy) [*32-11. *2059]
4 -*32131. F. R'x = y (xRy) [*32111. *20-59]
*32'132. F: ay. -. a =. a = (xRy) [*32-1-13. *20-57]
4- 4 -*32133. F: iRx. =. 3= R'x. =. / = 9 (xRy) [*32-101-131. *20'57]
The use of *20'57 will in general be tacit. It happens constantly that
we have propositions such as *32'13, in which a descriptive expression is
shown to be identical with a class. In such cases, whenever the properties
of the class are asserted of the descriptive expression, *20'57 is relevant.
*32-14. F:R=S.=-.R=S
Dem.. *21-43. F:: R= S. -. aRy. -,,. aSy:.
[*32-1]. a = x (xRy).,y a (Sy)
[*11.2] -:. (y):. a = x (xRy). —. a = x (xSy):.
[*20-25] -:. (y):. (.Ry) = (xSy):.
[*20-15]: (y):. (x):.Ry. =. Sy:.
[*11 2] -:. (, y): xRy. -. Sy:.
[*21-43]:. R=:: D F. Prop
R. & w. 17